https://www.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Andreic&feedformat=atomUW-Math Wiki - User contributions [en]2019-10-13T22:36:10ZUser contributionsMediaWiki 1.30.1https://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17523Colloquia2019-07-16T14:28:53Z<p>Andreic: /* Mathematics Colloquium */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6<br />
|<br />
|<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| Alicia Dickenstein (Buenos Aires)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| Jianfeng Lu (Duke)<br />
|[[#TBA | TBA]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 27<br />
|Elchnanan Mossel (MIT) Distinguished Lecture<br />
|-<br />
|Oct 4<br />
| Matt Baker (Georgia Tech)<br />
|<br />
|-<br />
|Oct 11<br />
|<br />
|-<br />
|Oct 18<br />
|<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Possibly reserved for job talk?<br />
|<br />
|-<br />
|Nov 8<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 15<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 22<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|<br />
|-<br />
|Jan 31<br />
|<br />
|-<br />
|Feb 7<br />
|<br />
|-<br />
|Feb 14<br />
|<br />
|-<br />
|Feb 21<br />
|<br />
|-<br />
|Feb 28<br />
|<br />
|-<br />
|March 6<br />
|<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|<br />
|-<br />
|April 3<br />
|<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|<br />
|-<br />
|April 24<br />
|<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17522Colloquia2019-07-16T14:27:23Z<p>Andreic: </p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6<br />
|<br />
|<br />
|-<br />
|Sept 13<br />
| [https://www.math.ksu.edu/~soibel/ Yan Soibelman] (Kansas State)<br />
|[[#Yan Soibelman (Kansas State)| Riemann-Hilbert correspondence and Fukaya categories. ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| Alicia Dickenstein (Buenos Aires)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| Jianfeng Lu (Duke)<br />
|[[#TBA | TBA]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 27<br />
|Elchnanan Mossel (MIT) Distinguished Lecture<br />
|-<br />
|Oct 4<br />
| Matt Baker (Georgia Tech)<br />
|<br />
|-<br />
|Oct 11<br />
|<br />
|-<br />
|Oct 18<br />
|<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Possibly reserved for job talk?<br />
|<br />
|-<br />
|Nov 8<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 15<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 22<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|<br />
|-<br />
|Jan 31<br />
|<br />
|-<br />
|Feb 7<br />
|<br />
|-<br />
|Feb 14<br />
|<br />
|-<br />
|Feb 21<br />
|<br />
|-<br />
|Feb 28<br />
|<br />
|-<br />
|March 6<br />
|<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|<br />
|-<br />
|April 3<br />
|<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|<br />
|-<br />
|April 24<br />
|<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Yan Soibelman (Kansas State)===<br />
<br />
Title: Riemann-Hilbert correspondence and Fukaya categories<br />
<br />
Abstract: In this talk I am going to discuss the role of Fukaya categories in the Riemann-Hilbert correspondence<br />
for differential, q-difference and elliptic difference equations in dimension one.<br />
This approach not only gives a unified answer for several versions of the Riemann-Hilbert correspondence but also leads to a natural formulation<br />
of the non-abelian Hodge theory in dimension one. It also explains why periodic monopoles<br />
should appear as harmonic objects in this generalized non-abelian Hodge theory.<br />
All that is a part of the bigger project ``Holomorphic Floer theory",<br />
joint with Maxim Kontsevich.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=17521Colloquia2019-07-16T14:23:21Z<p>Andreic: </p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
<br />
<br />
==Fall 2019==<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Sept 6<br />
|<br />
|<br />
|-<br />
|Sept 13<br />
| Yan Soibelman (Kansas State)<br />
|[[# TBA| TBA ]]<br />
| Caldararu<br />
|<br />
|-<br />
|Sept 16 '''Monday Room 911'''<br />
| Alicia Dickenstein (Buenos Aires)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
|Sept 20<br />
| Jianfeng Lu (Duke)<br />
|[[#TBA | TBA]]<br />
| Qin<br />
|<br />
|-<br />
|Sept 27<br />
|Elchnanan Mossel (MIT) Distinguished Lecture<br />
|-<br />
|Oct 4<br />
| Matt Baker (Georgia Tech)<br />
|<br />
|-<br />
|Oct 11<br />
|<br />
|-<br />
|Oct 18<br />
|<br />
|-<br />
|Oct 25<br />
|<br />
|-<br />
|Nov 1<br />
|Possibly reserved for job talk?<br />
|<br />
|-<br />
|Nov 8<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 15<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 22<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Nov 29<br />
|Thanksgiving<br />
|<br />
|-<br />
|Dec 6<br />
|Reserved for job talk<br />
|<br />
|-<br />
|Dec 13<br />
|Reserved for job talk<br />
|<br />
|}<br />
<br />
==Spring 2020==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|<br />
|-<br />
|Jan 24<br />
|<br />
|-<br />
|Jan 31<br />
|<br />
|-<br />
|Feb 7<br />
|<br />
|-<br />
|Feb 14<br />
|<br />
|-<br />
|Feb 21<br />
|<br />
|-<br />
|Feb 28<br />
|<br />
|-<br />
|March 6<br />
|<br />
|-<br />
|March 13<br />
|<br />
|-<br />
|March 20<br />
|Spring break<br />
|<br />
|-<br />
|March 27<br />
|<br />
|-<br />
|April 3<br />
|<br />
|-<br />
|April 10<br />
| Sarah Koch (Michigan)<br />
|<br />
| Bruce (WIMAW)<br />
|-<br />
|April 17<br />
|<br />
|-<br />
|April 24<br />
|<br />
|-<br />
|May 1<br />
|Robert Lazarsfeld (Stony Brook)<br />
|Distinguished lecture<br />
|Erman<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Person (Institution)===<br />
<br />
Title:<br />
<br />
Abstract:<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Spring2019|Spring 2019]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2019&diff=16835Algebra and Algebraic Geometry Seminar Spring 20192019-02-05T20:34:08Z<p>Andreic: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Fall 2018 | the previous semester]], for [[Algebra and Algebraic Geometry Seminar Fall 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
<br />
== Spring 2019 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 25<br />
|[http://www.math.utah.edu/~smolkin/ Daniel Smolkin (Utah)]<br />
|Symbolic Powers in Rings of Positive Characteristic<br />
|Daniel<br />
|-<br />
|February 1<br />
|Juliette Bruce<br />
|Asymptotic Syzgies for Products of Projective Spaces<br />
|Local<br />
|-<br />
|February 8<br />
|[http://www.mit.edu/~ivogt/ Isabel Vogt (MIT)]<br />
| Low degree points on curves<br />
|Wanlin and Juliette<br />
|-<br />
|February 15<br />
|Pavlo Pylyavskyy (U. Minn)<br />
|Zamolodchikov periodicity and integrability<br />
|Paul Terwilliger<br />
|-<br />
|February 22<br />
|Michael Brown (Wisconsin)<br />
|Chern-Weil theory for matrix factorizations<br />
|Local<br />
|-<br />
|March 1<br />
|Shamgar Gurevich (Wisconsin)<br />
|Harmonic Analysis on GLn over finite fields, and Random Walks<br />
|Local<br />
|-<br />
|March 8<br />
|Jay Kopper (UIC)<br />
|TBD<br />
|Daniel<br />
|-<br />
|March 15<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|March 22<br />
|No Meeting<br />
|Spring Break<br />
|TBD<br />
|-<br />
|March 29<br />
|[https://math.berkeley.edu/~ceur/ Chris Eur (UC Berkeley)]<br />
|TBD<br />
|Daniel<br />
|-<br />
|April 5<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|April 12<br />
|[http://www-personal.umich.edu/~ecanton/ Eric Canton (Michigan)]<br />
|TBD<br />
|Michael<br />
|-<br />
|April 19<br />
|[http://www-personal.umich.edu/~grifo/ Elo&iacute;sa Grifo (Michigan)]<br />
|TBD<br />
|TBD<br />
|-<br />
|April 26<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|May 3<br />
|TBD<br />
|TBD<br />
|TBD<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Daniel Smolkin===<br />
'''Symbolic Powers in Rings of Positive Characteristic'''<br />
<br />
The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend!<br />
<br />
===Juliette Bruce===<br />
<br />
'''Title: Asymptotic Syzygies for Products of Projective Spaces'''<br />
<br />
I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.<br />
<br />
===Isabel Vogt===<br />
<br />
'''Title: Low degree points on curves'''<br />
<br />
In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.<br />
<br />
===Pavlo Pylyavskyy===<br />
<br />
'''Zamolodchikov periodicity and integrability'''<br />
<br />
T-systems are certain discrete dynamical systems associated with quivers. They appear in several different contexts: quantum affine algebras and Yangians, commuting transfer matrices of vertex models, character theory of quantum groups, analytic Bethe ansatz, Wronskian-Casoratian duality in ODE, gauge/string theories, etc. Periodicity of certain T-systems was the main conjecture in the area until it was proven by Keller in 2013 using cluster categories. In this work we completely classify periodic T-systems, which turn out to consist of 5 infinite families and 4 exceptional cases, only one of the infinite families being known previously. We then proceed to classify T-systems that exhibit two forms of integrability: linearization and zero algebraic entropy. All three classifications rely on reduction of the problem to study of commuting Cartan matrices, either of finite or affine types. The finite type classification was obtained by Stembridge in his study of Kazhdan-Lusztig theory for dihedral groups, the other two classifications are new. This is joint work with Pavel Galashin. <br />
<br />
===Shamgar Gurevich===<br />
<br />
'''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2019&diff=16834Algebra and Algebraic Geometry Seminar Spring 20192019-02-05T20:33:17Z<p>Andreic: /* Spring 2019 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Fall 2018 | the previous semester]], for [[Algebra and Algebraic Geometry Seminar Fall 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
<br />
== Spring 2019 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
|-<br />
|January 25<br />
|[http://www.math.utah.edu/~smolkin/ Daniel Smolkin (Utah)]<br />
|Symbolic Powers in Rings of Positive Characteristic<br />
|Daniel<br />
|-<br />
|February 1<br />
|Juliette Bruce<br />
|Asymptotic Syzgies for Products of Projective Spaces<br />
|Local<br />
|-<br />
|February 8<br />
|[http://www.mit.edu/~ivogt/ Isabel Vogt (MIT)]<br />
| Low degree points on curves<br />
|Wanlin and Juliette<br />
|-<br />
|February 15<br />
|Pavlo Pylyavskyy (U. Minn)<br />
|Zamolodchikov periodicity and integrability<br />
|Paul Terwilliger<br />
|-<br />
|February 22<br />
|Michael Brown (Wisconsin)<br />
|Chern-Weil theory for matrix factorizations<br />
|Local<br />
|-<br />
|March 1<br />
|Shamgar Gurevich (Wisconsin)<br />
|Harmonic Analysis on GLn over finite fields, and Random Walks<br />
|Local<br />
|-<br />
|March 8<br />
|Jay Kopper (UIC)<br />
|TBD<br />
|Daniel<br />
|-<br />
|March 15<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|March 22<br />
|No Meeting<br />
|Spring Break<br />
|TBD<br />
|-<br />
|March 29<br />
|[https://math.berkeley.edu/~ceur/ Chris Eur (UC Berkeley)]<br />
|TBD<br />
|Daniel<br />
|-<br />
|April 5<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|April 12<br />
|[http://www-personal.umich.edu/~ecanton/ Eric Canton (Michigan)]<br />
|TBD<br />
|Michael<br />
|-<br />
|April 19<br />
|[http://www-personal.umich.edu/~grifo/ Elo&iacute;sa Grifo (Michigan)]<br />
|TBD<br />
|TBD<br />
|-<br />
|April 26<br />
|TBD<br />
|TBD<br />
|TBD<br />
|-<br />
|May 3<br />
|TBD<br />
|TBD<br />
|TBD<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Daniel Smolkin===<br />
'''Symbolic Powers in Rings of Positive Characteristic'''<br />
<br />
The n-th power of an ideal is easy to compute, though difficult to describe geometrically. In contrast, symbolic powers of ideals are difficult to compute while having a natural geometric description. In this talk, I will describe how to compare ordinary and symbolic powers of ideals using the techniques of positive-characteristic commutative algebra, especially in toric rings and Hibi rings. This is based on joint work with Javier Carvajal-Rojas, Janet Page, and Kevin Tucker. Graduate students are encouraged to attend!<br />
<br />
===Juliette Bruce===<br />
<br />
'''Title: Asymptotic Syzygies for Products of Projective Spaces'''<br />
<br />
I will discuss results describing the asymptotic syzygies of products of projective space, in the vein of the explicit methods of Ein, Erman, and Lazarsfeld’s non-vanishing results on projective space.<br />
<br />
===Isabel Vogt===<br />
<br />
'''Title: Low degree points on curves'''<br />
<br />
In this talk we will discuss an arithmetic analogue of the gonality of a curve over a number field: the smallest positive integer e such that the points of residue degree bounded by e are infinite. By work of Faltings, Harris--Silverman and Abramovich--Harris, it is well-understood when this invariant is 1, 2, or 3; by work of Debarre--Fahlaoui these criteria do not generalize to e at least 4. We will study this invariant using the auxiliary geometry of a surface containing the curve and devote particular attention to scenarios under which we can guarantee that this invariant is actually equal to the gonality . This is joint work with Geoffrey Smith.<br />
<br />
<br />
===Shamgar Gurevich===<br />
<br />
'''Harmonic Analysis on GLn over finite fields, and Random Walks'''<br />
<br />
There are many formulas that express interesting properties of a group G in terms of sums over its characters. For evaluating or estimating these sums, one of the most salient quantities to understand is the {\it character ratio}: <br />
<br />
$$<br />
trace(\rho(g))/dim(\rho),<br />
$$<br />
<br />
for an irreducible representation $\rho$ of G and an element g of G. For example, Diaconis and Shahshahani stated a formula of this type for analyzing G-biinvariant random walks on G. It turns out that, for classical groups G over finite fields (which provide most examples of finite simple groups), there is a natural invariant of representations that provides strong information on the character ratio. We call this invariant {\it rank}. This talk will discuss the notion of rank for GLn over finite fields, and apply the results to random walks. This is joint work with Roger Howe (Yale and Texas AM).</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=16813Colloquia2019-02-04T18:24:12Z<p>Andreic: </p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | Twisting things in topology and symplectic topology by pth powers ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[# TBA| TBA ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Evitar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===David Treumann (Boston College)===<br />
<br />
Title: Twisting things in topology and symplectic topology by pth powers<br />
<br />
Abstract: There's an old and popular analogy between circles and finite fields. I'll describe some constructions you can make in Lagrangian Floer theory and in microlocal sheaf theory by taking this analogy extremely literally, the main ingredient is an "F-field." An F-field on a manifold M is a local system of algebraically closed fields of characteristic p. When M is symplectic, maybe an F-field should remind you of a B-field, it can be used to change the Fukaya category in about the same way. On M = S^1 times R^3, this version of the Fukaya category is related to Deligne-Lusztig theory, and I found something like a cluster structure on the Deligne-Lusztig pairing varieties by studying it. On M = S^1 times S^1, Yanki Lekili and I have found that this version of the Fukaya category is related to the equal-characteristic version of the Fargues-Fontaine curve; the relationship is homological mirror symmetry.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia&diff=16810Colloquia2019-02-04T14:57:31Z<p>Andreic: /* Spring 2019 */</p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
==Spring 2019==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|Jan 25 '''Room 911'''<br />
| [http://www.users.miamioh.edu/randrib/ Beata Randrianantoanina] (Miami University Ohio) WIMAW<br />
|[[#Beata Randrianantoanina (Miami University Ohio) | Some nonlinear problems in the geometry of Banach spaces and their applications ]]<br />
| Tullia Dymarz<br />
|<br />
|-<br />
|Jan 30 '''Wednesday'''<br />
| Talk rescheduled to Feb 15<br />
|<br />
|-<br />
|Jan 31 '''Thursday'''<br />
| Talk rescheduled to Feb 13<br />
|<br />
|-<br />
|Feb 1<br />
| Talk cancelled due to weather<br />
|<br />
| <br />
|<br />
|-<br />
|Feb 5 '''Tuesday'''<br />
| [http://www.math.tamu.edu/~alexei.poltoratski/ Alexei Poltoratski] (Texas A&M University)<br />
|[[#Alexei Poltoratski (Texas A&M)| Completeness of exponentials: Beurling-Malliavin and type problems ]]<br />
| Denisov<br />
|<br />
|-<br />
|Feb 8<br />
| [https://sites.math.northwestern.edu/~anaber/ Aaron Naber] (Northwestern)<br />
|[[#Aaron Naber (Northwestern) | A structure theory for spaces with lower Ricci curvature bounds ]]<br />
| Street<br />
|<br />
|-<br />
|Feb 11 '''Monday'''<br />
| [https://www2.bc.edu/david-treumann/materials.html David Treumann] (Boston College)<br />
|[[#David Treumann (Boston College) | TBA ]]<br />
| Caldararu<br />
|<br />
|-<br />
| Feb 13 '''Wednesday'''<br />
| [http://www.math.tamu.edu/~dbaskin/ Dean Baskin] (Texas A&M)<br />
|[[#Dean Baskin (Texas A&M) | Radiation fields for wave equations ]]<br />
| Street<br />
<br />
|-<br />
| Feb 15 <br />
| [https://services.math.duke.edu/~pierce/ Lillian Pierce] (Duke University)<br />
| [[#Lillian Pierce (Duke University) | Short character sums ]]<br />
| Boston and Street<br />
|<br />
|-<br />
|Feb 22<br />
| [https://people.math.osu.edu/cueto.5/ Angelica Cueto] (Ohio State)<br />
|[[# TBA| TBA ]]<br />
| Erman and Corey<br />
|<br />
|-<br />
|March 4<br />
| [http://www-users.math.umn.edu/~sverak/ Vladimir Sverak] (Minnesota) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Kim<br />
|<br />
|-<br />
|March 8<br />
| [https://orion.math.iastate.edu/jmccullo/index.html Jason McCullough] (Iowa State)<br />
|[[# TBA| TBA ]]<br />
| Erman<br />
|<br />
|-<br />
|March 15<br />
| Maksym Radziwill (Caltech)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|March 29<br />
| Jennifer Park (OSU)<br />
|[[# TBA| TBA ]]<br />
| Marshall<br />
|<br />
|-<br />
|April 5<br />
| Ju-Lee Kim (MIT)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 12<br />
| Evitar Procaccia (TAMU)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|-<br />
|April 19<br />
| [http://www.math.rice.edu/~jkn3/ Jo Nelson] (Rice University)<br />
|[[# TBA| TBA ]]<br />
| Jean-Luc<br />
|<br />
|-<br />
|April 26<br />
| [https://www.brown.edu/academics/applied-mathematics/faculty/kavita-ramanan/home Kavita Ramanan] (Brown University)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|May 3<br />
| Tomasz Przebinda (Oklahoma)<br />
|[[# TBA| TBA ]]<br />
| Gurevich<br />
|<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Beata Randrianantoanina (Miami University Ohio)===<br />
<br />
Title: Some nonlinear problems in the geometry of Banach spaces and their applications.<br />
<br />
Abstract: Nonlinear problems in the geometry of Banach spaces have been studied since the inception of the field. In this talk I will outline some of the history, some of modern applications, and some open directions of research. The talk will be accessible to graduate students of any field of mathematics.<br />
<br />
===Lillian Pierce (Duke University)===<br />
<br />
Title: Short character sums <br />
<br />
Abstract: A surprisingly diverse array of problems in analytic number theory have at their heart a problem of bounding (from above) an exponential sum, or its multiplicative cousin, a so-called character sum. For example, both understanding the Riemann zeta function or Dirichlet L-functions inside the critical strip, and also counting solutions to Diophantine equations via the circle method or power sieve methods, involve bounding such sums. In general, the sums of interest fall into one of two main regimes: complete sums or incomplete sums, with this latter regime including in particular “short sums.” Short sums are particularly useful, and particularly resistant to almost all known methods. In this talk, we will see what makes a sum “short,” sketch why it would be incredibly powerful to understand short sums, and discuss a curious proof from the 1950’s which is still the best way we know to bound short sums. We will end by describing new work which extends the ideas of this curious proof to bound short sums in much more general situations.<br />
<br />
===Dean Baskin (Texas A&M)===<br />
<br />
Title: Radiation fields for wave equations<br />
<br />
Abstract: Radiation fields are rescaled limits of solutions of wave equations near "null infinity" and capture the radiation pattern seen by a distant observer. They are intimately connected with the Fourier and Radon transforms and with scattering theory. In this talk, I will define and discuss radiation fields in a few contexts, with an emphasis on spacetimes that look flat near infinity. The main result is a connection between the asymptotic behavior of the radiation field and a family of quantum objects on an associated asymptotically hyperbolic space.<br />
<br />
===Jianfeng Lu (Duke University)===<br />
<br />
Title: Density fitting: Analysis, algorithm and applications<br />
<br />
Abstract: Density fitting considers the low-rank approximation of pair products of eigenfunctions of Hamiltonian operators. It is a very useful tool with many applications in electronic structure theory. In this talk, we will discuss estimates of upper bound of the numerical rank of the pair products of eigenfunctions. We will also introduce the interpolative separable density fitting (ISDF) algorithm, which reduces the computational scaling of the low-rank approximation and can be used for efficient algorithms for electronic structure calculations. Based on joint works with Chris Sogge, Stefan Steinerberger, Kyle Thicke, and Lexing Ying.<br />
<br />
===Alexei Poltoratski (Texas A&M)===<br />
<br />
Title: Completeness of exponentials: Beurling-Malliavin and type problems<br />
<br />
Abstract: This talk is devoted to two old problems of harmonic analysis mentioned in the title. Both<br />
problems ask when a family of complex exponentials is complete (spans) an L^2-space. The Beruling-Malliavin<br />
problem was solved in the early 1960s and I will present its classical solution along with modern generalizations<br />
and applications. I will then discuss history and recent progress in the type problem, which stood open for<br />
more than 70 years.<br />
<br />
===Aaron Naber (Northwestern)===<br />
<br />
Title: A structure theory for spaces with lower Ricci curvature bounds.<br />
<br />
Abstract: One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The talk will require little background, and our time will be spent on understanding the basic statements and examples. The work discussed is joint with Cheeger, Jiang and with Li.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank]]<br />
<br />
[[Colloquia/Fall2018|Fall 2018]]<br />
<br />
[[Colloquia/Spring2018|Spring 2018]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=16217Algebra and Algebraic Geometry Seminar Fall 20182018-10-16T03:30:23Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]], [[Algebra and Algebraic Geometry Seminar Spring 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|Kaledin's noncommutative degeneration theorem and topological Hochschild homology<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|Categorical Gromov-Witten invariants beyond genus 1<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|Conjecture D for matrix factorizations<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|Modified quantum difference Toda systems<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|[https://juliettebruce.github.io Juliette Bruce]<br />
|Covering Abelian Varieties and Effective Bertini<br />
|Local<br />
|-<br />
|November 2<br />
|[http://sites.nd.edu/b-taji/ Behrouz Taji] (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|[http://www-personal.umich.edu/~rohitna/ Rohit Nagpal (Michigan)]<br />
|TBD<br />
|John WG<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|TBD (this date is now open again!)<br />
|TBD<br />
|<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===<br />
<br />
'''Title: Kaledin's noncommutative degeneration theorem and topological Hochschild homology'''<br />
<br />
For a smooth proper variety over a field of characteristic<br />
zero, the Hodge-to-de Rham spectral sequence (relating the cohomology<br />
of differential forms to de Rham cohomology) is well-known to<br />
degenerate, via Hodge theory. A "noncommutative" version of this<br />
theorem has been proved by Kaledin for smooth proper dg categories<br />
over a field of characteristic zero, based on the technique of<br />
reduction mod p. I will describe a short proof of this theorem using<br />
the theory of topological Hochschild homology, which provides a<br />
canonical one-parameter deformation of Hochschild homology in<br />
characteristic p.<br />
<br />
===Andrei Caldararu===<br />
'''Categorical Gromov-Witten invariants beyond genus 1'''<br />
<br />
In a seminal work from 2005 Kevin Costello defined numerical invariants associated to a <br />
Calabi-Yau A-infinity category. These invariants are supposed to generalize the classical<br />
Gromov-Witten invariants (counting curves in a target symplectic manifold) when the category<br />
is taken to be the Fukaya category. In my talk I shall describe some of the ideas involved in Costello's<br />
approach and recent progress (with Junwu Tu) on extending computations of these invariants<br />
past genus 1.<br />
<br />
===Mark Walker===<br />
'''Conjecture D for matrix factorizations'''<br />
<br />
Matrix factorizations form a dg category whose associated homotopy category is equivalent to the stable category of maximum Cohen-Macaulay modules over a hypersurface ring. In the isolated singularity case, the dg category of matrix factorizations is "smooth" and "proper" --- non-commutative analogues of the same-named properties of algebraic varieties. In general, for any smooth and proper dg category, there exist non-commutative analogues of Grothendieck's Standard Conjectures for cycles on smooth and projective varieties. In particular, the non-commutative version of Standard Conjecture D predicts that numerical equivalence and homological equivalence coincide for such a dg category. Recently, Michael Brown and I have proven the non-commutative analogue of Conjecture D for the category of matrix factorizations of an isolated singularity over a field of characteristic 0. In this talk, I will describe our theorem in more detail and give a sense of its proof.<br />
<br />
===Oleksandr Tsymbaliuk===<br />
'''Modified quantum difference Toda systems'''<br />
<br />
The q-version of a Toda system associated with any Lie algebra was introduced independently by Etingof and Sevostyanov in 1999. In this talk, we shall discuss the generalization of this construction which naturally produces a family of 3^{rk(g)-1} similar integrable systems. One of the key ingredients in the proof is played by the fermionic formula for the J-factors (defined as pairing of two Whittaker vectors in Verma modules), due to Feigin-Feigin-Jimbo-Miwa-Mukhin. In types A and C, our construction admits an alternative presentation via local Lax matrices, similar to the classical construction of Faddeev-Takhtajan for the classical type A Toda system. Finally, we shall discuss the geometric interpretation of Whittaker vectors in type A. <br />
<br />
This talk is based on the joint work with M. Finkelberg and R. Gonin.<br />
<br />
===Juliette Bruce===<br />
'''Covering Abelian Varieties and Effective Bertini'''<br />
<br />
I will discuss recent work showing that every abelian variety is covered by a Jacobian whose dimension is bounded. This is joint with Wanlin Li.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=16005Algebra and Algebraic Geometry Seminar Fall 20182018-09-17T21:38:34Z<p>Andreic: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]], [[Algebra and Algebraic Geometry Seminar Spring 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|Kaledin's noncommutative degeneration theorem and topological Hochschild homology<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|Categorical Gromov-Witten invariants beyond genus 1<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|Juliette Bruce<br />
|TBD<br />
|Local<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|[http://www-personal.umich.edu/~grifo/ Eloísa Grifo] (Michigan)<br />
|TBD<br />
|Daniel<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===<br />
<br />
'''Title: Kaledin's noncommutative degeneration theorem and topological Hochschild homology'''<br />
<br />
For a smooth proper variety over a field of characteristic<br />
zero, the Hodge-to-de Rham spectral sequence (relating the cohomology<br />
of differential forms to de Rham cohomology) is well-known to<br />
degenerate, via Hodge theory. A "noncommutative" version of this<br />
theorem has been proved by Kaledin for smooth proper dg categories<br />
over a field of characteristic zero, based on the technique of<br />
reduction mod p. I will describe a short proof of this theorem using<br />
the theory of topological Hochschild homology, which provides a<br />
canonical one-parameter deformation of Hochschild homology in<br />
characteristic p.<br />
<br />
===Andrei Caldararu===<br />
'''Categorical Gromov-Witten invariants beyond genus 1'''<br />
<br />
In a seminal work from 2005 Kevin Costello defined numerical invariants associated to a <br />
Calabi-Yau A-infinity category. These invariants are supposed to generalize the classical<br />
Gromov-Witten invariants (counting curves in a target symplectic manifold) when the category<br />
is taken to be the Fukaya category. In my talk I shall describe some of the ideas involved in Costello's<br />
approach and recent progress (with Junwu Tu) on extending computations of these invariants<br />
past genus 1.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=16004Algebra and Algebraic Geometry Seminar Fall 20182018-09-17T21:35:07Z<p>Andreic: /* Fall 2018 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]], [[Algebra and Algebraic Geometry Seminar Spring 2019 | the next semester]], and for [[Algebra and Algebraic Geometry Seminar | this semester]].<br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|Kaledin's noncommutative degeneration theorem and topological Hochschild homology<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|Categorical Gromov-Witten invariants beyond genus 1<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|Juliette Bruce<br />
|TBD<br />
|Local<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|[http://www-personal.umich.edu/~grifo/ Eloísa Grifo] (Michigan)<br />
|TBD<br />
|Daniel<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===<br />
<br />
'''Title: Kaledin's noncommutative degeneration theorem and topological Hochschild homology'''<br />
<br />
For a smooth proper variety over a field of characteristic<br />
zero, the Hodge-to-de Rham spectral sequence (relating the cohomology<br />
of differential forms to de Rham cohomology) is well-known to<br />
degenerate, via Hodge theory. A "noncommutative" version of this<br />
theorem has been proved by Kaledin for smooth proper dg categories<br />
over a field of characteristic zero, based on the technique of<br />
reduction mod p. I will describe a short proof of this theorem using<br />
the theory of topological Hochschild homology, which provides a<br />
canonical one-parameter deformation of Hochschild homology in<br />
characteristic p.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15924Algebra and Algebraic Geometry Seminar Fall 20182018-09-07T19:22:27Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|TBA<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|TBA<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===<br />
<br />
'''Title: Kaledin's noncommutative degeneration theorem and topological<br />
Hochschild homology'''<br />
<br />
For a smooth proper variety over a field of characteristic<br />
zero, the Hodge-to-de Rham spectral sequence (relating the cohomology<br />
of differential forms to de Rham cohomology) is well-known to<br />
degenerate, via Hodge theory. A "noncommutative" version of this<br />
theorem has been proved by Kaledin for smooth proper dg categories<br />
over a field of characteristic zero, based on the technique of<br />
reduction mod p. I will describe a short proof of this theorem using<br />
the theory of topological Hochschild homology, which provides a<br />
canonical one-parameter deformation of Hochschild homology in<br />
characteristic p.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15923Algebra and Algebraic Geometry Seminar Fall 20182018-09-07T19:20:22Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|TBA<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|TBA<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Akhil Mathew===</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15918Algebra and Algebraic Geometry Seminar Fall 20182018-09-07T17:35:07Z<p>Andreic: /* Fall 2018 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|Akhil Mathew (U Chicago)<br />
|TBA<br />
|Andrei<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|TBA<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15906Algebra and Algebraic Geometry Seminar Fall 20182018-09-06T21:16:59Z<p>Andreic: /* Fall 2018 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|September 7<br />
|Daniel Erman<br />
|Big Polynomial Rings<br />
|Local<br />
|-<br />
|September 14<br />
|<br />
|<br />
|<br />
|-<br />
|September 21<br />
|Andrei Caldararu<br />
|TBA<br />
|Local<br />
|-<br />
|September 28<br />
|Mark Walker (Nebraska)<br />
|TBD<br />
|Michael and Daniel<br />
|-<br />
|October 5<br />
|-<br />
|-<br />
|-<br />
|-<br />
|October 12<br />
|Jose Rodriguez (Wisconsin)<br />
|TBD<br />
|Local<br />
|-<br />
|October 19<br />
|Oleksandr Tsymbaliuk (Yale)<br />
|TBD<br />
|Paul Terwilliger<br />
|-<br />
|October 26<br />
|<br />
|<br />
|<br />
|-<br />
|November 2<br />
|Behrouz Taji (Notre Dame)<br />
|TBD<br />
|Botong Wang<br />
|-<br />
|November 9<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|November 16<br />
|Wanlin Li<br />
|TBD<br />
|Local<br />
|-<br />
|November 23<br />
|Thanksgiving<br />
|No Seminar<br />
|<br />
|-<br />
|November 30<br />
|John Wiltshire-Gordon<br />
|TBD<br />
|Local<br />
|-<br />
|December 7<br />
|Michael Brown<br />
|TBD<br />
|Local<br />
|-<br />
|December 14<br />
|-TBD<br />
|-TBD<br />
|-TBD<br />
|-<br />
|}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&diff=15817Algebra and Algebraic Geometry Seminar2018-09-01T20:49:36Z<p>Andreic: Redirected page to Algebra and Algebraic Geometry Seminar Fall 2018</p>
<hr />
<div>#REDIRECT [[Algebra and Algebraic Geometry Seminar Fall 2018]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15742Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T12:20:14Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|- }<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15741Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:51:07Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebra and Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
| September 7<br />
| TBA<br />
| TBA<br />
| TBA<br />
|- }<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15740Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:43:00Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
| September 7<br />
| TBA<br />
| TBA<br />
| TBA<br />
|- }<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15739Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:42:37Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
| September 7<br />
| TBA<br />
| TBA<br />
| TBA}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15738Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:41:28Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
| September 7<br />
| TBA<br />
| TBA<br />
| TBA<br />
|-}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15737Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:40:44Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Fall 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
| September 7<br />
| TBA<br />
| TBA<br />
| TBA<br />
|-}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15736Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:40:23Z<p>Andreic: /* Fall 2018 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Fall 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
| September 7<br />
| TBA<br />
| TBA<br />
| TBA<br />
|-<br />
}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15735Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:40:07Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Fall 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15734Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:39:42Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Fall 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
| September 7<br />
| TBA<br />
| TBA<br />
| TBA<br />
|-<br />
}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15733Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:39:10Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Fall 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
| September 7<br />
| TBA<br />
| TBA<br />
| TBA<br />
}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Fall_2018&diff=15732Algebra and Algebraic Geometry Seminar Fall 20182018-08-22T11:38:09Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B235.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Fall 2018 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|-<br />
|-<br />
|-<br />
}<br />
<br />
== Abstracts ==</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_750_--_Homological_algebra_--_Homeworks&diff=15472Math 750 -- Homological algebra -- Homeworks2018-04-26T21:06:43Z<p>Andreic: </p>
<hr />
<div><!-- The homework assignments refer to the following file, which also contains easier and harder exercises.<br />
<br />
[[Media:Homological_algebra_problems.pdf|Math 750 problems and exercises]]<br />
<br />
;Homework 1: Section 1.1: 4, 5, 8, 10, 20, 21, 22, due Thursday, February 6th.<br />
;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. --><br />
<br />
Homework 1, due Tuesday, February 27:<br />
<br />
a) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.) <br />
<br />
b) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)<br />
<br />
c) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:<br />
<br />
-- the cone of a morphism between two objects in F is again in F;<br />
<br />
-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);<br />
<br />
-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.<br />
<br />
d) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?<br />
<br />
Homework 2, due Thursday, April 5:<br />
<br />
a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)<br />
<br />
b) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)<br />
<br />
c) Consider a short exact sequence of R-modules <br />
<br />
0 -> M' -> M -> M" -> 0.<br />
<br />
Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M', M') under the map <br />
<br />
Hom_R(M', M') -> Ext^1_R(M", M')<br />
<br />
obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.<br />
<br />
d) Prove directly from the definitions (without using Baer's criterion) that Ext^1_Z(M, I) = 0 for every finitely generated Z-module M if and only if I is divisible.<br />
<br />
Homework 3, due Tuesday April 24:<br />
<br />
Do the following exercises from Gelfand-Manin, "Methods of Homological Algebra":<br />
<br />
a) Exercise 1 on p. 163<br />
<br />
b) Exercises 1 and 3 in the section beginning on p. 183<br />
<br />
c) Exercises 1 (parts a and b only) and 5 in the section beginning on p. 214<br />
<br />
Homework 4, due Thursday, May 3:<br />
<br />
a) Exercises 5.1.2 and 5.1.3 from Weibel's book.<br />
<br />
b) Exercise A3.45 from Eisenbud's Commutative Algebra book.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_750_--_Homological_algebra_--_Homeworks&diff=15399Math 750 -- Homological algebra -- Homeworks2018-04-12T18:43:51Z<p>Andreic: </p>
<hr />
<div><!-- The homework assignments refer to the following file, which also contains easier and harder exercises.<br />
<br />
[[Media:Homological_algebra_problems.pdf|Math 750 problems and exercises]]<br />
<br />
;Homework 1: Section 1.1: 4, 5, 8, 10, 20, 21, 22, due Thursday, February 6th.<br />
;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. --><br />
<br />
Homework 1, due Tuesday, February 27:<br />
<br />
a) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.) <br />
<br />
b) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)<br />
<br />
c) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:<br />
<br />
-- the cone of a morphism between two objects in F is again in F;<br />
<br />
-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);<br />
<br />
-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.<br />
<br />
d) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?<br />
<br />
Homework 2, due Thursday, April 5:<br />
<br />
a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)<br />
<br />
b) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)<br />
<br />
c) Consider a short exact sequence of R-modules <br />
<br />
0 -> M' -> M -> M" -> 0.<br />
<br />
Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M', M') under the map <br />
<br />
Hom_R(M', M') -> Ext^1_R(M", M')<br />
<br />
obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.<br />
<br />
d) Prove directly from the definitions (without using Baer's criterion) that Ext^1_Z(M, I) = 0 for every finitely generated Z-module M if and only if I is divisible.<br />
<br />
Homework 3, due Tuesday April 24:<br />
<br />
Do the following exercises from Gelfand-Manin, "Methods of Homological Algebra":<br />
<br />
a) Exercise 1 on p. 163<br />
<br />
b) Exercises 1 and 3 in the section beginning on p. 183<br />
<br />
c) Exercises 1 (parts a and b only) and 5 in the section beginning on p. 214<br />
<!--<br />
Homework 3, due Tuesday, Nov. 24:<br />
<br />
a) Exercises 5.1.2 and 5.1.3 from Weibel's book.<br />
<br />
b) Exercises A3.45, A3.49, A3.50 (parts a, b, c only) from Eisenbud's Commutative Algebra book.<br />
<br />
c) Show that the Koszul complex associated to a regular sequence (f1,...,fn) in a ''local'' commutative ring R is self-dual after a degree shift, in the sense that K(f1,...,fn)^* is isomorphic to K(f1,...fn)[-n]. Here (...)^* denotes the dual complex, that is, the complex obtained by taking Hom_R(..., R). <br />
<br />
Conclude that if the ideal I = (f1,...,fn) is cut out by a regular sequence of length n, we have Ext^n_R(R/I, R) = R/I, and all other Ext's are zero. (Hint: show that the complex K(f1) is self-dual in the above sense, and duality commutes with tensor products for complexes of free modules.)<br />
<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_750_--_Homological_algebra_--_Homeworks&diff=15266Math 750 -- Homological algebra -- Homeworks2018-03-16T18:46:24Z<p>Andreic: </p>
<hr />
<div><!-- The homework assignments refer to the following file, which also contains easier and harder exercises.<br />
<br />
[[Media:Homological_algebra_problems.pdf|Math 750 problems and exercises]]<br />
<br />
;Homework 1: Section 1.1: 4, 5, 8, 10, 20, 21, 22, due Thursday, February 6th.<br />
;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. --><br />
<br />
Homework 1, due Tuesday, February 27:<br />
<br />
a) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.) <br />
<br />
b) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)<br />
<br />
c) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:<br />
<br />
-- the cone of a morphism between two objects in F is again in F;<br />
<br />
-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);<br />
<br />
-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.<br />
<br />
d) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?<br />
<br />
Homework 2, due Thursday, April 5:<br />
<br />
a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)<br />
<br />
b) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)<br />
<br />
c) Consider a short exact sequence of R-modules <br />
<br />
0 -> M' -> M -> M" -> 0.<br />
<br />
Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M', M') under the map <br />
<br />
Hom_R(M', M') -> Ext^1_R(M", M')<br />
<br />
obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.<br />
<br />
d) Prove directly from the definitions (without using Baer's criterion) that Ext^1_Z(M, I) = 0 for every finitely generated Z-module M if and only if I is divisible.<br />
<br />
<br />
<!--<br />
Homework 3, due Tuesday, Nov. 24:<br />
<br />
a) Exercises 5.1.2 and 5.1.3 from Weibel's book.<br />
<br />
b) Exercises A3.45, A3.49, A3.50 (parts a, b, c only) from Eisenbud's Commutative Algebra book.<br />
<br />
c) Show that the Koszul complex associated to a regular sequence (f1,...,fn) in a ''local'' commutative ring R is self-dual after a degree shift, in the sense that K(f1,...,fn)^* is isomorphic to K(f1,...fn)[-n]. Here (...)^* denotes the dual complex, that is, the complex obtained by taking Hom_R(..., R). <br />
<br />
Conclude that if the ideal I = (f1,...,fn) is cut out by a regular sequence of length n, we have Ext^n_R(R/I, R) = R/I, and all other Ext's are zero. (Hint: show that the complex K(f1) is self-dual in the above sense, and duality commutes with tensor products for complexes of free modules.)<br />
<br />
Homework 4, due Tuesday Dec. 15:<br />
<br />
Do the following exercises from Gelfand-Manin:<br />
<br />
a) Exercise 1 on p. 163<br />
<br />
b) Exercises 1 and 3 in the section beginning on p. 183<br />
<br />
c) Exercises 1 and 5 in the section beginning on p. 214<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_750_--_Homological_algebra_--_Homeworks&diff=15263Math 750 -- Homological algebra -- Homeworks2018-03-16T01:40:11Z<p>Andreic: </p>
<hr />
<div><!-- The homework assignments refer to the following file, which also contains easier and harder exercises.<br />
<br />
[[Media:Homological_algebra_problems.pdf|Math 750 problems and exercises]]<br />
<br />
;Homework 1: Section 1.1: 4, 5, 8, 10, 20, 21, 22, due Thursday, February 6th.<br />
;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. --><br />
<br />
Homework 1, due Tuesday, February 27:<br />
<br />
a) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.) <br />
<br />
b) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)<br />
<br />
c) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:<br />
<br />
-- the cone of a morphism between two objects in F is again in F;<br />
<br />
-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);<br />
<br />
-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.<br />
<br />
d) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?<br />
<br />
Homework 2, due Thursday, April 5:<br />
<br />
a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)<br />
<br />
b) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)<br />
<br />
c) Consider a short exact sequence of R-modules <br />
<br />
0 -> M' -> M -> M" -> 0.<br />
<br />
Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M", M") under the map <br />
<br />
Hom_R(M', M') -> Ext^1_R(M", M')<br />
<br />
obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.<br />
<br />
d) Prove directly from the definitions (without using Baer's criterion) that Ext^1_Z(M, I) = 0 for every finitely generated Z-module M if and only if I is divisible.<br />
<br />
<br />
<!--<br />
Homework 3, due Tuesday, Nov. 24:<br />
<br />
a) Exercises 5.1.2 and 5.1.3 from Weibel's book.<br />
<br />
b) Exercises A3.45, A3.49, A3.50 (parts a, b, c only) from Eisenbud's Commutative Algebra book.<br />
<br />
c) Show that the Koszul complex associated to a regular sequence (f1,...,fn) in a ''local'' commutative ring R is self-dual after a degree shift, in the sense that K(f1,...,fn)^* is isomorphic to K(f1,...fn)[-n]. Here (...)^* denotes the dual complex, that is, the complex obtained by taking Hom_R(..., R). <br />
<br />
Conclude that if the ideal I = (f1,...,fn) is cut out by a regular sequence of length n, we have Ext^n_R(R/I, R) = R/I, and all other Ext's are zero. (Hint: show that the complex K(f1) is self-dual in the above sense, and duality commutes with tensor products for complexes of free modules.)<br />
<br />
Homework 4, due Tuesday Dec. 15:<br />
<br />
Do the following exercises from Gelfand-Manin:<br />
<br />
a) Exercise 1 on p. 163<br />
<br />
b) Exercises 1 and 3 in the section beginning on p. 183<br />
<br />
c) Exercises 1 and 5 in the section beginning on p. 214<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Colloquia/Fall18&diff=15183Colloquia/Fall182018-02-24T18:14:09Z<p>Andreic: </p>
<hr />
<div>= Mathematics Colloquium =<br />
<br />
All colloquia are on Fridays at 4:00 pm in Van Vleck B239, '''unless otherwise indicated'''.<br />
<br />
== Spring 2018 ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date <br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s)<br />
|-<br />
|January 29 (Monday)<br />
| [http://www.math.columbia.edu/~chaoli/ Li Chao] (Columbia)<br />
|[[#January 29 Li Chao (Columbia)| Elliptic curves and Goldfeld's conjecture ]]<br />
| Jordan Ellenberg<br />
|<br />
|-<br />
|February 2 (Room: 911)<br />
| [https://scholar.harvard.edu/tfai/home Thomas Fai] (Harvard)<br />
|[[#February 2 Thomas Fai (Harvard)| The Lubricated Immersed Boundary Method ]]<br />
| Spagnolie, Smith<br />
|<br />
|-<br />
|February 5 (Monday, Room: 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 5 Alex Lubotzky (Hebrew University)| High dimensional expanders: From Ramanujan graphs to Ramanujan complexes ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 6 (Tuesday 2 pm, Room 911)<br />
| [http://www.ma.huji.ac.il/~alexlub/ Alex Lubotzky] (Hebrew University) <br />
|[[#February 6 Alex Lubotzky (Hebrew University)| Groups' approximation, stability and high dimensional expanders ]]<br />
| Ellenberg, Gurevitch<br />
|<br />
|-<br />
|February 9<br />
| [http://www.math.cmu.edu/~wes/ Wes Pegden] (CMU)<br />
|[[#February 9 Wes Pegden (CMU)| The fractal nature of the Abelian Sandpile ]]<br />
| Roch<br />
|<br />
|-<br />
|March 2<br />
| [http://www.math.utah.edu/~bertram/ Aaron Bertram] (University of Utah)<br />
|[[#March 2 Aaron Bertram (Utah)| Stability in Algebraic Geometry ]]<br />
| Caldararu<br />
|<br />
|-<br />
| March 16<br />
|[https://math.dartmouth.edu/~annegelb/ Anne Gelb] (Dartmouth)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
|April 4 (Wednesday)<br />
| [http://math.ucr.edu/home/baez/ John Baez] (UC Riverside)<br />
|[[# TBA| TBA ]]<br />
| Craciun<br />
|<br />
|-<br />
| April 6<br />
| Reserved<br />
|[[# TBA| TBA ]]<br />
| Melanie<br />
|<br />
|-<br />
| April 13<br />
| [https://www.math.brown.edu/~jpipher/ Jill Pipher] (Brown)<br />
|[[# TBA| TBA ]]<br />
| WIMAW<br />
|<br />
|-<br />
| April 20<br />
| Xiuxiong Chen(Stony Brook University)<br />
|[[# Xiuxiong Chen| TBA ]]<br />
| Bing Wang<br />
|<br />
|-<br />
| April 25 (Wednesday)<br />
| [http://www.f.waseda.jp/hitoshi.ishii/ Hitoshi Ishii] (Waseda University) Wasow lecture<br />
|[[# TBA| TBA ]]<br />
| Tran<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
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|date<br />
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|[[# TBA| TBA ]]<br />
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|-<br />
|date<br />
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|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
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|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
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|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
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|<br />
|-<br />
|date<br />
| person (institution)<br />
|[[# TBA| TBA ]]<br />
| hosting faculty<br />
|<br />
|}<br />
<br />
== Spring Abstracts ==<br />
<br />
<br />
===January 29 Li Chao (Columbia)===<br />
<br />
Title: Elliptic curves and Goldfeld's conjecture<br />
<br />
Abstract: <br />
An elliptic curve is a plane curve defined by a cubic equation. Determining whether such an equation has infinitely many rational solutions has been a central problem in number theory for centuries, which lead to the celebrated conjecture of Birch and Swinnerton-Dyer. Within a family of elliptic curves (such as the Mordell curve family y^2=x^3-d), a conjecture of Goldfeld further predicts that there should be infinitely many rational solutions exactly half of the time. We will start with a history of this problem, discuss our recent work (with D. Kriz) towards Goldfeld's conjecture and illustrate the key ideas and ingredients behind these new progresses.<br />
<br />
=== February 2 Thomas Fai (Harvard) ===<br />
<br />
Title: The Lubricated Immersed Boundary Method<br />
<br />
Abstract:<br />
Many real-world examples of fluid-structure interaction, including the transit of red blood cells through the narrow slits in the spleen, involve the near-contact of elastic structures separated by thin layers of fluid. The separation of length scales between these fine lubrication layers and the larger elastic objects poses significant computational challenges. Motivated by the challenge of resolving such multiscale problems, we introduce an immersed boundary method that uses elements of lubrication theory to resolve thin fluid layers between immersed boundaries. We apply this method to two-dimensional flows of increasing complexity, including eccentric rotating cylinders and elastic vesicles near walls in shear flow, to show its increased accuracy compared to the classical immersed boundary method. We present preliminary simulation results of cell suspensions, a problem in which near-contact occurs at multiple levels, such as cell-wall, cell-cell, and intracellular interactions, to highlight the importance of resolving thin fluid layers in order to obtain the correct overall dynamics.<br />
<br />
===February 5 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: High dimensional expanders: From Ramanujan graphs to Ramanujan complexes<br />
<br />
Abstract: <br />
<br />
Expander graphs in general, and Ramanujan graphs , in particular, have played a major role in computer science in the last 5 decades and more recently also in pure math. The first explicit construction of bounded degree expanding graphs was given by Margulis in the early 70's. In mid 80' Margulis and Lubotzky-Phillips-Sarnak provided Ramanujan graphs which are optimal such expanders. <br />
<br />
In recent years a high dimensional theory of expanders is emerging. A notion of topological expanders was defined by Gromov in 2010 who proved that the complete d-dimensional simplical complexes are such. He raised the basic question of existence of such bounded degree complexes of dimension d>1. <br />
<br />
This question was answered recently affirmatively (by T. Kaufman, D. Kazdhan and A. Lubotzky for d=2 and by S. Evra and T. Kaufman for general d) by showing that the d-skeleton of (d+1)-dimensional Ramanujan complexes provide such topological expanders. We will describe these developments and the general area of high dimensional expanders. <br />
<br />
<br />
===February 6 Alex Lubotzky (Hebrew University)===<br />
<br />
Title: Groups' approximation, stability and high dimensional expanders<br />
<br />
Abstract: <br />
<br />
Several well-known open questions, such as: are all groups sofic or hyperlinear?, have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups Sym(n) (in the sofic case) or the unitary groups U(n) (in the hyperlinear case)? In the case of U(n), the question can be asked with respect to different metrics and norms. We answer, for the first time, one of these versions, showing that there exist fintely presented groups which are not approximated by U(n) with respect to the Frobenius (=L_2) norm.<br />
<br />
The strategy is via the notion of "stability": some higher dimensional cohomology vanishing phenomena is proven to imply stability and using high dimensional expanders, it is shown that some non-residually finite groups (central extensions of some lattices in p-adic Lie groups) are Frobenious stable and hence cannot be Frobenius approximated. <br />
<br />
All notions will be explained. Joint work with M, De Chiffre, L. Glebsky and A. Thom.<br />
<br />
===February 9 Wes Pegden (CMU)===<br />
<br />
Title: The fractal nature of the Abelian Sandpile <br />
<br />
Abstract: The Abelian Sandpile is a simple diffusion process on the integer lattice, in which configurations of chips disperse according to a simple rule: when a vertex has at least 4 chips, it can distribute one chip to each neighbor. <br />
<br />
Introduced in the statistical physics community in the 1980s, the Abelian sandpile exhibits striking fractal behavior which long resisted rigorous mathematical analysis (or even a plausible explanation). We now have a relatively robust mathematical understanding of this fractal nature of the sandpile, which involves surprising connections between integer superharmonic functions on the lattice, discrete tilings of the plane, and Apollonian circle packings. In this talk, we will survey our work in this area, and discuss avenues of current and future research.<br />
<br />
===March 3 Aaron Bertram (Utah)===<br />
<br />
Title: Stability in Algebraic Geometry<br />
<br />
Abstract: Stability was originally introduced in algebraic geometry in the context of finding a projective quotient space for the action of an algebraic group on a projective manifold. This, in turn, led in the 1960s to a notion of slope-stability for vector bundles on a Riemann surface, which was an important tool in the classification of vector bundles. In the 1990s, mirror symmetry considerations led Michael Douglas to notions of stability for "D-branes" (on a higher-dimensional manifold) that corresponded to no previously known mathematical definition. We now understand each of these notions of stability as a distinct point of a complex "stability manifold" that is an important invariant of the (derived) category of complexes of vector bundles of a projective manifold. In this talk I want to give some examples to illustrate the various stabilities, and also to describe some current work in the area.<br />
<br />
== Past Colloquia ==<br />
<br />
[[Colloquia/Blank|Blank Colloquia]]<br />
<br />
[[Colloquia/Fall2017|Fall 2017]]<br />
<br />
[[Colloquia/Spring2017|Spring 2017]]<br />
<br />
[[Archived Fall 2016 Colloquia|Fall 2016]]<br />
<br />
[[Colloquia/Spring2016|Spring 2016]]<br />
<br />
[[Colloquia/Fall2015|Fall 2015]]<br />
<br />
[[Colloquia/Spring2014|Spring 2015]]<br />
<br />
[[Colloquia/Fall2014|Fall 2014]]<br />
<br />
[[Colloquia/Spring2014|Spring 2014]]<br />
<br />
[[Colloquia/Fall2013|Fall 2013]]<br />
<br />
[[Colloquia 2012-2013|Spring 2013]]<br />
<br />
[[Colloquia 2012-2013#Fall 2012|Fall 2012]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_750_--_Homological_algebra_--_Homeworks&diff=15102Math 750 -- Homological algebra -- Homeworks2018-02-13T04:49:56Z<p>Andreic: </p>
<hr />
<div><!-- The homework assignments refer to the following file, which also contains easier and harder exercises.<br />
<br />
[[Media:Homological_algebra_problems.pdf|Math 750 problems and exercises]]<br />
<br />
;Homework 1: Section 1.1: 4, 5, 8, 10, 20, 21, 22, due Thursday, February 6th.<br />
;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. --><br />
<br />
Homework 1, due Tuesday, February 27:<br />
<br />
a) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.) <br />
<br />
b) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)<br />
<br />
c) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:<br />
<br />
-- the cone of a morphism between two objects in F is again in F;<br />
<br />
-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);<br />
<br />
-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.<br />
<br />
d) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?<br />
<br />
<!-- Homework 2, due Thursday, October 22:<br />
<br />
a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)<br />
<br />
b) Show that for any Z-modules M and N we have Ext^i_Z(M, N) = 0 for i >=2. We say that Z (the integers) has homological dimension 1.<br />
<br />
c) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)<br />
<br />
d) If F: A -> B is a right exact functor between abelian categories, and if A has enough projectives (so that LF is defined) then we shall say that an object X of A is F-acyclic if it satisfies R^i F(X) = 0 for i>0. For example any projective in A is F-acyclic.<br />
<br />
-- show that if F is in fact exact, then any X in A is F-acyclic.<br />
<br />
-- an F-acyclic resolution of an object Y of A is a complex<br />
<br />
... -> X_n -> X_{n-1} -> ... -> X_0 -> 0 <br />
<br />
such that the complex is exact except at the last spot, where the homology is Y, and each X_i is F-acyclic. <br />
<br />
Show that the homology H_i(F(X.)) is naturally isomorphic to L_i F(Y). (Thus derived functors can be computed using F-acylic resolutions, not only using projective resolutions.) In some situations this allows us to construct left derived functors even when there are not enough projectives, if we can identify enough F-acyclics. (Hint: break up the resolution into short exact sequences.)<br />
<br />
e) (More difficult) Consider a short exact sequence of R-modules <br />
<br />
0 -> M' -> M -> M" -> 0.<br />
<br />
Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M", M") under the map <br />
<br />
Hom_R(M', M') -> Ext^1_R(M", M')<br />
<br />
obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.<br />
<br />
Homework 3, due Tuesday, Nov. 24:<br />
<br />
a) Exercises 5.1.2 and 5.1.3 from Weibel's book.<br />
<br />
b) Exercises A3.45, A3.49, A3.50 (parts a, b, c only) from Eisenbud's Commutative Algebra book.<br />
<br />
c) Show that the Koszul complex associated to a regular sequence (f1,...,fn) in a ''local'' commutative ring R is self-dual after a degree shift, in the sense that K(f1,...,fn)^* is isomorphic to K(f1,...fn)[-n]. Here (...)^* denotes the dual complex, that is, the complex obtained by taking Hom_R(..., R). <br />
<br />
Conclude that if the ideal I = (f1,...,fn) is cut out by a regular sequence of length n, we have Ext^n_R(R/I, R) = R/I, and all other Ext's are zero. (Hint: show that the complex K(f1) is self-dual in the above sense, and duality commutes with tensor products for complexes of free modules.)<br />
<br />
Homework 4, due Tuesday Dec. 15:<br />
<br />
Do the following exercises from Gelfand-Manin:<br />
<br />
a) Exercise 1 on p. 163<br />
<br />
b) Exercises 1 and 3 in the section beginning on p. 183<br />
<br />
c) Exercises 1 and 5 in the section beginning on p. 214<br />
--></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_750_--_Homological_algebra_--_Homeworks&diff=15101Math 750 -- Homological algebra -- Homeworks2018-02-13T04:44:07Z<p>Andreic: </p>
<hr />
<div><!-- The homework assignments refer to the following file, which also contains easier and harder exercises.<br />
<br />
[[Media:Homological_algebra_problems.pdf|Math 750 problems and exercises]]<br />
<br />
;Homework 1: Section 1.1: 4, 5, 8, 10, 20, 21, 22, due Thursday, February 6th.<br />
;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th. --><br />
<br />
Homework 1, due Thursday, October 1:<br />
<br />
a) Show by example that HoCh(A) is not always an abelian category.<br />
<br />
b) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.) <br />
<br />
c) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)<br />
<br />
d) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:<br />
<br />
-- the cone of a morphism between two objects in F is again in F;<br />
<br />
-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);<br />
<br />
-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.<br />
<br />
e) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?<br />
<br />
Homework 2, due Thursday, October 22:<br />
<br />
a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)<br />
<br />
b) Show that for any Z-modules M and N we have Ext^i_Z(M, N) = 0 for i >=2. We say that Z (the integers) has homological dimension 1.<br />
<br />
c) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)<br />
<br />
d) If F: A -> B is a right exact functor between abelian categories, and if A has enough projectives (so that LF is defined) then we shall say that an object X of A is F-acyclic if it satisfies R^i F(X) = 0 for i>0. For example any projective in A is F-acyclic.<br />
<br />
-- show that if F is in fact exact, then any X in A is F-acyclic.<br />
<br />
-- an F-acyclic resolution of an object Y of A is a complex<br />
<br />
... -> X_n -> X_{n-1} -> ... -> X_0 -> 0 <br />
<br />
such that the complex is exact except at the last spot, where the homology is Y, and each X_i is F-acyclic. <br />
<br />
Show that the homology H_i(F(X.)) is naturally isomorphic to L_i F(Y). (Thus derived functors can be computed using F-acylic resolutions, not only using projective resolutions.) In some situations this allows us to construct left derived functors even when there are not enough projectives, if we can identify enough F-acyclics. (Hint: break up the resolution into short exact sequences.)<br />
<br />
e) (More difficult) Consider a short exact sequence of R-modules <br />
<br />
0 -> M' -> M -> M" -> 0.<br />
<br />
Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M", M") under the map <br />
<br />
Hom_R(M', M') -> Ext^1_R(M", M')<br />
<br />
obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.<br />
<br />
Homework 3, due Tuesday, Nov. 24:<br />
<br />
a) Exercises 5.1.2 and 5.1.3 from Weibel's book.<br />
<br />
b) Exercises A3.45, A3.49, A3.50 (parts a, b, c only) from Eisenbud's Commutative Algebra book.<br />
<br />
c) Show that the Koszul complex associated to a regular sequence (f1,...,fn) in a ''local'' commutative ring R is self-dual after a degree shift, in the sense that K(f1,...,fn)^* is isomorphic to K(f1,...fn)[-n]. Here (...)^* denotes the dual complex, that is, the complex obtained by taking Hom_R(..., R). <br />
<br />
Conclude that if the ideal I = (f1,...,fn) is cut out by a regular sequence of length n, we have Ext^n_R(R/I, R) = R/I, and all other Ext's are zero. (Hint: show that the complex K(f1) is self-dual in the above sense, and duality commutes with tensor products for complexes of free modules.)<br />
<br />
Homework 4, due Tuesday Dec. 15:<br />
<br />
Do the following exercises from Gelfand-Manin:<br />
<br />
a) Exercise 1 on p. 163<br />
<br />
b) Exercises 1 and 3 in the section beginning on p. 183<br />
<br />
c) Exercises 1 and 5 in the section beginning on p. 214</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_750_--_Homological_algebra_--_Homeworks&diff=15100Math 750 -- Homological algebra -- Homeworks2018-02-13T04:39:01Z<p>Andreic: </p>
<hr />
<div>The homework assignments refer to the following file, which also contains easier and harder exercises.<br />
<br />
[[Media:Homological_algebra_problems.pdf|Math 750 problems and exercises]]<br />
<br />
;Homework 1: Section 1.1: 4, 5, 8, 10, 20, 21, 22, due Thursday, February 6th.<br />
;Homework 2: Section 1.2: 2, 4, 6, 7, 8, 14, 15, due Thursday, February 20th.<br />
<br />
Homework 1, due Thursday, October 1:<br />
<br />
a) Show by example that HoCh(A) is not always an abelian category.<br />
<br />
b) Give an example of a bounded chain complex in Ch(Z-mod) which is acyclic but not split exact. (So both conditions of the Theorem we proved in class are needed.) <br />
<br />
c) Prove that any acyclic complex of vector spaces over a field is split exact. (Do not assume that the complex is bounded!)<br />
<br />
d) Prove that the full subcategory F of Ch^-(R-Mod) consisting of bounded above complexes of free R-modules satisfies the properties below:<br />
<br />
-- the cone of a morphism between two objects in F is again in F;<br />
<br />
-- if G : R-Mod -> A is an additive functor to an abelian category A, then G takes quasi-isomorphisms in F to quasi-isomorphisms in Ch(A);<br />
<br />
-- if M is an R-module, a resolution of M is an object C of F together with a map C -> M[0] which is a quasi-isomorphism. Prove that maps M -> N of R-modules lift to maps of resolutions of the modules, uniquely up to homotopy.<br />
<br />
e) Let f: M' -> M be an injective map of R-modules, and let M" denote the cokernel of f. Show that cone(f) is quasi-isomorphic to M"[0]. Are they always isomorphic in HoCh(R-mod)?<br />
<br />
Homework 2, due Thursday, October 22:<br />
<br />
a) Compute Ext^i_{Z/4Z}(Z/2Z, Z/2Z) and Tor_i^{Z/4Z}(Z/2Z, Z/2Z) for all i. (For Ext use projective resolutions in the first variable.)<br />
<br />
b) Show that for any Z-modules M and N we have Ext^i_Z(M, N) = 0 for i >=2. We say that Z (the integers) has homological dimension 1.<br />
<br />
c) Compute Ext^i_R(R, k) and Ext^i_R(k,k) where R = k[x1,...,xn] is the polynomial ring in n variables, k is a field, and k is regarded as an R module by the identification k = R/(x1, ..., xn). (Hint: you may want to look up the following commutative algebra topics on Wikipedia or in any standard textbook: regular sequence, Koszul resolution.) So the homological dimension of R is >=n. (It is in fact n.)<br />
<br />
d) If F: A -> B is a right exact functor between abelian categories, and if A has enough projectives (so that LF is defined) then we shall say that an object X of A is F-acyclic if it satisfies R^i F(X) = 0 for i>0. For example any projective in A is F-acyclic.<br />
<br />
-- show that if F is in fact exact, then any X in A is F-acyclic.<br />
<br />
-- an F-acyclic resolution of an object Y of A is a complex<br />
<br />
... -> X_n -> X_{n-1} -> ... -> X_0 -> 0 <br />
<br />
such that the complex is exact except at the last spot, where the homology is Y, and each X_i is F-acyclic. <br />
<br />
Show that the homology H_i(F(X.)) is naturally isomorphic to L_i F(Y). (Thus derived functors can be computed using F-acylic resolutions, not only using projective resolutions.) In some situations this allows us to construct left derived functors even when there are not enough projectives, if we can identify enough F-acyclics. (Hint: break up the resolution into short exact sequences.)<br />
<br />
e) (More difficult) Consider a short exact sequence of R-modules <br />
<br />
0 -> M' -> M -> M" -> 0.<br />
<br />
Let eta in Ext^1_R(M", M') be the image of the identity in Hom(M", M") under the map <br />
<br />
Hom_R(M', M') -> Ext^1_R(M", M')<br />
<br />
obtained by applying the Hom_R(--, M') functor to the above short exact sequence. Prove that eta=0 if and only if the short exact sequence we started with is split.<br />
<br />
Homework 3, due Tuesday, Nov. 24:<br />
<br />
a) Exercises 5.1.2 and 5.1.3 from Weibel's book.<br />
<br />
b) Exercises A3.45, A3.49, A3.50 (parts a, b, c only) from Eisenbud's Commutative Algebra book.<br />
<br />
c) Show that the Koszul complex associated to a regular sequence (f1,...,fn) in a ''local'' commutative ring R is self-dual after a degree shift, in the sense that K(f1,...,fn)^* is isomorphic to K(f1,...fn)[-n]. Here (...)^* denotes the dual complex, that is, the complex obtained by taking Hom_R(..., R). <br />
<br />
Conclude that if the ideal I = (f1,...,fn) is cut out by a regular sequence of length n, we have Ext^n_R(R/I, R) = R/I, and all other Ext's are zero. (Hint: show that the complex K(f1) is self-dual in the above sense, and duality commutes with tensor products for complexes of free modules.)<br />
<br />
Homework 4, due Tuesday Dec. 15:<br />
<br />
Do the following exercises from Gelfand-Manin:<br />
<br />
a) Exercise 1 on p. 163<br />
<br />
b) Exercises 1 and 3 in the section beginning on p. 183<br />
<br />
c) Exercises 1 and 5 in the section beginning on p. 214</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_750_--_Homological_algebra&diff=15099Math 750 -- Homological algebra2018-02-13T04:37:32Z<p>Andreic: /* Spring 2015 */</p>
<hr />
<div><br />
===Spring 2018===<br />
<br />
[[Math 750 -- Homological algebra -- Homeworks|Homework assignments]]<br />
<br />
== Course content (probably overly optimistic) ==<br />
<br />
This course is an introduction to homological algebra. I hope to cover the following topics.<br />
<br />
;Derived functors: Many constructions in mathematics lead to functors that fail to be exact (do not respect exact sequences). This issue can often be corrected by introducing the so called derived functors; we will look at the construction and numerous examples from commutative algebra/algebraic geometry, representation theory, and topology. Important ideas: injective/projective resolutions, acyclic objects, (co)homological dimension, spectral sequences.<br />
;Derived categories: Derived functors can be computed as cohomology objects of certain complexes. However, in this process we lose some information: it is often important to remember the complex itself. This path naturally leads to derived categories; roughly speaking, the idea is to identify an object with all of its resolutions. Important ideas: quasi-isomorphism, cone of morphism, triangulated categories, resolutions of complexes, homotopy category of complexes.<br />
;Beyond triangulated categories: As we will see, viewing the derived categories as triangulated categories is sometimes less than ideal. To resolve this issue, one has to replace traingulated categories by better frameworks, such as dg-categories or infinity-categories. Important ideas: dg-algebras and dg-categories, homotopy category of a dg-category.<br />
<br />
There are many other important and interesting topics, which I would be happy to discuss if the time permits. However, this seems incredibly unlikely, so perhaps they may be more appropriate for, say, a talk at a seminar for graduate students. Some such topics are compact objects, compactly generated categories, the Brown Representability Theorem, model categories, dg-Lie algberas and deformation theory, and others.<br />
<br />
== Plan (probably overly optimistic) ==<br />
<ol><br />
<li>''Naive'' homological algebra: derived functors.<br />
<ol style="list-style-type:lower-roman"> <br />
<li>Ext for abelian groups</li><br />
<li>Derived functors in category of modules. Examples: Ext, Tor</li><br />
<li>Abelian categories. Examples.</li><br />
<li>Spectral sequences.</li> <br />
</ol><br />
</li><br />
<li>''Modern'' homological algebra: derived categories.<br />
<ol style="list-style-type:lower-roman"><br />
<li>Resolution of complexes.</li><br />
<li>Derived categories and derived functors. Examples</li><br />
<li>Triangulated categories.</li><br />
</ol><br />
</li><br />
<li>''Sophisticated'' homological algebra: beyond triangulated categories.<br />
<ol style="list-style-type:lower-roman"><br />
<li> DG categories.</li><br />
</ol><br />
</li><br />
</ol></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2018&diff=15052Algebra and Algebraic Geometry Seminar Spring 20182018-02-07T23:10:31Z<p>Andreic: /* Spring 2018 Schedule */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B113.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2017 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|January 26<br />
|[http://homepages.math.uic.edu/~tmoulinos/ Tasos Moulinos (UIC)] <br />
|[[#Tasos Moulinos|Derived Azumaya Algebras and Twisted K-theory]]<br />
|Michael<br />
|-<br />
|February 2<br />
|Daniel Erman (Wisconsin) <br />
|[[#Daniel Erman|TBA]]<br />
|Local<br />
|-<br />
|'''February 8''' 2:30-3:30 in VV B113<br />
|[http://www.mathematics.pitt.edu/person/roman-fedorov/ Roman Fedorov (University of Pittsburgh)]<br />
|[[#Roman Fedorov|A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic]]<br />
|Dima<br />
|-<br />
|February 9<br />
|Juliette Bruce (Wisconsin) <br />
|[[#Juliette Bruce|Asymptotic Syzygies in the Semi-Ample Setting ]]<br />
|Local<br />
|-<br />
|February 16<br />
|[http://www.math.wisc.edu/~andreic/ Andrei Caldararu (Wisconsin)]<br />
|[[#Andrei Caldararu|Computing a categorical Gromov-Witten invariant]]<br />
|Local<br />
|-<br />
|February 23<br />
|Aron Heleodoro (Northwestern) <br />
|[[#Aron Heleodoro|TBA]]<br />
|Dima<br />
|-<br />
|March 2<br />
|Moisés Herradón Cueto (Wisconsin)<br />
|[[#Moisés Herradón Cueto|TBA]]<br />
|Local<br />
|-<br />
|March 16<br />
|[https://math.berkeley.edu/~chenhi/ Harrison Chen (Berkeley)]<br />
|[[#Harrison Chen|Equivariant localization for periodic cyclic homology and derived loop spaces]]<br />
|Andrei<br />
|-<br />
|April 6<br />
|[http://www-personal.umich.edu/~ptoste/ Phil Tosteson (Michigan)]<br />
|[[#Phil Tosteson|TBA]]<br />
|Steven<br />
|-<br />
|-<br />
|April 13<br />
|Reserved<br />
|<br />
|Daniel<br />
|-<br />
|April 20<br />
|Alena Pirutka (NYU)<br />
|[[#Alena Pirutka|TBA]]<br />
|Jordan<br />
|-<br />
|April 27<br />
|Alexander Yom Din (Caltech) <br />
|[[#Alexander Yom Din|TBA]]<br />
|Dima<br />
|-<br />
|May 4<br />
|John Lesieutre (UIC) <br />
|[[#John Lesieutre|TBA]]<br />
|Daniel<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Tasos Moulinos===<br />
<br />
'''Derived Azumaya Algebras and Twisted K-theory'''<br />
<br />
Topological K-theory of dg-categories is a localizing invariant of dg-categories over <math> \mathbb{C} </math><br />
taking values in the <math> \infty </math>-category of <math> KU </math>-modules. In this talk I describe a relative version<br />
of this construction; namely for <math>X</math> a quasi-compact, quasi-separated <math> \mathbb{C} </math>-scheme I construct a<br />
functor valued in the <math> \infty </math>-category of sheaves of spectra on <math> X(\mathbb{C}) </math>, the complex points of <math>X</math>. For inputs<br />
of the form <math>\operatorname{Perf}(X, A)</math> where <math>A</math> is an Azumaya algebra over <math>X</math>, I characterize the values<br />
of this functor in terms of the twisted topological K-theory of <math> X(\mathbb{C}) </math>. From this I deduce<br />
a certain decomposition, for <math> X </math> a finite CW-complex equipped with a bundle <math> P </math> of projective<br />
spaces over <math> X </math>, of <math> KU(P) </math> in terms of the twisted topological K-theory of <math> X </math> ; this is<br />
a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer<br />
schemes.<br />
<br />
===Roman Fedorov===<br />
<br />
'''A conjecture of Grothendieck and Serre on principal bundles in mixed<br />
characteristic'''<br />
<br />
Let G be a reductive group scheme over a regular local ring R. An old<br />
conjecture of Grothendieck and Serre predicts that such a principal<br />
bundle is trivial, if it is trivial over the fraction field of R. The<br />
conjecture has recently been proved in the "geometric" case, that is,<br />
when R contains a field. In the remaining case, the difficulty comes<br />
from the fact, that the situation is more rigid, so that a certain<br />
general position argument does not go through. I will discuss this<br />
difficulty and a way to circumvent it to obtain some partial results.<br />
<br />
===Juliette Bruce===<br />
<br />
'''Asymptotic Syzygies in the Semi-Ample Setting'''<br />
<br />
In recent years numerous conjectures have been made describing the asymptotic Betti numbers of a projective variety as the embedding line bundle becomes more ample. I will discuss recent work attempting to generalize these conjectures to the case when the embedding line bundle becomes more semi-ample. (Recall a line bundle is semi-ample if a sufficiently large multiple is base point free.) In particular, I will discuss how the monomial methods of Ein, Erman, and Lazarsfeld used to prove non-vanishing results on projective space can be extended to prove non-vanishing results for products of projective space.<br />
<br />
===Andrei Caldararu===<br />
<br />
'''Computing a categorical Gromov-Witten invariant'''<br />
<br />
In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-inifinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical invariance of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other categories like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.<br />
<br />
In my talk I shall describe the details of the computation (joint with Junwu Tu) of the invariant, at g=1, n=1, for elliptic curves. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.<br />
<br />
===Aron Heleodoro===<br />
<br />
'''TBA'''<br />
<br />
===Harrison Chen===<br />
<br />
'''Equivariant localization for periodic cyclic homology and derived loop spaces'''<br />
<br />
There is a close relationship between derived loop spaces, a geometric object, and (periodic) cyclic homology, a categorical invariant. In this talk we will discuss this relationship and how it leads to an equivariant localization result, which has an intuitive interpretation using the language of derived loop spaces. We discuss ongoing generalizations and potential applications in computing the periodic cyclic homology of categories of equivariant (coherent) sheaves on algebraic varieties.<br />
<br />
===Alexander Yom Din===<br />
<br />
'''TBA'''</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2018&diff=15051Algebra and Algebraic Geometry Seminar Spring 20182018-02-07T23:08:58Z<p>Andreic: /* Abstracts */</p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B113.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2017 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|January 26<br />
|[http://homepages.math.uic.edu/~tmoulinos/ Tasos Moulinos (UIC)] <br />
|[[#Tasos Moulinos|Derived Azumaya Algebras and Twisted K-theory]]<br />
|Michael<br />
|-<br />
|February 2<br />
|Daniel Erman (Wisconsin) <br />
|[[#Daniel Erman|TBA]]<br />
|Local<br />
|-<br />
|'''February 8''' 2:30-3:30 in VV B113<br />
|[http://www.mathematics.pitt.edu/person/roman-fedorov/ Roman Fedorov (University of Pittsburgh)]<br />
|[[#Roman Fedorov|A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic]]<br />
|Dima<br />
|-<br />
|February 9<br />
|Juliette Bruce (Wisconsin) <br />
|[[#Juliette Bruce|Asymptotic Syzygies in the Semi-Ample Setting ]]<br />
|Local<br />
|-<br />
|February 16<br />
|[http://www.math.wisc.edu/~andreic/ Andrei Caldararu (Wisconsin)]<br />
|[[#Andrei Caldararu|Computing a categorical Gromov-Witten invariant]]<br />
|Local<br />
|-<br />
|February 23<br />
|Aron Heleodoro (Northwestern) <br />
|[[#Aron Heleodoro|TBA]]<br />
|Dima<br />
|-<br />
|March 2<br />
|Moisés Herradón Cueto (Wisconsin)<br />
|[[#Moisés Herradón Cueto|TBA]]<br />
|Local<br />
|-<br />
|March 16<br />
|[https://math.berkeley.edu/~chenhi/ Harrison Chen (Berkeley)]<br />
|[[#Harrison Chen|TBA]]<br />
|Andrei<br />
|-<br />
|April 6<br />
|[http://www-personal.umich.edu/~ptoste/ Phil Tosteson (Michigan)]<br />
|[[#Phil Tosteson|TBA]]<br />
|Steven<br />
|-<br />
|-<br />
|April 13<br />
|Reserved<br />
|<br />
|Daniel<br />
|-<br />
|April 20<br />
|Alena Pirutka (NYU)<br />
|[[#Alena Pirutka|TBA]]<br />
|Jordan<br />
|-<br />
|April 27<br />
|Alexander Yom Din (Caltech) <br />
|[[#Alexander Yom Din|TBA]]<br />
|Dima<br />
|-<br />
|May 4<br />
|John Lesieutre (UIC) <br />
|[[#John Lesieutre|TBA]]<br />
|Daniel<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Tasos Moulinos===<br />
<br />
'''Derived Azumaya Algebras and Twisted K-theory'''<br />
<br />
Topological K-theory of dg-categories is a localizing invariant of dg-categories over <math> \mathbb{C} </math><br />
taking values in the <math> \infty </math>-category of <math> KU </math>-modules. In this talk I describe a relative version<br />
of this construction; namely for <math>X</math> a quasi-compact, quasi-separated <math> \mathbb{C} </math>-scheme I construct a<br />
functor valued in the <math> \infty </math>-category of sheaves of spectra on <math> X(\mathbb{C}) </math>, the complex points of <math>X</math>. For inputs<br />
of the form <math>\operatorname{Perf}(X, A)</math> where <math>A</math> is an Azumaya algebra over <math>X</math>, I characterize the values<br />
of this functor in terms of the twisted topological K-theory of <math> X(\mathbb{C}) </math>. From this I deduce<br />
a certain decomposition, for <math> X </math> a finite CW-complex equipped with a bundle <math> P </math> of projective<br />
spaces over <math> X </math>, of <math> KU(P) </math> in terms of the twisted topological K-theory of <math> X </math> ; this is<br />
a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer<br />
schemes.<br />
<br />
===Roman Fedorov===<br />
<br />
'''A conjecture of Grothendieck and Serre on principal bundles in mixed<br />
characteristic'''<br />
<br />
Let G be a reductive group scheme over a regular local ring R. An old<br />
conjecture of Grothendieck and Serre predicts that such a principal<br />
bundle is trivial, if it is trivial over the fraction field of R. The<br />
conjecture has recently been proved in the "geometric" case, that is,<br />
when R contains a field. In the remaining case, the difficulty comes<br />
from the fact, that the situation is more rigid, so that a certain<br />
general position argument does not go through. I will discuss this<br />
difficulty and a way to circumvent it to obtain some partial results.<br />
<br />
===Juliette Bruce===<br />
<br />
'''Asymptotic Syzygies in the Semi-Ample Setting'''<br />
<br />
In recent years numerous conjectures have been made describing the asymptotic Betti numbers of a projective variety as the embedding line bundle becomes more ample. I will discuss recent work attempting to generalize these conjectures to the case when the embedding line bundle becomes more semi-ample. (Recall a line bundle is semi-ample if a sufficiently large multiple is base point free.) In particular, I will discuss how the monomial methods of Ein, Erman, and Lazarsfeld used to prove non-vanishing results on projective space can be extended to prove non-vanishing results for products of projective space.<br />
<br />
===Andrei Caldararu===<br />
<br />
'''Computing a categorical Gromov-Witten invariant'''<br />
<br />
In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-inifinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical invariance of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other categories like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.<br />
<br />
In my talk I shall describe the details of the computation (joint with Junwu Tu) of the invariant, at g=1, n=1, for elliptic curves. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.<br />
<br />
===Aron Heleodoro===<br />
<br />
'''TBA'''<br />
<br />
===Harrison Chen===<br />
<br />
'''Equivariant localization for periodic cyclic homology and derived loop spaces'''<br />
<br />
There is a close relationship between derived loop spaces, a geometric object, and (periodic) cyclic homology, a categorical invariant. In this talk we will discuss this relationship and how it leads to an equivariant localization result, which has an intuitive interpretation using the language of derived loop spaces. We discuss ongoing generalizations and potential applications in computing the periodic cyclic homology of categories of equivariant (coherent) sheaves on algebraic varieties.<br />
<br />
===Alexander Yom Din===<br />
<br />
'''TBA'''</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Main_Page&diff=15043Main Page2018-02-06T22:15:00Z<p>Andreic: </p>
<hr />
<div><br />
== Welcome to the University of Wisconsin Math Department Wiki ==<br />
<br />
This site is by and for the faculty, students and staff of the UW Mathematics Department. It contains useful information about the department, not always available from other sources. Pages can only be edited by members of the department but are viewable by everyone. <br />
<br />
*[[Getting Around Van Vleck]]<br />
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== Research groups at UW-Madison ==<br />
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*[[Algebra]]<br />
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*[https://www.math.wisc.edu/wiki/index.php/Research_at_UW-Madison_in_DifferentialEquations Differential Equations]<br />
*[[Dynamics Special Lecture]]<br />
*[[Geometry and Topology]]<br />
* [http://www.math.wisc.edu/~lempp/logic.html Logic]<br />
*[[Probability]]<br />
<br />
== Math Seminars at UW-Madison ==<br />
<br />
*[[Colloquia|Colloquium]]<br />
*[[Algebra_and_Algebraic_Geometry_Seminar|Algebra and Algebraic Geometry Seminar]]<br />
*[[Analysis_Seminar|Analysis Seminar]]<br />
*[[Applied/ACMS|Applied and Computational Math Seminar]]<br />
*[http://www.math.wisc.edu/~zcharles/aas/index.html Applied Algebra Seminar]<br />
*[[Cookie_seminar|Cookie Seminar]]<br />
*[[Geometry_and_Topology_Seminar|Geometry and Topology Seminar]]<br />
*[[Group_Theory_Seminar|Group Theory Seminar]]<br />
*[[Networks_Seminar|Networks Seminar]]<br />
*[[NTS|Number Theory Seminar]]<br />
*[[PDE_Geometric_Analysis_seminar| PDE and Geometric Analysis Seminar]]<br />
*[[Probability_Seminar|Probability Seminar]]<br />
* [http://www.math.wisc.edu/~lempp/conf/swlc.html Southern Wisconsin Logic Colloquium]<br />
*[[Research Recruitment Seminar]]<br />
<br />
=== Graduate Student Seminars ===<br />
<br />
*[[AMS_Student_Chapter_Seminar|AMS Student Chapter Seminar]]<br />
*[[Graduate_Algebraic_Geometry_Seminar|Graduate Algebraic Geometry Seminar]]<br />
*[[Graduate_Applied_Algebra_Seminar|Graduate Applied Algebra Seminar]]<br />
*[[Applied/GPS| GPS Applied Math Seminar]]<br />
*[[NTSGrad_Spring_2018|Graduate Number Theory/Representation Theory Seminar]]<br />
*[[Symplectic_Geometry_Seminar|Symplectic Geometry Seminar]]<br />
*[[Math843Seminar| Math 843 Homework Seminar]]<br />
*[[Graduate_student_reading_seminar|Graduate Probability Reading Seminar]]<br />
*[[Summer_stacks|Summer 2012 Stacks Reading Group]]<br />
*[[Graduate_Student_Singularity_Theory]]<br />
*[[Graduate/Postdoc Topology and Singularities Seminar]]<br />
*[[Shimura Varieties Reading Group]]<br />
*[[Summer graduate harmonic analysis seminar]]<br />
<br />
=== Other ===<br />
*[[Madison Math Circle]]<br />
*[[High School Math Night]]<br />
*[http://www.siam-uw.org/ UW-Madison SIAM Student Chapter]<br />
*[http://www.math.wisc.edu/%7Emathclub/ UW-Madison Math Club]<br />
*[[Putnam Club]]<br />
*[[Undergraduate Math Competition]]<br />
*[[Basic Linux Seminar]]<br />
*[[Basic HTML Seminar]]<br />
<br />
== Graduate Program ==<br />
<br />
* [[Algebra Qualifying Exam]]<br />
* [[Analysis Qualifying Exam]]<br />
* [[Topology Qualifying Exam]]<br />
<br />
== Undergraduate Program ==<br />
<br />
* [[Overview of the undergraduate math program|Overview]]<br />
* [[Groups looking to hire students as tutors]]<br />
<br />
== Getting started with Wiki-stuff ==<br />
<br />
Consult the [http://meta.wikimedia.org/wiki/Help:Contents User's Guide] for information on using the wiki software.<br />
* [http://www.mediawiki.org/wiki/Manual:Configuration_settings Configuration settings list]<br />
* [http://www.mediawiki.org/wiki/Manual:FAQ MediaWiki FAQ]<br />
* [http://lists.wikimedia.org/mailman/listinfo/mediawiki-announce MediaWiki release mailing list]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&diff=15042Algebra and Algebraic Geometry Seminar2018-02-06T22:13:43Z<p>Andreic: Redirected page to Algebra and Algebraic Geometry Seminar Spring 2018</p>
<hr />
<div>#REDIRECT [[Algebra and Algebraic Geometry Seminar Spring 2018]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebraic_Geometry_Seminar&diff=15041Algebraic Geometry Seminar2018-02-06T22:13:12Z<p>Andreic: Redirected page to Algebra and Algebraic Geometry Seminar Spring 2018</p>
<hr />
<div>#REDIRECT [[Algebra and Algebraic Geometry Seminar Spring 2018]]</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2018&diff=15040Algebra and Algebraic Geometry Seminar Spring 20182018-02-06T22:07:52Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B113.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2017 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|January 26<br />
|[http://homepages.math.uic.edu/~tmoulinos/ Tasos Moulinos (UIC)] <br />
|[[#Tasos Moulinos|Derived Azumaya Algebras and Twisted K-theory]]<br />
|Michael<br />
|-<br />
|February 2<br />
|Daniel Erman (Wisconsin) <br />
|[[#Daniel Erman|TBA]]<br />
|Local<br />
|-<br />
|'''February 8''' 2:30-3:30 in VV B113<br />
|[http://www.mathematics.pitt.edu/person/roman-fedorov/ Roman Fedorov (University of Pittsburgh)]<br />
|[[#Roman Fedorov|A conjecture of Grothendieck and Serre on principal bundles in mixed characteristic]]<br />
|Dima<br />
|-<br />
|February 9<br />
|Juliette Bruce (Wisconsin) <br />
|[[#Juliette Bruce|Asymptotic Syzygies in the Semi-Ample Setting ]]<br />
|Local<br />
|-<br />
|February 16<br />
|[http://www.math.wisc.edu/~andreic/ Andrei Caldararu (Wisconsin)]<br />
|[[#Andrei Caldararu|Computing a categorical Gromov-Witten invariant]]<br />
|Local<br />
|-<br />
|February 23<br />
|Aron Heleodoro (Northwestern) <br />
|[[#Aron Heleodoro|TBA]]<br />
|Dima<br />
|-<br />
|March 2<br />
|Moisés Herradón Cueto (Wisconsin)<br />
|[[#Moisés Herradón Cueto|TBA]]<br />
|Local<br />
|-<br />
|March 16<br />
|[https://math.berkeley.edu/~chenhi/ Harrison Chen (Berkeley)]<br />
|[[#Harrison Chen|TBA]]<br />
|Andrei<br />
|-<br />
|April 6<br />
|[http://www-personal.umich.edu/~ptoste/ Phil Tosteson (Michigan)]<br />
|[[#Phil Tosteson|TBA]]<br />
|Steven<br />
|-<br />
|-<br />
|April 13<br />
|Reserved<br />
|<br />
|Daniel<br />
|-<br />
|April 20<br />
|Alena Pirutka (NYU)<br />
|[[#Alena Pirutka|TBA]]<br />
|Jordan<br />
|-<br />
|April 27<br />
|Alexander Yom Din (Caltech) <br />
|[[#Alexander Yom Din|TBA]]<br />
|Dima<br />
|-<br />
|May 4<br />
|John Lesieutre (UIC) <br />
|[[#John Lesieutre|TBA]]<br />
|Daniel<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Tasos Moulinos===<br />
<br />
'''Derived Azumaya Algebras and Twisted K-theory'''<br />
<br />
Topological K-theory of dg-categories is a localizing invariant of dg-categories over <math> \mathbb{C} </math><br />
taking values in the <math> \infty </math>-category of <math> KU </math>-modules. In this talk I describe a relative version<br />
of this construction; namely for <math>X</math> a quasi-compact, quasi-separated <math> \mathbb{C} </math>-scheme I construct a<br />
functor valued in the <math> \infty </math>-category of sheaves of spectra on <math> X(\mathbb{C}) </math>, the complex points of <math>X</math>. For inputs<br />
of the form <math>\operatorname{Perf}(X, A)</math> where <math>A</math> is an Azumaya algebra over <math>X</math>, I characterize the values<br />
of this functor in terms of the twisted topological K-theory of <math> X(\mathbb{C}) </math>. From this I deduce<br />
a certain decomposition, for <math> X </math> a finite CW-complex equipped with a bundle <math> P </math> of projective<br />
spaces over <math> X </math>, of <math> KU(P) </math> in terms of the twisted topological K-theory of <math> X </math> ; this is<br />
a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer<br />
schemes.<br />
<br />
===Roman Fedorov===<br />
<br />
'''A conjecture of Grothendieck and Serre on principal bundles in mixed<br />
characteristic'''<br />
<br />
Let G be a reductive group scheme over a regular local ring R. An old<br />
conjecture of Grothendieck and Serre predicts that such a principal<br />
bundle is trivial, if it is trivial over the fraction field of R. The<br />
conjecture has recently been proved in the "geometric" case, that is,<br />
when R contains a field. In the remaining case, the difficulty comes<br />
from the fact, that the situation is more rigid, so that a certain<br />
general position argument does not go through. I will discuss this<br />
difficulty and a way to circumvent it to obtain some partial results.<br />
<br />
===Juliette Bruce===<br />
<br />
'''Asymptotic Syzygies in the Semi-Ample Setting'''<br />
<br />
In recent years numerous conjectures have been made describing the asymptotic Betti numbers of a projective variety as the embedding line bundle becomes more ample. I will discuss recent work attempting to generalize these conjectures to the case when the embedding line bundle becomes more semi-ample. (Recall a line bundle is semi-ample if a sufficiently large multiple is base point free.) In particular, I will discuss how the monomial methods of Ein, Erman, and Lazarsfeld used to prove non-vanishing results on projective space can be extended to prove non-vanishing results for products of projective space.<br />
<br />
===Andrei Caldararu===<br />
<br />
'''Computing a categorical Gromov-Witten invariant'''<br />
<br />
In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-inifinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical invariance of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other categories like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.<br />
<br />
In my talk I shall describe the details of the computation (joint with Junwu Tu) of the invariant, at g=1, n=1, for elliptic curves. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.<br />
<br />
===Aron Heleodoro===<br />
<br />
'''TBA'''<br />
<br />
===Alexander Yom Din===<br />
<br />
'''TBA'''</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar_Spring_2018&diff=14836Algebra and Algebraic Geometry Seminar Spring 20182018-01-24T16:17:39Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B113.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2017 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebra and Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|January 26<br />
|[http://homepages.math.uic.edu/~tmoulinos/ Tasos Moulinos (UIC)] <br />
|[[#Tasos Moulinos|Derived Azumaya Algebras and Twisted K-theory]]<br />
|Michael<br />
|-<br />
|February 2<br />
|Daniel Erman (Wisconsin) <br />
|[[#Daniel Erman|TBA]]<br />
|Local<br />
|-<br />
|February 8 (unusual date!)<br />
|Roman Fedorov (University of Pittsburgh)<br />
|[[#Roman Fedorov|TBA]]<br />
|Dima<br />
|-<br />
|February 9<br />
|Juliette Bruce (Wisconsin) <br />
|[[#Juliette Bruce|TBA]]<br />
|Local<br />
|-<br />
|February 16<br />
|[http://www.math.wisc.edu/~andreic/ Andrei Caldararu (Wisconsin)]<br />
|[[#Andrei Caldararu|Computing a categorical Gromov-Witten invariant]]<br />
|Local<br />
|-<br />
|February 23<br />
|Aron Heleodoro (Northwestern) <br />
|[[#Aron Heleodoro|TBA]]<br />
|Dima<br />
|-<br />
|April 6<br />
|[http://www-personal.umich.edu/~ptoste/ Phil Tosteson (Michigan)]<br />
|[[#Phil Tosteson|TBA]]<br />
|Steven<br />
|-<br />
|-<br />
|April 13<br />
|Reserved<br />
|<br />
|Daniel<br />
|-<br />
|April 20<br />
|Alena Pirutka (NYU)<br />
|[[#Alena Pirutka|TBA]]<br />
|Jordan<br />
|-<br />
|April 27<br />
|Alexander Yom Din (Caltech) <br />
|[[#Alexander Yom Din|TBA]]<br />
|Dima<br />
|-<br />
|May 4<br />
|John Lesieutre (UIC) <br />
|[[#John Lesieutre|TBA]]<br />
|Daniel<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Tasos Moulinos===<br />
<br />
'''Derived Azumaya Algebras and Twisted K-theory'''<br />
<br />
Topological K-theory of dg-categories is a localizing invariant of dg-categories over <math> \mathbb{C} </math><br />
taking values in the <math> \infty </math>-category of <math> KU </math>-modules. In this talk I describe a relative version<br />
of this construction; namely for <math>X</math> a quasi-compact, quasi-separated <math> \mathbb{C} </math>-scheme I construct a<br />
functor valued in the <math> \infty </math>-category of sheaves of spectra on <math> X(\mathbb{C}) </math>, the complex points of <math>X</math>. For inputs<br />
of the form <math>\operatorname{Perf}(X, A)</math> where <math>A</math> is an Azumaya algebra over <math>X</math>, I characterize the values<br />
of this functor in terms of the twisted topological K-theory of <math> X(\mathbb{C}) </math>. From this I deduce<br />
a certain decomposition, for <math> X </math> a finite CW-complex equipped with a bundle <math> P </math> of projective<br />
spaces over <math> X </math>, of <math> KU(P) </math> in terms of the twisted topological K-theory of <math> X </math> ; this is<br />
a topological analogue of a result of Quillen’s on the algebraic K-theory of Severi-Brauer<br />
schemes.<br />
<br />
===Andrei Caldararu===<br />
<br />
'''Computing a categorical Gromov-Witten invariant'''<br />
<br />
In his 2005 paper "The Gromov-Witten potential associated to a TCFT" Kevin Costello described a procedure for recovering an analogue of the Gromov-Witten potential directly out of a cyclic A-inifinity algebra or category. Applying his construction to the derived category of sheaves of a complex projective variety provides a definition of higher genus B-model Gromov-Witten invariants, independent of the BCOV formalism. This has several advantages. Due to the categorical invariance of these invariants, categorical mirror symmetry automatically implies classical mirror symmetry to all genera. Also, the construction can be applied to other categories like categories of matrix factorization, giving a direct definition of FJRW invariants, for example.<br />
<br />
In my talk I shall describe the details of the computation (joint with Junwu Tu) of the invariant, at g=1, n=1, for elliptic curves. The result agrees with the predictions of mirror symmetry, matching classical calculations of Dijkgraaf. It is the first non-trivial computation of a categorical Gromov-Witten invariant.<br />
<br />
===Aron Heleodoro===<br />
<br />
'''TBA'''<br />
<br />
===Alexander Yom Din===<br />
<br />
'''TBA'''</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebraic_Geometry_Seminar_Spring_2018&diff=14766Algebraic Geometry Seminar Spring 20182018-01-13T10:39:33Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in room B113.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2017 | the previous semester]].<br />
<!--, [[Algebraic Geometry Seminar Spring 2018 | the next semester]], and for [[Algebraic Geometry Seminar | this semester]]. --><br />
<br />
==Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Spring 2018 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|January 26<br />
|[http://homepages.math.uic.edu/~tmoulinos/ Tasos Moulinos (UIC)] <br />
|[[#Tasos Moulinos|TBA]]<br />
|Michael<br />
|-<br />
|February 23<br />
|Aron Heleodoro (Northwestern) <br />
|[[#Aron Heleodoro|TBA]]<br />
|Dima<br />
|-<br />
|April 20<br />
|Alena Pirutka (NYU)<br />
|[[#Alena Pirutka|TBA]]<br />
|Jordan<br />
|-<br />
|April 27<br />
|Alexander Yom Din (Caltech) <br />
|[[#Alexander Yom Din|TBA]]<br />
|Dima<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Aron Heleodoro===<br />
<br />
'''TBA'''<br />
<br />
===Alexander Yom Din===<br />
<br />
'''TBA'''</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=14653Math 567 -- Elementary Number Theory2017-12-05T17:14:22Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
TR 9:30-10:45, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Tuesdays 1:30-3:00, Van Vleck 605. <br />
<br />
'''Grader:''' Shouwei Hui (shui5@wisc.edu)<br />
''Office Hours:'' Wednesdays 3-4pm, Van Vleck 903.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Thursdays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on November 9, in class. The final exam will be on 12/21/2017, 12:25-2:25PM in Social Sciences building, room 6240.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Oct 2-6: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers <br />
* Oct 16-20: Quadratic reciprocity (4.1-4.4)<br />
* Oct 23-27: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 31, Nov 2, Nov 7: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov. 9: ''Midterm exam''<br />
* Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)<br />
* Dec. 4-8: Applications of elliptic curves (6.3-6.4)<br />
* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Thursday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Sep 19''' (note this is Tuesday, not Thursday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? --><br />
<br />
'''December 12''' Book problems 6.1, 6.2, 6.5, 6.10. For extra credit attempt to do problem 6.9.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=14610Math 567 -- Elementary Number Theory2017-11-29T15:55:12Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
TR 9:30-10:45, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Tuesdays 1:30-3:00, Van Vleck 605. <br />
<br />
'''Grader:''' Shouwei Hui (shui5@wisc.edu)<br />
''Office Hours:'' Wednesdays 3-4pm, Van Vleck 903.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Thursdays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on November 9, in class. The final exam date and location will be announced by the University and posted here when available.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Oct 2-6: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers <br />
* Oct 16-20: Quadratic reciprocity (4.1-4.4)<br />
* Oct 23-27: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 31, Nov 2, Nov 7: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov. 9: ''Midterm exam''<br />
* Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)<br />
* Dec. 4-8: Applications of elliptic curves (6.3-6.4)<br />
* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Thursday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Sep 19''' (note this is Tuesday, not Thursday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''December 7'''<br />
<br />
Problem A. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D.1. Pick two values of a in F_11 = Z/11Z (a not equal to 3), such that the equation y^2 = x^3+ax+1 defines an elliptic curve (i.e., it is smooth). For each such a, determine the number of points #E(F_11) and check that it falls inside the interval described in class. <br />
<br />
Problem D.2. The point P = (0,1) lies on each of these curves. For a =3 (the curve discussed in class) determine the order of P in the elliptic curve group, that is, find the smallest positive integer n such that nP = (infinity) -- the identity element in the group law.<br />
<br />
<!-- '''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? --></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=14513Math 567 -- Elementary Number Theory2017-11-07T22:55:12Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
TR 9:30-10:45, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Tuesdays 1:30-3:00, Van Vleck 605. <br />
<br />
'''Grader:''' Shouwei Hui (shui5@wisc.edu)<br />
''Office Hours:'' Wednesdays 3-4pm, Van Vleck 903.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Thursdays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on November 9, in class. The final exam date and location will be announced by the University and posted here when available.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Oct 2-6: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers <br />
* Oct 16-20: Quadratic reciprocity (4.1-4.4)<br />
* Oct 23-27: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 31, Nov 2, Nov 7: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov. 9: ''Midterm exam''<br />
* Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)<br />
* Dec. 4-8: Applications of elliptic curves (6.3-6.4)<br />
* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Thursday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Sep 19''' (note this is Tuesday, not Thursday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
<!--<br />
'''November 12'''<br />
<br />
Problem A. When p is a prime congruent to 1 mod 4, prove that ((p-1)/2)! is a square root of -1 in (Z/pZ), along the lines described on the midterm (or by some other means, if you prefer.)<br />
Problem B. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
Problem C. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
Problem D. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
Problem E. An ''ideal'' of Z[i] is a subset I of Z[i] which is closed under addition (if x and y are in I, then x+y is in I) and multiplication by Gaussian integers (if x is in i, then zx is also in I for every Gaussian integer z.) The set of multiples of a Gaussian integer is always an ideal (you don't need to prove this.) List all the ideals of Z[i] containing 2Z[i]. (Because such ideals satisfy conditions 1 and 2 from Monday's lecture, this list will be a subset of the list of 5 subgroups we computed in class.) <br />
<br />
We will not go any further with the notion of ideals in Math 567, but it is worth saying that the language of ideals is absolutely essential for the understanding of contemporary number theory, in a sense "making up for" the failure of unique factorization.<br />
<br />
'''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? --></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=14512Math 567 -- Elementary Number Theory2017-11-07T22:54:24Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
TR 9:30-10:45, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Tuesdays 1:30-3:00, Van Vleck 605. <br />
<br />
'''Grader:''' Shouwei Hui (shui5@wisc.edu)<br />
''Office Hours:'' Wednesdays 3-4pm, Van Vleck 903.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Thursdays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on November 9, in class. The final exam date and location will be announced by the University and posted here when available.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Oct 2-6: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers <br />
* Oct 16-20: Quadratic reciprocity (4.1-4.4)<br />
* Oct 23-27: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 31, Nov 2, Nov 7: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov. 9: ''Midterm exam''<br />
* Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)<br />
* Dec. 4-8: Applications of elliptic curves (6.3-6.4)<br />
* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Thursday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Sep 19''' (note this is Tuesday, not Thursday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
*'''November 16'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
Problem B. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
Problem C. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
Problem D. Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
<!--<br />
'''November 12'''<br />
<br />
Problem A. When p is a prime congruent to 1 mod 4, prove that ((p-1)/2)! is a square root of -1 in (Z/pZ), along the lines described on the midterm (or by some other means, if you prefer.)<br />
Problem B. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
Problem C. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
Problem D. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
Problem E. An ''ideal'' of Z[i] is a subset I of Z[i] which is closed under addition (if x and y are in I, then x+y is in I) and multiplication by Gaussian integers (if x is in i, then zx is also in I for every Gaussian integer z.) The set of multiples of a Gaussian integer is always an ideal (you don't need to prove this.) List all the ideals of Z[i] containing 2Z[i]. (Because such ideals satisfy conditions 1 and 2 from Monday's lecture, this list will be a subset of the list of 5 subgroups we computed in class.) <br />
<br />
We will not go any further with the notion of ideals in Math 567, but it is worth saying that the language of ideals is absolutely essential for the understanding of contemporary number theory, in a sense "making up for" the failure of unique factorization.<br />
<br />
'''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? --></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=14487Math 567 -- Elementary Number Theory2017-10-31T16:20:15Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
TR 9:30-10:45, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Tuesdays 1:30-3:00, Van Vleck 605. <br />
<br />
'''Grader:''' Shouwei Hui (shui5@wisc.edu)<br />
''Office Hours:'' Wednesdays 3-4pm, Van Vleck 903.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Thursdays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on November 9, in class. The final exam date and location will be announced by the University and posted here when available.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Oct 2-6: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers <br />
* Oct 16-20: Quadratic reciprocity (4.1-4.4)<br />
* Oct 23-27: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 31, Nov 2, Nov 7: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov. 9: ''Midterm exam''<br />
* Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)<br />
* Dec. 4-8: Applications of elliptic curves (6.3-6.4)<br />
* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Thursday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Sep 19''' (note this is Tuesday, not Thursday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are irreducible in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into irreducibles. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
<br />
*'''October 22'''<br />
<br />
Book problems: 4.3, 4.4, 4.6 (quite involved, counts as two problems) 4.9a<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
<br />
*'''November 5'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A. Let p = 58741. Use the method of section 4.5 (discussed in class) to find an r in (Z/pZ) with r^2 = -1.<br />
Problem B. Use the result of problem A, and the method of section 5.7, to find integers a,b such that a^2 + b^2 = p.<br />
Problem C. As discussed in class, Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''November 12'''<br />
<br />
Problem A. When p is a prime congruent to 1 mod 4, prove that ((p-1)/2)! is a square root of -1 in (Z/pZ), along the lines described on the midterm (or by some other means, if you prefer.)<br />
Problem B. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
Problem C. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
Problem D. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
Problem E. An ''ideal'' of Z[i] is a subset I of Z[i] which is closed under addition (if x and y are in I, then x+y is in I) and multiplication by Gaussian integers (if x is in i, then zx is also in I for every Gaussian integer z.) The set of multiples of a Gaussian integer is always an ideal (you don't need to prove this.) List all the ideals of Z[i] containing 2Z[i]. (Because such ideals satisfy conditions 1 and 2 from Monday's lecture, this list will be a subset of the list of 5 subgroups we computed in class.) <br />
<br />
We will not go any further with the notion of ideals in Math 567, but it is worth saying that the language of ideals is absolutely essential for the understanding of contemporary number theory, in a sense "making up for" the failure of unique factorization.<br />
<br />
'''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? --></div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Algebraic_Geometry_Seminar_Fall_2017&diff=14463Algebraic Geometry Seminar Fall 20172017-10-27T04:28:55Z<p>Andreic: </p>
<hr />
<div>The seminar meets on Fridays at 2:25 pm in Van Vleck B321.<br />
<br />
Here is the schedule for [[Algebraic Geometry Seminar Spring 2017 | the previous semester]].<br />
<!--and for [[Algebraic Geometry Seminar Spring 2017 | the next semester]].---><br />
<!-- and for [[Algebraic Geometry Seminar | this semester]].---><br />
<br />
==Algebraic Geometry Mailing List==<br />
<br />
<br />
*Please join the [https://admin.lists.wisc.edu/index.php?p=11&l=ags AGS Mailing List] to hear about upcoming seminars, lunches, and other algebraic geometry events in the department (it is possible you must be on a math department computer to use this link).<br />
<br />
== Fall 2017 Schedule ==<br />
<br />
{| cellpadding="8"<br />
!align="left" | date<br />
!align="left" | speaker<br />
!align="left" | title<br />
!align="left" | host(s) <br />
<br />
|-<br />
|October 6<br />
|[http://www.math.wisc.edu/~mkbrown5// Michael Brown (UW-Madison)] <br />
|[[#Michael Brown|Topological K-theory of equivariant singularity categories]]<br />
|local<br />
<br />
|-<br />
|October 9 (Monday!!, 3:30-4:30, B119)<br />
|[http://www.math.utah.edu/~bertram/ Aaron Bertram (Utah)] <br />
|[[#Aaron Bertram|LePotier's Strange Duality and Quot Schemes]]<br />
|Andrei<br />
<br />
|-<br />
|October 27<br />
|Mao Li (UW-Madison) <br />
|[[#Mao Li|Poincare sheaf on the stack of rank two Higgs bundles]]<br />
|local<br />
<br />
|-<br />
|November 3<br />
|Dario Beraldo (Oxford) <br />
|[[#Dario Beraldo|tba]]<br />
|Dima<br />
<br />
|-<br />
|December 1<br />
|Jonathan Wang (IAS)<br />
|<br />
|Dima<br />
<br />
|-<br />
|December 8<br />
|[https://www.math.brown.edu/~mtchan/ Melody Chan (Brown)] <br />
|[[#Melody Chan|tba]]<br />
|Jordan<br />
<br />
|-<br />
|December 15<br />
|[http://math.rice.edu/~jrc9/ John Calabrese (Rice)] <br />
|[[# John Calabrese|Towards the crepant resolution conjecture for Donaldson-Thomas invariants]]<br />
|Andrei<br />
|}<br />
<br />
== Abstracts ==<br />
<br />
===Michael Brown===<br />
<br />
'''Topological K-theory of equivariant singularity categories'''<br />
<br />
This is joint work with Tobias Dyckerhoff. Topological K-theory of complex-linear dg categories is a notion introduced by Blanc in his recent article "Topological K-theory of complex noncommutative spaces". In this talk, I will discuss a calculation of the topological K-theory of the dg category of graded matrix factorizations associated to a quasi-homogeneous polynomial with complex coefficients in terms of a classical topological invariant of a complex hypersurface singularity: the Milnor fiber and its monodromy.<br />
<br />
===Aaron Bertram===<br />
<br />
'''LePotier's Strange Duality and Quot Schemes'''<br />
<br />
Finite schemes of quotients over a smooth curve are a vehicle<br />
for proving strange duality for determinant line bundles on<br />
moduli spaces of vector bundles on Riemann surfaces. This was observed<br />
by Marian and Oprea. In work with Drew Johnson and Thomas Goller,<br />
we extend this idea to del Pezzo surfaces, where we are able to use it to prove cases of Le Potier's strange duality conjecture.<br />
<br />
===Mao Li===<br />
<br />
'''Poincare sheaf on the stack of rank two Higgs bundles'''<br />
<br />
It is well known that for a smooth projective curve, there exists a Poincare line bundle on the product of the Jacobian of the curve which is the universal family of topologically trivial line bundles of the Jacobian. It is natural to ask whether similar results still hold for the compactified Jacobian of singular curves. There has been a lot of work in this problem. In this talk I will sketch the construction of the Poincare sheaf for spectral curves in Hitchin fibration. The new feature is that we are able to extend the construction of Poincare sheaf to nonreduced spectral curves.</div>Andreichttps://www.math.wisc.edu/wiki/index.php?title=Math_567_--_Elementary_Number_Theory&diff=14444Math 567 -- Elementary Number Theory2017-10-25T15:52:12Z<p>Andreic: </p>
<hr />
<div>'''MATH 567<br />
<br />
Elementary Number Theory'''<br />
<br />
TR 9:30-10:45, Van Vleck B119<br />
<br />
'''Professor:''' [http://www.math.wisc.edu/~andreic/ Andrei Caldararu] (andreic@math.wisc.edu)<br />
''Office Hours:'' Tuesdays 1:30-3:00, Van Vleck 605. <br />
<br />
'''Grader:''' Shouwei Hui (shui5@wisc.edu)<br />
''Office Hours:'' Wednesdays 3-4pm, Van Vleck 903.<br />
<br />
Math 567 is a course in elementary number theory, aimed at undergraduates majoring in math or other quantitative disciplines. A general familiarity with abstract algebra at the level of Math 541 will be assumed, but students who haven't taken 541 are welcome to attend if they're willing to play a little catchup. We will be using William Stein's new (and cheap) textbook [http://www.amazon.com/Elementary-Number-Theory-Computational-Undergraduate/dp/0387855246 Elementary Number Theory: Primes, Congruences, and Secrets], which emphasizes computational approaches to the subject. If you don't need a physical copy of the book, [http://www.williamstein.org/ent/ it is available as a free legal .pdf.] We will be using the (free, public-domain) mathematical software [http://www.sagemath.org/ SAGE], developed largely by Stein, as an integral component of our coursework. There is a [http://doc.sagemath.org/pdf/en/tutorial/SageTutorial.pdf useful online tutorial.] You can download SAGE to your own computer or [http://www.sagenb.org use it online].<br />
<br />
Topics include some subset of, but are not limited to: Divisibility, the Euclidean algorithm and the GCD, linear Diophantine equations, prime numbers and uniqueness of factorization. Congruences, Chinese remainder theorem, Fermat's "little" theorem, Wilson's theorem, Euler's theorem and totient function, the RSA cryptosystem, Rabin's encryption scheme, Diffie-Hellman key exchange protocol. Number-theoretic functions, multiplicative functions, Möbius inversion. Primitive roots and indices. Quadratic reciprocity and the Legendre symbol. Perfect numbers, Mersenne primes, Fermat primes. Pythagorean triples, Special cases of Fermat's "last" theorem. Fibonacci numbers. Continued fractions. Distribution of primes, discussion of prime number theorem. Primality testing and factoring algorithms.<br />
<br />
'''Course Policies:''' Homework will be due on Thursdays. It can be turned in late only with ''advance'' permission from your grader. It is acceptable to use calculators and computers on homework (indeed, some of it will require a computer) but calculators are not allowed during exams. You are encouraged to work together on homework, but writeups must be done individually.<br />
<br />
Many of the problems in this course will ask you to prove things. I expect proofs to be written in English sentences; the proofs in Stein's book are a good model for the level of verbosity I am looking for. <br />
<br />
'''Grading:''' The grade in Math 567 will be composed of 50% homework, 25% midterm, 25% final. The midterm will be on November 9, in class. The final exam date and location will be announced by the University and posted here when available.<br />
<br />
'''Syllabus:''' <br />
(This may change as we see what pace works well for the course. All section numbers refer to Stein's book.)<br />
<br />
* Sep 7 + Sep 11-15: Prime numbers, prime factorizations, Euclidean algorithm and GCD (1.1-1.2)<br />
* Sep 18-22: The integers mod n, Euler's theorem, the phi function (2.1-2.2)<br />
* Sep 25-29: Modular exponentiation, primality testing, and primitive roots (2.4-2.5)<br />
* Oct 2-6: Public-key cryptography and RSA (3.1-3.4)<br />
* Oct 9-13: Rabin's algorithm (not in the book); algebraic numbers <br />
* Oct 16-20: Quadratic reciprocity (4.1-4.4)<br />
* Oct 23-27: Finite and infinite continued fractions (5.1-5.3)<br />
* Oct 31, Nov 2, Nov 7: Continued fractions and diophantine approximation (5.4-5.5)<br />
* Nov. 9: ''Midterm exam''<br />
* Nov 13-17: Diophantine equations I: Pell's equation and Lagrange's theorem<br />
* Nov 21 and Nov 28-30: Elliptic curves (6.1-6.2)<br />
* Dec. 4-8: Applications of elliptic curves (6.3-6.4)<br />
* Dec. 12: advanced topic TBD: maybe additional discussion of cryptographic techniques?<br />
<br />
'''Homework:'''<br />
Homework is due at the beginning of class on the specified Thursday. Typing your homework is not a requirement, but if you don't already know LaTeX I highly recommend that you learn it and use it to typeset your homework. I will sometimes assign extra problems, which I will e-mail to the class list and include here.<br />
<br />
* '''Sep 19''' (note this is Tuesday, not Thursday!): 1.1, 1.3, 1.5, 1.7 (use SAGE), 1.8, 1.14.<br />
Problem A: Use SAGE to compute the number of x in [1..N] such that x^2 + 1 is prime, for N = 100, N = 1000, and N = 10000. Let f(N) be the number of such x. <br />
<br />
a) Can you formulate a conjecture about the relationship between f(N) and N/log N? <br />
<br />
b) What if x^2 + 1 is replaced with x^2 + 2? Can you explain why x^2 + 2 appears less likely to be prime? (Hint: consider x mod 3.)<br />
<br />
c) Prove that f(N) is at most (1/2)N+1. (Hint: consider x mod 2.) <br />
<br />
d) Give as good an upper bound as you can for f(N).<br />
<br />
Note that, despite the evident regularities you'll observe in this problem, we do not even know whether there are infinitely many primes of the form x^2 + 1! You would become very famous if you proved this.<br />
<br />
* '''Sep 28''': 2.6 (the formulation of numerical evidence should be done by Sage if you've got Sage working, and by calculator if not; you can use an online tool like [http://primes.utm.edu/curios/includes/primetest.php this] to test whether a number is prime.) 2.8,2.9,2.11,2.12,2.14,2.19<br />
<br />
* '''Oct 5''': 2.15, 2.18, 2.23, 2.26, 2.30.<br />
<br />
Problem A: Prove that if n=pq, with p,q prime, then n is not a Carmichael number.<br />
<br />
* '''Oct 12''': Book problems: 3.4, 3.5, 3.6<br />
<br />
Problem A. Prove that there are infinitely many primes p such that 2 is '''not''' a primitive root in Z/pZ. We break this up into steps.<br />
<br />
Problem A.1. Prove that, if x is an element of Z/nZ, then x^2 is not a primitive root.<br />
<br />
Problem A.2. Prove that there are infinitely many primes p such that 2 is a square in Z/pZ. Hint: suppose there are only finitely many such primes p_1, .. p_r, and define N = (p_1 .. p_r)^2 - 2. Where can you go from here...?<br />
<br />
Problem A.3. Give a list of five primes p such that 2 is not a primitive root in Z/pZ (you can use the method of this proof or any other.)<br />
<br />
Problem B. Prove that 24 is the largest integer n such that every element of (Z/nZ)^* is a root of x^2-1.<br />
<br />
* '''Oct 19''': Problem A. Using p = 23 and q=31, show how to encrypt the message x=240 with Rabin's algorithm. Find all possible decryptions of your encrypted message.<br />
<br />
Problem B. Give a prime factorization of the Gaussian integer 7+9i.<br />
<br />
Problem C. Read the notes from [http://www.math.ucsd.edu/~jverstra/Gaussian1.pdf here]. Note that Z[i] satisfies a ''reduction theorem'': if n and d are Gaussian integers, then there exists integers q and r such that n = qd + r and Norm(r) < Norm(d). But (by contrast with the case of Z) this d may not be unique. In some contexts it is better to be able to choose r uniquely, even if this means letting r have norm greater than Norm(d). <br />
<br />
C.1. When d = 1+2i, show that, for each n in Z[i], there is a '''unique''' pair (q,r) in Z[i] such that n = qd+r and r is contained in the set {0,1,2,3,4}. For instance, i can be written as i(1+2i) + 2, so we say i reduces to 2 mod (1+2i).<br />
(Hint: Suppose q(1+2i)+r = q'(1+2i) + r'. What can we say about (r-r'), and why is this incompatible with both r and r' being in {0,1,2,3,4}?<br />
<br />
C.2. Show that if n is an integer in Z, the reduction of n mod (1+2i) is equal to its reduction mod 5.<br />
<br />
Problem D. Let's try to figure out how to define "phi(d)" for a Gaussian integer d. Suppose S is a set of Gaussian integers such that every n in Z[i] can be written uniquely as qd+r, with q in Z[i] and r in S. (So for instance when d=1+2i, we showed in problem B that we can take S to be {0...4}. It would also be OK to take S to be {1..5} or {0,i,2i,1+i,1+2i}. In fact, it turns out that S has to be a set of size Norm(d) (I might or might not prove this in class; if not, feel free just to accept it.)<br />
<br />
Now define phi(d) to be the number of elements s of S such that s and d are coprime.<br />
<br />
D.1. Compute phi(1+2i) and phi(3).<br />
<br />
D.2. Prove that the value of phi(d) does not depend on the choice of S.<br />
<br />
D.3. Prove that every n in Z[i] which is coprime to 3 satisfies n^phi(3) = 1 mod 3; that is, Euler's theorem holds in this case. (You can prove this by direct computation; of course, if you want, you are welcome to prove that Euler's theorem holds for Z[i] in general, adapting the proof in Stein's book or the one we gave in class.)<br />
<br />
*'''Nov 2:''' 4.1 from the book.<br />
<br />
Problem A. Express 50005 as the sum of two squares.<br />
<br />
In the next two problems we denote by r(n) the number of ways to express n as the sum of two squares (i.e. the number of pairs (a,b) such that a^2 + b^2 = n.) For instance, r(5) = 8 (as shown on the midterm.)<br />
<br />
Problem B. Prove that, for any N, there exists an integer n such that r(n) > N. (I.E., the function r(n) is ''unbounded.''<br />
<br />
Problem C. If you like Sage, write a short program in Sage to compute r(n) and compute the average of r(n) as n ranges from 1 to 1000. Whether or not you like Sage, make a guess as to how this average would behave if you replaced "1000" by a larger and larger number. (Feel free to ask the Sage-lovers what answer they got in the optional first part of the question.) Can you prove this guess is correct?<br />
<br />
Problem D. We saw in class that the ring Z[sqrt(-5)] doesn't have unique factorization; 6 can be factored as 2*3 or (1+sqrt(-5))(1-sqrt(-5)). In this problem, we will prove that Z[sqrt(-d)] fails to have unique factorization for EVERY odd d >= 5. (Actually it's true for all d >= 5 but to make the proof manageable we'll restrict to the odd case.]<br />
<br />
D.1. Show that (1+sqrt(-d)), (1-sqrt(-d)) and 2 are primes in Z[sqrt(-d)].<br />
<br />
D.2. Now give an element of Z[sqrt(-d)] that has two distinct factorizations into primes. (Hint: imitate the example we used for Z[sqrt(-5)].)<br />
''Remark:'' The rings Z[sqrt(d)], where d is positive, are quite different -- here we believe that there are infinitely many with unique factorization, though this conjecture has remained unproved for many decades!<br />
<br />
<!--<br />
<br />
*'''October 22'''<br />
<br />
Book problems: 4.3, 4.4, 4.6 (quite involved, counts as two problems) 4.9a<br />
<br />
Problem A: We discussed in class the problem of studying which positive integers are the sum of two squares. In this problem we prove that every element n of (Z/pZ) is the sum of two squares. We argue as follows. Let S be the set of squares in (Z/pZ), and let T be the set of (Z/pZ) consisting of all elements of the form n - x^2, for some x in (Z/pZ).<br />
<br />
A.1. What is the size of S and of T? Use this to show that S and T are not disjoint.<br />
A.2. Given that S and T are not disjoint, prove that n is the sum of two squares.<br />
<br />
<br />
*'''November 5'''<br />
<br />
Book problems: 5.3,5.4,5.5<br />
<br />
Problem A. Let p = 58741. Use the method of section 4.5 (discussed in class) to find an r in (Z/pZ) with r^2 = -1.<br />
Problem B. Use the result of problem A, and the method of section 5.7, to find integers a,b such that a^2 + b^2 = p.<br />
Problem C. As discussed in class, Stuffy Stirnweiss finished the 1945 season with a batting average of .3085443. Using continued fractions, guess how many at-bats he had. Tony Cuccinello had a batting average of .3084577. Given that he had more than 200 and fewer than 600 at-bats, can you estimate the number of at-bats he had?<br />
<br />
'''November 12'''<br />
<br />
Problem A. When p is a prime congruent to 1 mod 4, prove that ((p-1)/2)! is a square root of -1 in (Z/pZ), along the lines described on the midterm (or by some other means, if you prefer.)<br />
Problem B. When p is a prime congruent to 3 mod 4, prove that ((p-1)/2)! is either 1 or -1 in (Z/pZ). '''OPTIONAL:''' Use sage to compute whether this factorial is 1 or -1 for many primes p. Is there any pattern? Does it seem to be 1 half the time and -1 half the time? Note that I have no idea what the answer to this question is.<br />
Problem C. Using the continued fraction expansion, find a solution to the Pell equation x^2 - 13 y^2 = 1.<br />
Problem D. Show that the modified Pell equation x^2 - 7y^2 = -1 has no solutions in integers x,y. (Hint: reduce the equation modulo a suitably chosen prime.)<br />
Problem E. An ''ideal'' of Z[i] is a subset I of Z[i] which is closed under addition (if x and y are in I, then x+y is in I) and multiplication by Gaussian integers (if x is in i, then zx is also in I for every Gaussian integer z.) The set of multiples of a Gaussian integer is always an ideal (you don't need to prove this.) List all the ideals of Z[i] containing 2Z[i]. (Because such ideals satisfy conditions 1 and 2 from Monday's lecture, this list will be a subset of the list of 5 subgroups we computed in class.) <br />
<br />
We will not go any further with the notion of ideals in Math 567, but it is worth saying that the language of ideals is absolutely essential for the understanding of contemporary number theory, in a sense "making up for" the failure of unique factorization.<br />
<br />
'''November 19'''<br />
<br />
Problem A. Using the method discussed in class (which is also the method of problem 4.6 in Stein, which in retrospect I think was too hard to assign with no preparation) find a nontrivial solution with y > 0 to the equation x^2 + 11y^2 = z^2. <br />
Problem B. Let f(X) be the number of solutions of A^2 + B^2 = C^3 such that A^2, B^2, and C^3 are all at most X. Use the heuristic described in class to explain why one might expect f(X) to grow more or less like X^{1/3}.<br />
<br />
Problem C. Show that if A+Bi is the cube of a Gaussian integer, then A^2 + B^2 is a perfect cube.<br />
<br />
Do ONE of problem D1 and D2; D1 is for people who like Sage, D2 is for people who like proving things.<br />
<br />
D1. Use Sage to compute f(X) for X = 1000, 10000, 100000. Does the answer look consistent with the heuristic prediction you made in B? Does f(X)/X^{1/3} appear to be approaching a limit?<br />
D2. We will use problem C to give a lower bound for f(X). Use this to show that there are at least X^{1/3} Gaussian integers with norm at most X that are perfect cubes. From here, show that f(X) > X^{1/3}.<br />
<br />
OPTIONAL: Show that the converse of problem C also holds -- A+Bi has norm a perfect cube if and only if A+Bi is a perfect cube. (You will need unique factorization of Gaussian integers.) Using this fact, prove that f(X) is bounded above by C X^{1/3} for some constant C.<br />
<br />
'''December 8 (note nonstandard due date)'''<br />
<br />
Problem A: Hyperbolas, ellipses, and "magic slopes"<br />
<br />
In the usual analytic geometry, both hyperbolas and ellipses are given by equations of the form<br />
<br />
ax^2 + bxy + cy^2 = d (*)<br />
<br />
<br />
(we always assume d is nonzero.)<br />
<br />
How can we tell whether such an equation describes an ellipse or a<br />
hyperbola? Well, in the geometric setting we all know and love (i.e. in<br />
R^2) a hyperbola has asymptotes and an ellipse does not. What is an <br />
asymptote? You could say it's "a line which doesn't intersect the curve", <br />
but ellipses have such lines too. You might want to say "a line which <br />
doesn't intersect the curve but comes closer and closer to the curve," but <br />
the problem here is that this a) is somewhat imprecise, and b) relies on a <br />
notion of "closer" that is not going to be very clear in Z/pZ, which is <br />
our ultimate goal!<br />
<br />
<br />
So let me state it a different way: let's say the asymptotes of a <br />
hyperbola have slopes m_1 and m_2. These are "magic slopes" in the <br />
following sense: ANY line of slope m_1 (and ditto for m_2) intersects the <br />
hyperbola in at most one point. An ellipse doesn't have "magic slopes" -- <br />
you can convince yourself by drawing pictures that, for any slope m, you <br />
can find a line of slope m striking the ellipse twice.<br />
<br />
<br />
[REMARK: For technical reasons we're going to ignore hyperbolae with a <br />
vertical asymptote, since vertical lines don't exactly have slope.]<br />
<br />
<br />
Now you might know the criterion that, over the real numbers.<br />
<br />
<br />
ax^2 + bxy + cy^2 = d<br />
<br />
<br />
is a hyperbola if b^2 - 4ac > 0 and an ellipse if b^2 - 4ac < 0. (For <br />
instance, x^2 - y^2 = 1 is a hyperbola with magic slopes 1 and -1, and x^2 <br />
+ y^2 = 1 is an ellipse.) <br />
This criterion does not make sense in Z/pZ, where there is no notion of <br />
"greater" or "less."<br />
<br />
<br />
QUESTION A.1.<br />
<br />
For which primes p is it the case that <br />
<br />
x^2 + y^2 = 1<br />
<br />
has magic slopes in Z/pZ? <br />
<br />
QUESTION A.2. For which primes p is it the case that<br />
<br />
<br />
x^2 - y^2 = 1<br />
<br />
<br />
has magic slopes in Z/pZ?<br />
<br />
<br />
QUESTION B: Back in the very first week of this course we studied the set of integers x such that x^2 + 1 is prime. Based on the heuristics we discussed in class, about how many x between 1 and N would you expect to satisfy "x^2 + 1 is prime?"<br />
<br />
<br />
QUESTION C: Let z be a complex number. Then the set of all complex numbers of the form a + bz, with a and b integers, forms a lattice in the complex plane (thought of as R^2.) What is the covolume of this lattice, in terms of z?<br />
<br />
<br />
QUESTION D: Let z be a complex number. Using Minkowski's theorem and the result of question C, show that there exist integers a,b, not both zero, such that Norm(a+bz) is at most (4/pi) Im(z). (Hint: let Omega be the set of all complex numbers of norm at most (4/pi) Im(z).)<br />
<br />
<br />
OPTIONAL EXTRA: Can you give an exact formula for the minimal norm of any nonzero complex number of the form a+bz? --></div>Andreic