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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__
 
== Fall 2019 ==


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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"| September 18
| bgcolor="#E0E0E0"| January 29
| bgcolor="#C6D46E"| David Wagner
| bgcolor="#C6D46E"| Colin Crowley
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 18| M_g Potpourri]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]
|-
|-
| bgcolor="#E0E0E0"| September 25
| bgcolor="#E0E0E0"| February 5
| bgcolor="#C6D46E"| Shengyuan Huang
| bgcolor="#C6D46E"| Asvin Gothandaraman
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 25| Derived Groups and Groupoids]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]
|-
|-
| bgcolor="#E0E0E0"| October 9
| bgcolor="#E0E0E0"| February 12
| bgcolor="#C6D46E"| Brandon Boggess
| bgcolor="#C6D46E"| Qiao He
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 9| Geometry of Generalized Fermat Curves ]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]
|-
|-
| bgcolor="#E0E0E0"| October 16
| bgcolor="#E0E0E0"| February 19
| bgcolor="#C6D46E"| Soumya Sankar
| bgcolor="#C6D46E"| Dima Arinkin
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 16| Brauer groups and obstruction problems]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]
|-
|-
| bgcolor="#E0E0E0"| October 23
| bgcolor="#E0E0E0"| February 26
| bgcolor="#C6D46E"| Alex Mine
| bgcolor="#C6D46E"| Connor Simpson
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 23| The Ax-Grothendieck theorem and other fun stuff]]
| 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"| October 30
| bgcolor="#E0E0E0"| March 25
| bgcolor="#C6D46E"| Steven He
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Title]]
|-
| bgcolor="#E0E0E0"| April 1
| bgcolor="#C6D46E"| Vlad Sotirov
| bgcolor="#C6D46E"| Vlad Sotirov
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 30| Buildings and algebraic groups]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]
|-
|-
| bgcolor="#E0E0E0"| November 6
| bgcolor="#E0E0E0"| April 8
| bgcolor="#C6D46E"| Connor Simpson
| bgcolor="#C6D46E"| Maya Banks
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 6| Lorentzian Polynomials]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]
|-
|-
| bgcolor="#E0E0E0"| November 13
| bgcolor="#E0E0E0"| April 15
| bgcolor="#C6D46E"| Alex Hof
| bgcolor="#C6D46E"| Alex Hof
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 13| Tropicalization Blues]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]
|-
|-
| bgcolor="#E0E0E0"| November 20
| bgcolor="#E0E0E0"| April 22
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#C6D46E"| Ruofan
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 20| Computing Gr<span>&#246;</span>bner Bases of Submodules]]
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 22| Birational geometry: existence of rational curves]]
|-
| bgcolor="#E0E0E0"| April 29
| bgcolor="#C6D46E"| John Cobb
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 29| Title]]
|}
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== January 29 ==
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{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#E0E0E0"| November 27
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''
| bgcolor="#C6D46E"| Thanksgiving Break
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 27| ]]
|-
|-
| bgcolor="#E0E0E0"| December 4
| bgcolor="#BCD2EE" align="center" | Title: Lefschetz hyperplane section theorem via Morse theory
| bgcolor="#C6D46E"| Colin Crowley
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 4| Title TBD]]
|-
|-
| bgcolor="#E0E0E0"| December 11
| 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.
| bgcolor="#C6D46E"| Erika Pirnes
|}                                                                      
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 11| Title TBD]]
|}
</center>
</center>


== September 18 ==
== 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"  align="center" | Title: M_g Potpourri
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to unirationality
|-
|-
| bgcolor="#BCD2EE"  |  
| 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.  
Abstract: In 1983, David Mumford proposed that the rational cohomology ring of Mg should be a polynomial algebra. I will discuss some of the history of Mumford's conjecture, possibly indicating a few ideas from the 2007 proof as the Madsen-Weiss theorem. If all goes well, the talk will take us through such diverse places as homotopy theory, representation stability, combinatorics of ribbon graph complexes, and deformations of algebras.
 
|}                                                                         
|}                                                                         
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</center>


== September 25 ==
== 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%" | '''Shengyuan Huang'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Derived Groups and Groupoids
| bgcolor="#BCD2EE"  align="center" | Title:  
|-
|-
| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract:  
Abstract: In this talk, we will discuss groups and groupoids in the derived category of dg schemes.  I will focus on examples instead of the abstract theory.  If X is a smooth subscheme of a smooth scheme S over the field of complex numbers, then the derived self-intersection of X in S is a groupoid.  We will investigate the corresponding Lie algebroid of the groupoid I mentioned above, and exponential map between them.
 
|}                                                                         
|}                                                                         
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</center>


== October 9 ==
== 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%" | '''Brandon Boggess'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Dima Arinkin'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Geometry of Generalized Fermat Curves
| bgcolor="#BCD2EE"  align="center" | Title: Blowing down, blowing up: surface geometry
|-
|-
| bgcolor="#BCD2EE"  |  
| 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?
Abstract: We will look at the generalized Fermat equation, and see how studying its integral points leads one to study quotient stacks. We will then very quickly turn and run away from the general picture to a particularly simple example of these quotient stacks, the M-curves of Darmon-Granville, and how they can be used to say something about integral points without having to actually know what the hell a stack is.
 
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.)
|}                                                                         
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</center>
</center>


== October 16 ==
== 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"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Soumya Sankar'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Brauer groups and obstruction problems
| bgcolor="#BCD2EE"  align="center" | Title: Intro to Toric Varieties
|-
|-
| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.
Abstract: Brauer groups are ubiquitous in arithmetic and algebraic geometry. I will try to describe different contexts in which they appear, ranging from Brauer groups of fields and class field theory, to obstructions to moduli problems and derived equivalences.  
|}                                                                         
|}                                                                         
</center>
</center>


== October 23 ==
== March 4 ==
<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 Mine'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Peter Wei'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: The Ax-Grothendieck theorem and other fun stuff
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem
|-
|-
| bgcolor="#BCD2EE"  |  
| 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.
Abstract: The Ax-Grothendieck theorem says that any polynomial map from C^n to C^n that is injective is also surjective. The way this is proven is to note that the statement is trivial over finite fields, and somehow use this to work up to the complex numbers. We'll talk about this and other ways of translating information between finite fields and C.
 
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</center>
</center>


== October 30 ==
== 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%" | '''Vlad Sotirov'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Buildings and algebraic groups
| bgcolor="#BCD2EE"  align="center" | Title: Intro to Stanley-Reisner Theory
|-
|-
| bgcolor="#BCD2EE"  |  
| 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.
Abstract: I will give a concrete introduction to the notion of a Tits building and its relationship to algebraic groups.
 
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</center>
</center>


== November 6 ==
== 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%" | '''Connor Simpson'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Steven He'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Lorentzian Polynomials
| bgcolor="#BCD2EE"  align="center" | Title:  
|-
|-
| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract:  
Abstract:
Lorentzian polynomials are a family of multivariate polynomials recently introduced by Branden and Huh. We will define Lorentzian polynomials and survey some of their applications to combinatorics, representation theory, and computer  science. The first 20 minutes of this talk should not require more than the ability to take partial derivatives of polynomials and basic linear algebra.
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</center>
</center>


== November 13 ==
== 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%" | '''Alex Hof'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vlad Sotirov'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Tropicalization Blues
| bgcolor="#BCD2EE"  align="center" | Title:  
|-
|-
| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract:  
Abstract: Tropicalization turns algebro-geometric objects into piecewise linear ones which can then be studied through the lens of combinatorics. In this talk, I will introduce the basic construction, then discuss some of the recent efforts to generalize and improve upon it, touching upon the Giansiracusa tropicalization and <s>developing</s> gazing wistfully in the direction of the machinery of ordered blueprints necessary for the Lorscheid tropicalization.
 
|}                                                                         
|}                                                                         
</center>
</center>


== November 20 ==
== 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%" | '''Caitlyn Booms'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Maya Banks'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Computing Gr<span>&#246;</span>bner Bases of Submodules
| bgcolor="#BCD2EE"  align="center" | Title:  
|-
|-
| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract:  
Abstract: In this talk, we will give motivation for and define Gr<span>&#246;</span>bner bases of submodules of finitely generated free modules over a polynomial ring S=k[x_1,...,x_r]. Not only are such bases extremely useful in constructive module theory and elimination theory, they are actually computable thanks to Buchberger's Algorithm. Further, they have a wide variety of applications in algebraic geometry including aiding in the computation of syzygies (kernels of maps of finitely generated, free S-modules), Hilbert functions, intersections of submodules, saturations, annihilators, projective closures, and elimination ideals. We will work through several examples and discuss some of these applications.
 
|}                                                                         
|}                                                                         
</center>
</center>


== November 28 ==
== 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%" | '''Thanksgiving Break'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Alex Hof'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title:
| bgcolor="#BCD2EE"  align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory
|-
|-
| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract: Early on in the semester, Colin told us a bit about Morse
Abstract:  
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.
|}                                                                         
|}                                                                         
</center>
</center>


== December 4 ==
== 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%" | '''Colin Crowley'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ruofan'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title:
| bgcolor="#BCD2EE"  align="center" | Title: Birational geometry: existence of rational curves
|-
|-
| bgcolor="#BCD2EE"  |  
| 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.
Abstract:
 
|}                                                                         
|}                                                                         
</center>
</center>


== December 11 ==
== 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%" | '''Erika Pirnes'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Buchsbaum-Eisenbud-Horrocks conjecture
| bgcolor="#BCD2EE"  align="center" | Title:  
|-
|-
| bgcolor="#BCD2EE"  |  
| bgcolor="#BCD2EE"  | Abstract:  
Abstract: Betti numbers are defined to be the ranks of the free modules in the free resolution of a module. The Buchsbaum-Eisenbud-Horrocks conjecture gives upper bounds for the Betti numbers. I'll state the conjecture and give some examples.
 
|}                                                                         
|}                                                                         
</center>
</center>


== Organizers' Contact Info ==
== Organizers' Contact Info ==


[https://sites.google.com/view/colincrowley/home Colin Crowley]
[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 ;).)


[https://sites.google.com/wisc.edu/cbooms/ Caitlyn Booms]
* ''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.)


[http://www.math.wisc.edu/~drwagner/ David Wagner]
* ''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_Spring_2019 Spring 2019]



Revision as of 19:09, 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

Fall 2019

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015