https://hilbert.math.wisc.edu/wiki/api.php?action=feedcontributions&user=Ogoff&feedformat=atomUW-Math Wiki - User contributions [en]2021-03-06T11:47:31ZUser contributionsMediaWiki 1.30.1https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar&diff=20127Graduate Algebraic Geometry Seminar2020-10-13T18:28:46Z<p>Ogoff: </p>
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<div>'''<br />
'''When:''' Wednesdays 4:00pm<br />
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'''Where:''' https://uwmadison.zoom.us/j/92877740706?pwd=OVo0QmxRVEdUQ3RnUWpoWmFRRUI3dz09<br />
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]<br />
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'''Who:''' All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.<br />
<br />
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Talks are typically accessible to beginning graduate students and take many different forms. Sometimes people present an interesting paper they find. Other times people give a prep talk for the Friday Algebraic Geometry Seminar. Other times people give a series of talks on a topic they have been studying in-depth. Regardless the goal of GAGS is to provide a supportive and inclusive place for all to learn more about algebraic geometry and commutative algebra.<br />
<br />
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].<br />
'''<br />
<br />
== Give a talk! ==<br />
We need volunteers to give talks this semester. If you're interested contact [mailto:cwcrowley@wisc.edu Colin] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.<br />
<br />
== Being an audience member ==<br />
The goal of GAGS is to create a safe and comfortable space inclusive of all who wish to expand their knowledge of algebraic geometry and commutative algebra. In order to promote such an environment in addition to the standard expectations of respect/kindness all participants are asked to following the following guidelines:<br />
* Do Not Speak For/Over the Speaker: <br />
* Ask Questions Appropriately: <br />
<br />
== Fall 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#E0E0E0"| September 30<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 30| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 5<br />
| bgcolor="#C6D46E"| Yifan Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 5| On the Analytic Side (GAGA)]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 14<br />
| bgcolor="#C6D46E"| Owen Goff<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 14| The Magic and Comagic of Hopf Algebras]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 21<br />
| bgcolor="#C6D46E"| Roufan Jiang<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 21| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| October 28<br />
| bgcolor="#C6D46E"| Erika Pirnes<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 28| Introduction to representation theory via an example]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 4<br />
| bgcolor="#C6D46E"| Alex Mine<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 4| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 11<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 11| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November 18<br />
| bgcolor="#C6D46E"| Yunfan He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 18| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| November November 25<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 25| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 2<br />
| bgcolor="#C6D46E"| Peter Wei<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 2| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 9<br />
| bgcolor="#C6D46E"| Wendy Cheng<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 9| TBD]]<br />
|-<br />
| bgcolor="#E0E0E0"| December 16<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 16| TBD]]<br />
|}<br />
</center><br />
<br />
== Spring 2020 ==<br />
<br />
<center><br />
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"<br />
|-<br />
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''<br />
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''<br />
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''<br />
|-<br />
| bgcolor="#E0E0E0"| January 29<br />
| bgcolor="#C6D46E"| Colin Crowley<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 5<br />
| bgcolor="#C6D46E"| Asvin Gothandaraman<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 12<br />
| bgcolor="#C6D46E"| Qiao He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 19<br />
| bgcolor="#C6D46E"| Dima Arinkin<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]<br />
|-<br />
| bgcolor="#E0E0E0"| February 26<br />
| bgcolor="#C6D46E"| Connor Simpson<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 4<br />
| bgcolor="#C6D46E"| Peter<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 11<br />
| bgcolor="#C6D46E"| Caitlyn Booms<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| March 25<br />
| bgcolor="#C6D46E"| Steven He<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 1<br />
| bgcolor="#C6D46E"| Vlad Sotirov<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 8<br />
| bgcolor="#C6D46E"| Maya Banks<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 15<br />
| bgcolor="#C6D46E"| Alex Hof<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 22<br />
| bgcolor="#C6D46E"| Ruofan<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]<br />
|-<br />
| bgcolor="#E0E0E0"| April 29<br />
| bgcolor="#C6D46E"| John Cobb<br />
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]<br />
|}<br />
</center><br />
<br />
== January 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.<br />
|} <br />
</center><br />
<br />
== February 5 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to unirationality<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected. <br />
|} <br />
</center><br />
<br />
== February 12 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== February 19 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Blowing down, blowing up: surface geometry<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?<br />
<br />
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the<br />
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.<br />
<br />
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case<br />
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.) <br />
|} <br />
</center><br />
<br />
== February 26 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Toric Varieties<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.<br />
|} <br />
</center><br />
<br />
== March 4 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.<br />
|} <br />
</center><br />
<br />
== March 11 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Intro to Stanley-Reisner Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.<br />
|} <br />
</center><br />
<br />
== March 25 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 1 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 8 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
== April 15 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Early on in the semester, Colin told us a bit about Morse<br />
Theory, and how it lets us get a handle on the (classical) topology of<br />
smooth complex varieties. As we all know, however, not everything in<br />
life goes smoothly, and so too in algebraic geometry. Singular<br />
varieties, when given the classical topology, are not manifolds, but<br />
they can be described in terms of manifolds by means of something called<br />
a Whitney stratification. This allows us to develop a version of Morse<br />
Theory that applies to singular spaces (and also, with a bit of work, to<br />
smooth spaces that fail to be nice in other ways, like non-compact<br />
manifolds!), called Stratified Morse Theory. After going through the<br />
appropriate definitions and briefly reviewing the results of classical<br />
Morse Theory, we'll discuss the so-called Main Theorem of Stratified<br />
Morse Theory and survey some of its consequences.<br />
|} <br />
</center><br />
<br />
== April 22 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: Birational geometry: existence of rational curves<br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight. <br />
|} <br />
</center><br />
<br />
== April 29 ==<br />
<center><br />
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"<br />
|-<br />
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''<br />
|-<br />
| bgcolor="#BCD2EE" align="center" | Title: <br />
|-<br />
| bgcolor="#BCD2EE" | Abstract: <br />
|} <br />
</center><br />
<br />
<br />
== Organizers' Contact Info ==<br />
<br />
[https://sites.google.com/view/colincrowley/home Colin Crowley]<br />
<br />
[http://www.math.wisc.edu/~drwagner/ David Wagner]<br />
<br />
==The List of Topics that we Made February 2018==<br />
<br />
On February 21st of the Month of February of The 2018th Year of the Seventh Age of The Sun, the People Present at GAGS Compiled Ye Followinge Liste of Topics They Wished to Hear Aboute:<br />
<br />
Feel free to edit the list and/or add references to learn this stuff from. Since then, we've succeeded in talking about some of these, which doesn't mean there shouldn't be another talk. Ask around or look at old semester's websites.<br />
<br />
* Schubert Calculus, aka how many lines intersect four given lines in three-dimensional space? The answer to this question is prettiest when you think about it as a problem of intersecting subvarieties in the Grassmanian. ''What is the Grassmanian, you say?'' That's probably a talk we should have every year, so you should give it!<br />
<br />
* Kindergarten GAGA. GAGA stands for Algebraic Geometry - Analytic Geometry. Serre wrote a famous paper explaining how the two are related, and you could give an exposition suitable to kindergardeners.<br />
<br />
* Katz and Mazur explanation of what a modular form is. What is it?<br />
<br />
* Kindergarten moduli of curves.<br />
<br />
* What is a dualizing sheaf? What is a dualizing complex? What is Serre duality? What is local duality? Can local duality help us understand Serre duality?<br />
<br />
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)<br />
<br />
* Hodge theory for babies<br />
<br />
* What is a Néron model?<br />
<br />
* What is a crystal? What does it have to do with D-modules? [http://www.math.harvard.edu/~gaitsgde/grad_2009/SeminarNotes/Nov17-19(Crystals).pdf Here's an encouragingly short set of notes on it].<br />
<br />
* What and why is a dessin d'enfants?<br />
<br />
* DG Schemes.<br />
<br />
==Ed Dewey's Wish List Of Olde==__NOTOC__<br />
<br />
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.<br />
<br />
Here are the topics we're '''DYING''' to learn about! Please consider looking into one of these topics and giving one or two GAGS talks.<br />
<br />
===Specifically Vague Topics===<br />
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.<br />
<br />
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)<br />
<br />
===Interesting Papers & Books===<br />
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.<br />
<br />
* ''Residues and Duality'' - Robin Hatshorne.<br />
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)<br />
<br />
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.<br />
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)<br />
<br />
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.<br />
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off! <br />
<br />
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.<br />
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!<br />
<br />
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.<br />
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.<br />
<br />
* ''Esquisse d’une programme'' - Alexander Grothendieck.<br />
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.<br />
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)<br />
<br />
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.<br />
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.<br />
<br />
* ''Picard Groups of Moduli Problems'' - David Mumford.<br />
** This paper is essentially the origin of algebraic stacks.<br />
<br />
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar<br />
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties. <br />
<br />
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.<br />
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.<br />
<br />
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.<br />
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.<br />
<br />
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.<br />
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)<br />
<br />
<br />
== Past Semesters ==<br />
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]<br />
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[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]<br />
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[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]<br />
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[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]<br />
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[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2017 Fall 2017]<br />
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[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2017 Spring 2017]<br />
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[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2016 Fall 2016]<br />
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[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2016 Spring 2016]<br />
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[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_(Fall_2015) Fall 2015]</div>Ogoffhttps://hilbert.math.wisc.edu/wiki/index.php?title=AMS_Student_Chapter_Seminar&diff=19044AMS Student Chapter Seminar2020-02-17T18:31:36Z<p>Ogoff: </p>
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<div>The AMS Student Chapter Seminar (aka Donut Seminar) is an informal, graduate student seminar on a wide range of mathematical topics. Pastries (usually donuts) will be provided.<br />
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* '''When:''' Wednesdays, 3:20 PM – 3:50 PM<br />
* '''Where:''' Van Vleck, 9th floor lounge (unless otherwise announced)<br />
* '''Organizers:''' [https://www.math.wisc.edu/~malexis/ Michel Alexis], [https://www.math.wisc.edu/~drwagner/ David Wagner], [http://www.math.wisc.edu/~nicodemus/ Patrick Nicodemus], [http://www.math.wisc.edu/~thaison/ Son Tu], Carrie Chen<br />
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Everyone is welcome to give a talk. To sign up, please contact one of the organizers with a title and abstract. Talks are 25 minutes long and should avoid assuming significant mathematical background beyond first-year graduate courses.<br />
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The schedule of talks from past semesters can be found [[AMS Student Chapter Seminar, previous semesters|here]].<br />
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== Spring 2020 ==<br />
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=== February 5, Alex Mine===<br />
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Title: Khinchin's Constant<br />
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Abstract: I'll talk about a really weird fact about continued fractions.<br />
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=== February 12, Xiao Shen===<br />
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Title: Coalescence estimates for the corner growth model with exponential weights<br />
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Abstract: (Joint with Timo Seppalainen) I will talk about estimates for the coalescence time of semi-infinite directed geodesics in the planar corner growth model. Not much probability background is needed.<br />
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=== February 19, Hyun Jong Kim===<br />
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Title: Orbifolds for Music<br />
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Abstract: TBD<br />
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=== February 26, Solly Parenti===<br />
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Title: TBD<br />
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Abstract: TBD<br />
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=== March 4, ===<br />
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=== March 11, Ivan Aidun===<br />
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Title: TBD<br />
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Abstract: TBD<br />
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=== March 24 - Visit Day===<br />
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==== Brandon Boggess, Time TBD====<br />
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Title: TBD<br />
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Abstract: TBD<br />
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==== Yandi Wu, Time TBD====<br />
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=== April 1, Ying Li===<br />
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Title: TBD<br />
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Abstract: TBD<br />
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=== April 8, TBD===<br />
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Title: TBD<br />
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=== April 15, Owen Goff===<br />
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Title: TBD<br />
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Abstract: TBD<br />
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=== April 22, TBD===<br />
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Title: TBD<br />
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== Fall 2019 ==<br />
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=== October 9, Brandon Boggess===<br />
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Title: An Application of Elliptic Curves to the Theory of Internet Memes<br />
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Abstract: Solve polynomial equations with this one weird trick! Math teachers hate him!!!<br />
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[[File:Thumbnail fruit meme.png]]<br />
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=== October 16, Jiaxin Jin===<br />
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Title: Persistence and global stability for biochemical reaction-diffusion systems<br />
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Abstract: The investigation of the dynamics of solutions of nonlinear reaction-diffusion PDE systems generated by biochemical networks is a great challenge; in general, even the existence of classical solutions is difficult to establish. On the other hand, these kinds of problems appear very often in biological applications, e.g., when trying to understand the role of spatial inhomogeneities in living cells. We discuss the persistence and global stability properties of special classes of such systems, under additional assumptions such as: low number of species, complex balance or weak reversibility.<br />
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=== October 23, Erika Pirnes===<br />
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(special edition: carrot seminar)<br />
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Title: Why do ice hockey players fall in love with mathematicians? (Behavior of certain number string sequences)<br />
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Abstract: Starting with some string of digits 0-9, add the adjacent numbers pairwise to obtain a new string. Whenever the sum is 10 or greater, separate its digits. For example, 26621 would become 81283 and then 931011. Repeating this process with different inputs gives varying behavior. In some cases the process terminates (becomes a single digit), or ends up in a loop, like 999, 1818, 999... The length of the strings can also start growing very fast. I'll discuss some data and conjectures about classifying the behavior.<br />
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=== October 30, Yunbai Cao===<br />
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Title: Kinetic theory in bounded domains<br />
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Abstract: In 1900, David Hilbert outlined 23 important problems in the International Congress of Mathematics. One of them is the Hilbert's sixth problem which asks the mathematical linkage between the mechanics from microscopic view and the macroscopic view. A relative new mesoscopic point of view at that time which is "kinetic theory" was highlighted by Hilbert as the bridge to link the two. In this talk, I will talk about the history and basic elements of kinetic theory and Boltzmann equation, and the role boundary plays for such a system, as well as briefly mention some recent progress.<br />
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=== November 6, Tung Nguyen===<br />
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Title: Introduction to Chemical Reaction Network<br />
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Abstract: Reaction network models are often used to investigate the dynamics of different species from various branches of chemistry, biology and ecology. The study of reaction network has grown significantly and involves a wide range of mathematics and applications. In this talk, I aim to show a big picture of what is happening in reaction network theory. I will first introduce the basic dynamical models for reaction network: the deterministic and stochastic models. Then, I will mention some big questions of interest, and the mathematical tools that are used by people in the field. Finally, I will make connection between reaction network and other branches of mathematics such as PDE, control theory, and random graph theory.<br />
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=== November 13, Jane Davis===<br />
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Title: Brownian Minions<br />
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Abstract: Having lots of small minions help you perform a task is often very effective. For example, if you need to grade a large stack of calculus problems, it is effective to have several TAs grade parts of the pile for you. We'll talk about how we can use random motions as minions to help us perform mathematical tasks. Typically, this mathematical task would be optimization, but we'll reframe a little bit and focus on art and beauty instead. We'll also try to talk about the so-called "storytelling metric," which is relevant here. There will be pictures and animations! 🎉<br />
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Sneak preview: some modern art generated with MATLAB.<br />
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[[File:Picpic.jpg]]<br />
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=== November 20, Colin Crowley===<br />
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Title: Matroid Bingo<br />
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Abstract: Matroids are combinatorial objects that generalize graphs and matrices. The famous combinatorialist Gian Carlo Rota once said that "anyone who has worked with matroids has come away with the conviction that matroids are one of the richest and most useful ideas of our day." Although his day was in the 60s and 70s, matroids remain an active area of current research with connections to areas such as algebraic geometry, tropical geometry, and parts of computer science. Since this is a doughnut talk, I will introduce matroids in a cute way that involves playing bingo, and then I'll show you some cool examples.<br />
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=== December 4, Xiaocheng Li===<br />
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Title: The method of stationary phase and Duistermaat-Heckman formula<br />
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Abstract: The oscillatory integral $\int_X e^{itf(x)}\mu=:I(t), t\in \mathbb{R}$ is a fundamental object in analysis. In general, $I(t)$ seldom has an explicit expression as Fourier transform is usually inexplicit. In practice, we are interested in the asymptotic behavior of $I(t)$, that is, for $|t|$ very large. A classical tool of getting an approximation is the method of stationary phase which gives the leading term of $I(t)$. Furthermore, there are rare instances for which the approximation coincides with the exact value of $I(t)$. One example is the Duistermaat-Heckman formula in which the Hamiltonian action and the momentum map are addressed. In the talk, I will start with basic facts in Fourier analysis, then discuss the method of stationary phase and the Duistermaat-Heckman formula.<br />
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=== December 11, Chaojie Yuan===<br />
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Title: Coupling and its application in stochastic chemical reaction network<br />
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Abstract: Stochastic models for chemical reaction networks have become increasingly popular in the past few decades. When the molecules are present in low numbers, the chemical system always displays randomness in their dynamics, and the randomness cannot be ignored as it can have a significant effect on the overall properties of the dynamics. In this talk, I will introduce the stochastic models utilized in the context of biological interaction network. Then I will discuss coupling in this context, and illustrate through examples how coupling methods can be utilized for numerical simulations. Specifically, I will introduce two biological models, which attempts to address the behavior of interesting real-world phenomenon.</div>Ogoff