Difference between revisions of "Graduate Algebraic Geometry Seminar"

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'''When:''' Wednesdays 4:25pm
 
'''When:''' Wednesdays 4:25pm
  
'''Where:''' Van Vleck B317 (Spring 2019)
+
'''Where:''' Van Vleck B317
 
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]
 
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]
  
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== Give a talk! ==
 
== Give a talk! ==
We need volunteers to give talks this semester. If you're interested contact [mailto:cbooms@wisc.edu Caitlyn] 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.
+
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.
  
 
== Being an audience member ==
 
== Being an audience member ==
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* Ask Questions Appropriately:  
 
* Ask Questions Appropriately:  
  
==The List of Topics that we Made February 2018==
+
== Spring 2020 ==
 
 
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:
 
 
 
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.
 
 
 
* 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!
 
 
 
* 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.
 
 
 
* Katz and Mazur explanation of what a modular form is. What is it?
 
 
 
* Kindergarten moduli of curves.
 
 
 
* 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?
 
 
 
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)
 
 
 
* Hodge theory for babies
 
 
 
* What is a Néron model?
 
 
 
* 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].
 
 
 
* What and why is a dessin d'enfants?
 
 
 
* DG Schemes.
 
 
 
 
 
==Ed Dewey's Wish List Of Olde==
 
 
 
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.
 
 
 
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.
 
 
 
===Specifically Vague Topics===
 
* 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.
 
 
 
* 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)
 
 
 
===Famous Theorems===
 
 
 
===Interesting Papers & Books===
 
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.
 
 
 
* ''Residues and Duality'' - Robin Hatshorne.
 
** 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 ;).)
 
 
 
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.
 
** 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.)
 
 
 
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.
 
** 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!
 
 
 
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.
 
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!
 
 
 
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.
 
** 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.
 
 
 
* ''Esquisse d’une programme'' - Alexander Grothendieck.
 
** 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.)
 
 
 
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.
 
** 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.)
 
 
 
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.
 
** 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.
 
 
 
* ''Picard Groups of Moduli Problems'' - David Mumford.
 
** This paper is essentially the origin of algebraic stacks.
 
 
 
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar
 
** 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.
 
 
 
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.
 
** 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?''.
 
 
 
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.
 
** 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.
 
 
 
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.
 
** 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.)
 
__NOTOC__
 
 
 
== Spring 2019 ==
 
  
 
<center>
 
<center>
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| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
 
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
 
|-
 
|-
| bgcolor="#E0E0E0"| February 6
+
| bgcolor="#E0E0E0"| January 29
| bgcolor="#C6D46E"| Vlad Sotirov
+
| bgcolor="#C6D46E"| Colin Crowley
| bgcolor="#BCE2FE"|[[#February 6| Heisenberg Groups and the Fourier Transform]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]
 
|-
 
|-
| bgcolor="#E0E0E0"| February 13
+
| bgcolor="#E0E0E0"| February 5
| bgcolor="#C6D46E"| David Wagner
+
| bgcolor="#C6D46E"| Asvin Gothandaraman
| bgcolor="#BCE2FE"|[[#February 13| DG potpourri]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]
 
|-
 
|-
| bgcolor="#E0E0E0"| February 20
+
| bgcolor="#E0E0E0"| February 12
| bgcolor="#C6D46E"| Caitlyn Booms
+
| bgcolor="#C6D46E"| Qiao He
| bgcolor="#BCE2FE"|[[#February 20| Completions of Noncatenary Local Domains and UFDs]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]
 
|-
 
|-
| bgcolor="#E0E0E0"| February 27
+
| bgcolor="#E0E0E0"| February 19
| bgcolor="#C6D46E"| Sun Woo Park
+
| bgcolor="#C6D46E"| Dima Arinkin
| bgcolor="#BCE2FE"|[[#February 27| Baker’s Theorem]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]
 
|-
 
|-
| bgcolor="#E0E0E0"| March 6
+
| bgcolor="#E0E0E0"| February 26
 
| bgcolor="#C6D46E"| Connor Simpson
 
| bgcolor="#C6D46E"| Connor Simpson
| bgcolor="#BCE2FE"|[[#March 6| Mason's Conjectures and Chow Rings of Matroids]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]
 +
|-
 +
| bgcolor="#E0E0E0"| March 4
 +
| bgcolor="#C6D46E"| Peter
 +
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]
 +
|-
 +
| bgcolor="#E0E0E0"| March 11
 +
| bgcolor="#C6D46E"| Caitlyn Booms
 +
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]
 
|-
 
|-
| bgcolor="#E0E0E0"| March 13
+
| bgcolor="#E0E0E0"| March 25
| bgcolor="#C6D46E"| Brandon Boggess
+
| bgcolor="#C6D46E"| Steven He
| bgcolor="#BCE2FE"|[[#March 13| Dial M_1,1 for moduli]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]
 
|-
 
|-
| bgcolor="#E0E0E0"| March 27
+
| bgcolor="#E0E0E0"| April 1
| bgcolor="#C6D46E"| Solly Parenti
+
| bgcolor="#C6D46E"| Vlad Sotirov
| bgcolor="#BCE2FE"|[[#March 27| Quadratic Forms]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]
 
|-
 
|-
| bgcolor="#E0E0E0"| April 3
+
| bgcolor="#E0E0E0"| April 8
| bgcolor="#C6D46E"| Colin Crowley
+
| bgcolor="#C6D46E"| Maya Banks
| bgcolor="#BCE2FE"|[[#April 3| Riemann-Roch and Abel-Jacobi theory on a finite graph]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]
 
|-
 
|-
| bgcolor="#E0E0E0"| April 10
+
| bgcolor="#E0E0E0"| April 15
 
| bgcolor="#C6D46E"| Alex Hof
 
| bgcolor="#C6D46E"| Alex Hof
| bgcolor="#BCE2FE"|[[#April 10| Kindergarten GAGA]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]
|-
 
| bgcolor="#E0E0E0"| April 17
 
| bgcolor="#C6D46E"| Soumya Sankar
 
| bgcolor="#BCE2FE"|[[#April 17| Inseparable maps and quotients of varieties]]
 
 
|-
 
|-
| bgcolor="#E0E0E0"| April 24
+
| bgcolor="#E0E0E0"| April 22
| bgcolor="#C6D46E"| Wendy Cheng
+
| bgcolor="#C6D46E"| Ruofan
| bgcolor="#BCE2FE"|[[#April 24| Introduction to Boij-S&#246;derberg Theory]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]
 
|-
 
|-
| bgcolor="#E0E0E0"| May 1
+
| bgcolor="#E0E0E0"| April 29
| bgcolor="#C6D46E"| Shengyuan Huang
+
| bgcolor="#C6D46E"| John Cobb
| bgcolor="#BCE2FE"|[[#May 1| Orbifold Singular Cohomology]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]
 
|}
 
|}
 
</center>
 
</center>
  
== February 6 ==
+
== January 29 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vladimir Sotirov'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Heisenberg Groups and the Fourier Transform
+
| bgcolor="#BCD2EE"  align="center" | Title: Lefschetz hyperplane section theorem via Morse theory
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
Abstract: I will discuss the relationship between Fourier transforms and the Heisenberg groups, with a view toward the discussion of line bundles on complex tori that appears in Polishchuk's book Abelian Varieties, Theta functions, and the Fourier transform.
 
 
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== February 13 ==
+
== February 5 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''David Wagner'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: DG potpourri
+
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to unirationality
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: I will explain how differential graded categories made their way into AG as a way to solve some of the inadequacies of the ordinary derived category. We will then give examples of the utility of DG techniques.
+
| 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 />
 
[[File:Dg-meme.png|center]]
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== February 20 ==
+
== February 12 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Completions of Noncatenary Local Domains and UFDs
+
| bgcolor="#BCD2EE"  align="center" | Title:  
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: A ring is called catenary if for any pair of prime ideals P contained in Q, all saturated chains of prime ideals between P and Q have the same length.
+
| bgcolor="#BCD2EE"  | Abstract:  
In this talk, I will introduce the necessary background about noncatenary rings and completions of local (Noetherian) domains, as well as the relevant history. Then, I will give the characterization of completions of noncatenary local domains and noncatenary local UFDs, which I will use to describe examples of very strange rings.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== February 27 ==
+
== February 19 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Sun Woo Park'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Baker's Theorem
+
| bgcolor="#BCD2EE"  align="center" | Title: Blowing down, blowing up: surface geometry
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: As a follow up talk to last semester, I will state and prove Baker’s theorem, a classical theorem which uses Newton Polygons to count the genus of a given curve. I will also briefly sketch how one can use Newton polygons to understand resolution of singularities of a given curve over valuation rings.
+
| 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 />
+
 
[[File:Sun_woo_baker.png|500px|center]]
+
The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the
 +
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.
 +
 
 +
In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case
 +
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.)
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== March 6 ==
+
== February 26 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
Line 225: Line 143:
 
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''
 
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Mason's Conjectures and Chow Rings of Matroids
+
| bgcolor="#BCD2EE"  align="center" | Title: Intro to Toric Varieties
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: A matroid is a combinatorial structure that abstracts many different notions of independence, including linear independence in a vector space. Mason's Conjectures are a series of three increasingly strong inequalities on certain numerical invariants of matroids, the weakest of which resisted proof for over 40 years until its resolution in 2015. However, in the years since, all of Mason's conjectures have become theorems!
+
| bgcolor="#BCD2EE"  | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.
 +
|}                                                                       
 +
</center>
  
In this talk, we will introduce matroids and the Chow ring of a matroid, the amazing algebraic gadget used to prove Mason's first conjecture. Finally, outline the proof of Mason's second conjecture (work of our very own Botong Wang & coauthors).
+
== March 4 ==
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 +
|-
 +
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem
 +
|-
 +
| 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.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== March 13 ==
+
== March 11 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Brandon Boggess'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Dial M_1,1 for moduli
+
| bgcolor="#BCD2EE"  align="center" | Title: Intro to Stanley-Reisner Theory
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: We will speculate wildly about what kind of object the moduli space of elliptic curves should be. I don't know what a stack is, and I promise not to try to define one.
+
| 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/>
 
[[File:Dial-M-For-Elliptic.png|400px|center]]
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== March 27 ==
+
== March 25 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Solly Parenti'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Quadratic Forms
+
| bgcolor="#BCD2EE"  align="center" | Title:  
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: Starting from the classical story of binary quadratic forms, we'll move on to more modern aspects of the theory of quadratic forms and try to make sense of some mass formulas.
+
| bgcolor="#BCD2EE"  | Abstract:  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== April 3 ==
+
== April 1 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Riemann-Roch and Abel-Jacobi theory on a finite graph
+
| bgcolor="#BCD2EE"  align="center" | Title:  
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: A finite graph is like a Riemann surface, or a smooth projective curve. Following the paper by Baker and Norine, we will investigate linear equivalence of divisors on graphs, the Jacobian of a graph, and a combinatorial interpretation of these as a chip firing game.
+
| bgcolor="#BCD2EE"  | Abstract:  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== April 10 ==
+
== April 8 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Kindergarten GAGA
+
| bgcolor="#BCD2EE"  align="center" | Title:  
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: Join me in regressing to an infantile state as we discuss Serre's 1956 paper ''Algebraic geometry and analytic geometry'', widely considered to be the most influential work ever authored by a mathematician under the age of five. We will define the notion of an analytic space, construct the analytic space associated to any algebraic variety over <math>\mathbb{C}</math>, and examine the relationships between the two, including the equivalence between coherent algebraic sheaves and coherent analytic sheaves in the projective case.
+
| bgcolor="#BCD2EE"  | Abstract:  
 
 
[[File:Badromancehof.png|500px|center]]
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== April 17 ==
+
== April 15 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Soumya Sankar'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Inseparable maps and quotients of varieties
+
| bgcolor="#BCD2EE"  align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: The theory of inseparable maps is inseparable from the study of varieties in positive characteristic, as are quotients of varieties by wonderfully non-reduced group schemes. I will talk about the theory of derivations and Lie algebras and how these are helpful in understanding both the structure of inseparable maps, as well as group-scheme actions on varieties.
+
| bgcolor="#BCD2EE"  | Abstract: Early on in the semester, Colin told us a bit about Morse
 
+
Theory, and how it lets us get a handle on the (classical) topology of
[[File:Prime_Characteristic.jpg|500px|center]]
+
smooth complex varieties. As we all know, however, not everything in
 +
life goes smoothly, and so too in algebraic geometry. Singular
 +
varieties, when given the classical topology, are not manifolds, but
 +
they can be described in terms of manifolds by means of something called
 +
a Whitney stratification. This allows us to develop a version of Morse
 +
Theory that applies to singular spaces (and also, with a bit of work, to
 +
smooth spaces that fail to be nice in other ways, like non-compact
 +
manifolds!), called Stratified Morse Theory. After going through the
 +
appropriate definitions and briefly reviewing the results of classical
 +
Morse Theory, we'll discuss the so-called Main Theorem of Stratified
 +
Morse Theory and survey some of its consequences.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== April 24 ==
+
== April 22 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Wendy Cheng'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Introduction to Boij-S&#246;derberg Theory
+
| bgcolor="#BCD2EE"  align="center" | Title: Birational geometry: existence of rational curves
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract:  
+
| 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.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== May 1 ==
+
== April 29 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Shengyuan Huang'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Orbifold Singular Cohomology
+
| bgcolor="#BCD2EE"  align="center" | Title:  
 
|-
 
|-
 
| bgcolor="#BCD2EE"  | Abstract:  
 
| bgcolor="#BCD2EE"  | Abstract:  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
 +
  
 
== Organizers' Contact Info ==
 
== Organizers' Contact Info ==
  
[https://sites.google.com/wisc.edu/cbooms/ Caitlyn Booms]
+
[https://sites.google.com/view/colincrowley/home Colin Crowley]
  
 
[http://www.math.wisc.edu/~drwagner/ David Wagner]
 
[http://www.math.wisc.edu/~drwagner/ David Wagner]
 +
 +
==The List of Topics that we Made February 2018==
 +
 +
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:
 +
 +
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.
 +
 +
* 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!
 +
 +
* 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.
 +
 +
* Katz and Mazur explanation of what a modular form is. What is it?
 +
 +
* Kindergarten moduli of curves.
 +
 +
* 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?
 +
 +
* Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)
 +
 +
* Hodge theory for babies
 +
 +
* What is a Néron model?
 +
 +
* 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].
 +
 +
* What and why is a dessin d'enfants?
 +
 +
* DG Schemes.
 +
 +
==Ed Dewey's Wish List Of Olde==__NOTOC__
 +
 +
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.
 +
 +
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.
 +
 +
===Specifically Vague Topics===
 +
* 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.
 +
 +
* 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)
 +
 +
===Interesting Papers & Books===
 +
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.
 +
 +
* ''Residues and Duality'' - Robin Hatshorne.
 +
** 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 ;).)
 +
 +
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.
 +
** 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.)
 +
 +
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.
 +
** 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!
 +
 +
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.
 +
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!
 +
 +
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.
 +
** 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.
 +
 +
* ''Esquisse d’une programme'' - Alexander Grothendieck.
 +
** 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.)
 +
 +
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.
 +
** 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.)
 +
 +
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.
 +
** 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.
 +
 +
* ''Picard Groups of Moduli Problems'' - David Mumford.
 +
** This paper is essentially the origin of algebraic stacks.
 +
 +
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar
 +
** 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.
 +
 +
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.
 +
** 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?''.
 +
 +
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.
 +
** 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.
 +
 +
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.
 +
** 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.)
 +
  
 
== Past Semesters ==
 
== Past Semesters ==
 +
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2019 Fall 2019]
 +
 +
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]
 +
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]
  

Latest revision as of 14:09, 21 April 2020

When: Wednesdays 4:25pm

Where: Van Vleck B317

Lizzie the OFFICIAL mascot of GAGS!!

Who: All undergraduate and graduate students interested in algebraic geometry, commutative algebra, and related fields are welcome to attend.

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.

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 here.

Give a talk!

We need volunteers to give talks this semester. If you're interested contact Colin or 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.

Being an audience member

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:

  • Do Not Speak For/Over the Speaker:
  • Ask Questions Appropriately:

Spring 2020

Date Speaker Title (click to see abstract)
January 29 Colin Crowley Lefschetz hyperplane section theorem via Morse theory
February 5 Asvin Gothandaraman An Introduction to Unirationality
February 12 Qiao He Title
February 19 Dima Arinkin Blowing down, blowing up: surface geometry
February 26 Connor Simpson Intro to toric varieties
March 4 Peter An introduction to Grothendieck-Riemann-Roch Theorem
March 11 Caitlyn Booms Intro to Stanley-Reisner Theory
March 25 Steven He Title
April 1 Vlad Sotirov Title
April 8 Maya Banks Title
April 15 Alex Hof Embrace the Singularity: An Introduction to Stratified Morse Theory
April 22 Ruofan Birational geometry: existence of rational curves
April 29 John Cobb Title

January 29

Colin Crowley
Title: Lefschetz hyperplane section theorem via Morse theory
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.

February 5

Asvin Gothandaraman
Title: An introduction to unirationality
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.

February 12

Qiao He
Title:
Abstract:

February 19

Dima Arinkin
Title: Blowing down, blowing up: surface geometry
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?

The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the 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.

In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case 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.)

February 26

Connor Simpson
Title: Intro to Toric Varieties
Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.

March 4

Peter Wei
Title: An introduction to Grothendieck-Riemann-Roch Theorem
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.

March 11

Caitlyn Booms
Title: Intro to Stanley-Reisner Theory
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.

March 25

Steven He
Title:
Abstract:

April 1

Vlad Sotirov
Title:
Abstract:

April 8

Maya Banks
Title:
Abstract:

April 15

Alex Hof
Title: Embrace the Singularity: An Introduction to Stratified Morse Theory
Abstract: Early on in the semester, Colin told us a bit about Morse

Theory, and how it lets us get a handle on the (classical) topology of smooth complex varieties. As we all know, however, not everything in life goes smoothly, and so too in algebraic geometry. Singular varieties, when given the classical topology, are not manifolds, but they can be described in terms of manifolds by means of something called a Whitney stratification. This allows us to develop a version of Morse Theory that applies to singular spaces (and also, with a bit of work, to smooth spaces that fail to be nice in other ways, like non-compact manifolds!), called Stratified Morse Theory. After going through the appropriate definitions and briefly reviewing the results of classical Morse Theory, we'll discuss the so-called Main Theorem of Stratified Morse Theory and survey some of its consequences.

April 22

Ruofan
Title: Birational geometry: existence of rational curves
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.

April 29

John Cobb
Title:
Abstract:


Organizers' Contact Info

Colin Crowley

David Wagner

The List of Topics that we Made February 2018

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:

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.

  • 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!
  • 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.
  • Katz and Mazur explanation of what a modular form is. What is it?
  • Kindergarten moduli of curves.
  • 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?
  • Generalizations of Riemann - Roch. (Grothendieck - Riemann - Roch? Hirzebruch - Riemann - Roch?)
  • Hodge theory for babies
  • What is a Néron model?
  • What and why is a dessin d'enfants?
  • DG Schemes.

Ed Dewey's Wish List Of Olde

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.

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.

Specifically Vague Topics

  • 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.
  • 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)

Interesting Papers & Books

  • Symplectic structure of the moduli space of sheaves on an abelian or K3 surface - Shigeru Mukai.
  • Residues and Duality - Robin Hatshorne.
    • 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 ;).)
  • Coherent sheaves on P^n and problems in linear algebra - A. A. Beilinson.
    • 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.)
  • Frobenius splitting and cohomology vanishing for Schubert varieties - V.B. Mehta and A. Ramanathan.
    • 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!
  • Schubert Calculus - S. L. Kleiman and Dan Laksov.
    • An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!
  • Rational Isogenies of Prime Degree - Barry Mazur.
    • 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.
  • Esquisse d’une programme - Alexander Grothendieck.
    • 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.)
  • Géométrie algébraique et géométrie analytique - J.P. Serre.
    • 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.)
  • Limit linear series: Basic theory- David Eisenbud and Joe Harris.
    • 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.
  • Picard Groups of Moduli Problems - David Mumford.
    • This paper is essentially the origin of algebraic stacks.
  • The Structure of Algebraic Threefolds: An Introduction to Mori's Program - Janos Kollar
    • 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.
  • Cayley-Bacharach Formulas - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.
    • 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?.
  • On Varieties of Minimal Degree (A Centennial Approach) - David Eisenbud and Joe Harris.
    • 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.
  • The Gromov-Witten potential associated to a TCFT - Kevin J. Costello.
    • 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.)


Past Semesters

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Fall 2017

Spring 2017

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Spring 2016

Fall 2015