Difference between revisions of "Graduate Algebraic Geometry Seminar"

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'''
 
'''
'''When:''' Wednesdays 4:10pm
+
'''When:''' Wednesdays 4:25pm
  
'''Where:''' Van Vleck B215 (Fall 2018)
+
'''Where:''' Van Vleck B317 (Spring 2019)
 
[[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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'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].
 
'''How:''' If you want to get emails regarding time, place, and talk topics ('''which are often assigned quite last minute''') add yourself to the gags mailing list: gags@lists.wisc.edu. The list registration page is [https://admin.lists.wisc.edu/index.php?p=11&l=gags here].
 
'''
 
'''
 
== Organize the seminar! ==
 
 
'''This could be you writing this wiki page! Soon (Spring 2019) we will need volunteers to organize the seminar!! Why not start now?'''
 
  
 
== Give a talk! ==
 
== Give a talk! ==
We need volunteers to give talks this semester. If you're interested contact [mailto:moises@math.wisc.edu Moisés], 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: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.
 
 
 
 
  
 
== Being an audience member ==
 
== Being an audience member ==
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__NOTOC__
 
__NOTOC__
  
== Autumn 2018 ==
+
== Spring 2019 ==
  
 
<center>
 
<center>
Line 120: Line 114:
 
| 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 12
+
| bgcolor="#E0E0E0"| February 6
| bgcolor="#C6D46E"| Moisés Herradón Cueto
+
| bgcolor="#C6D46E"| Vlad Sotirov
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 12| Hodge Theory: One hour closer to understanding what it's about]]
+
| bgcolor="#BCE2FE"|[[#February 6| Heisenberg Groups and the Fourier Transform]]
 +
|-
 +
| bgcolor="#E0E0E0"| February 13
 +
| bgcolor="#C6D46E"| David Wagner
 +
| bgcolor="#BCE2FE"|[[#February 13| DG potpourri]]
 
|-
 
|-
| bgcolor="#E0E0E0"| September 19
+
| bgcolor="#E0E0E0"| February 20
 
| bgcolor="#C6D46E"| Caitlyn Booms
 
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 12| Linear Resolutions of Edge Ideals]]
+
| bgcolor="#BCE2FE"|[[#February 20| Completions of Noncatenary Local Domains and UFDs]]
 
|-
 
|-
| bgcolor="#E0E0E0"| September 26
+
| bgcolor="#E0E0E0"| February 27
| bgcolor="#C6D46E"| Qiao He
+
| bgcolor="#C6D46E"| Sun Woo Park
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 26| An Elementary Introduction to Geometric Langlands]]
+
| bgcolor="#BCE2FE"|[[#February 27| Baker’s Theorem]]
 
|-
 
|-
| bgcolor="#E0E0E0"| October 3
+
| bgcolor="#E0E0E0"| March 6
| bgcolor="#C6D46E"| Wanlin Li
+
| bgcolor="#C6D46E"| Connor Simpson
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 3| Gonality of Curves and More]]
+
| bgcolor="#BCE2FE"|[[#March 6| Mason's Conjectures and Chow Rings of Matroids]]
 
|-
 
|-
| bgcolor="#E0E0E0"| October 10
+
| bgcolor="#E0E0E0"| March 13
| bgcolor="#C6D46E"| Ewan Dalby
+
| bgcolor="#C6D46E"| Brandon Boggess
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 10| A Gentle introduction to Grothendieck's Galois theory]]
+
| bgcolor="#BCE2FE"|[[#March 13| Dial M_1,1 for moduli]]
 
|-
 
|-
| bgcolor="#E0E0E0"| October 17
+
| bgcolor="#E0E0E0"| March 27
| bgcolor="#C6D46E"| Johnnie Han
 
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 17| Schubert Calculus]]
 
|-
 
| bgcolor="#E0E0E0"| October 24
 
 
| bgcolor="#C6D46E"| Solly Parenti
 
| bgcolor="#C6D46E"| Solly Parenti
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 24| Quadratic Polynomials]]
+
| bgcolor="#BCE2FE"|[[#March 27| Quadratic Forms]]
|-
 
| bgcolor="#E0E0E0"| October 31
 
| bgcolor="#C6D46E"| Brandon Boggess
 
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 31| How to Parameterize Elliptic Curves and Influence People]]
 
 
|-
 
|-
| bgcolor="#E0E0E0"| November 7
+
| bgcolor="#E0E0E0"| April 3
| bgcolor="#C6D46E"| Vladimir Sotirov
+
| bgcolor="#C6D46E"| Colin Crowley
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 7| Morita Duality and Local Duality]]
+
| bgcolor="#BCE2FE"|[[#April 3| Riemann-Roch and Abel-Jacobi theory on a finite graph]]
 
|-
 
|-
| bgcolor="#E0E0E0"| November 14
+
| bgcolor="#E0E0E0"| April 10
| bgcolor="#C6D46E"| David Wagner
+
| bgcolor="#C6D46E"| Alex Hof
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 14| Homological Projective Duality]]
+
| bgcolor="#BCE2FE"|[[#April 10| Kindergarten GAGA]]
 
|-
 
|-
| bgcolor="#E0E0E0"| November 21
+
| bgcolor="#E0E0E0"| April 17
| bgcolor="#C6D46E"| A turkey/Smallpox
+
| bgcolor="#C6D46E"| Soumya Sankar
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 21| TBD]]
+
| bgcolor="#BCE2FE"|[[#April 17| Inseparable maps and quotients of varieties]]
 
|-
 
|-
| bgcolor="#E0E0E0"| November 28
+
| bgcolor="#E0E0E0"| April 24
| bgcolor="#C6D46E"| Asvin Gothandaraman
+
| bgcolor="#C6D46E"| Wendy Cheng
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 30| Deformation Theory]]
+
| bgcolor="#BCE2FE"|[[#April 24| Introduction to Boij-S&#246;derberg Theory]]
|-
 
| bgcolor="#E0E0E0"| December 5
 
| bgcolor="#C6D46E"| Soumya Sankar
 
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 5| One Step Closer to <math>B_{cris}</math>]]
 
 
|-
 
|-
| bgcolor="#E0E0E0"| December 12
+
| bgcolor="#E0E0E0"| May 1
| bgcolor="#C6D46E"| Sun Woo Park
+
| bgcolor="#C6D46E"| Shengyuan Huang
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 12| A Survey of Newton Polygons]]
+
| bgcolor="#BCE2FE"|[[#May 1| Orbifold Singular Cohomology]]
 
|}
 
|}
 
</center>
 
</center>
  
== September 12 ==
+
== February 6 ==
 
<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%" | '''Moisés Herradón Cueto'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vladimir Sotirov'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Hodge Theory: One hour closer to understanding what it's about
+
| bgcolor="#BCD2EE" | Title: Heisenberg Groups and the Fourier Transform
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
Abstract:  
+
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.
  
Following the request for baby Hodge theory from our meeting last semester, I will speak for one hour about Hodge theory, starting from the beginning of times, as they say. There will be d's, dbar's, Kählers and Hodge structures, but that's the extent of my promises. It will be a joyful time!
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== September 19 ==
+
== February 13 ==
 
<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%" | '''David Wagner'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Linear Resolutions of Edge Ideals
+
| bgcolor="#BCD2EE"  | Title: DG potpourri
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
Abstract:  
+
<br />
 
+
[[File:Dg-meme.png|center]]
We will briefly discuss monomial ideals in the multivariate polynomial ring over an algebraically closed field and some of their properties, including what it means for an ideal to have a linear resolution. Then we will talk about graphs on n vertices and their corresponding edge ideals, which are a particular kind of monomial ideal. Together, these will help us understand Froberg's Theorem, which says exactly when an edge ideal has a linear resolution. This talk will focus on a few computational examples and will end with some open questions and conjectures related to the presented material.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== September 26 ==
+
== February 20 ==
 
<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%" | '''Qiao He'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: An Elementary Introduction to Geometric Langlands
+
| bgcolor="#BCD2EE"  | Title: Completions of Noncatenary Local Domains and UFDs
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
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.
 
 
I will start with the a rough statement of global langlands correspondence which suggest some correspondence between Galois representation and automorphic representation. Given this motivation, I will try to explain how can we replace both Galois side and Automorphic side with algebraic geometry objects. After that I will sketch what the geometric Langlands should be in this context.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== October 3 ==
+
== February 27 ==
 
<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%" | '''Wanlin Li'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Sun Woo Park'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Gonality of Curves and More
+
| bgcolor="#BCD2EE"  | Title: Baker's Theorem
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
Abstract:  
+
<br />
 
+
[[File:Sun_woo_baker.png|500px|center]]
I will introduce an invariant, gonality of curves, from the definition, properties to its applications on modular curves.  
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== October 10 ==
+
== March 6 ==
 
<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%" | '''Ewan Dalby'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: A Gentle introduction to Grothendiecks Galois theory
+
| bgcolor="#BCD2EE" | Title: Mason's Conjectures and Chow Rings of Matroids
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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!
Abstract:  
 
  
We all know and love Galois theory as it applies to fields and their extensions. Grothendieck, as always, showed how to lever the same ideas much more generally in algebraic geometry. I will try to explain how things work for the case of commutative rings in an "elementary" fashion.
+
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).
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== October 17 ==
+
== March 13 ==
 
<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%" | '''Johnnie Han'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Brandon Boggess'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Schubert Calculus
+
| bgcolor="#BCD2EE" | Title: Dial M_1,1 for moduli
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
Abstract:  
+
<br/>
 
+
[[File:Dial-M-For-Elliptic.png|400px|center]]
In this talk, we’ll go back to 19th-century Europe, when enumerative geometric questions like “how many lines intersect a quadric” or “how many lines lie on a cubic surface” were answered without even knowing the intersection pairing existed! We’ll go through the methods of Schubert calculus with examples and talk briefly about Steiner’s conics problem, when a famous mathematician was actually proven completely wrong.
 
 
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== October 24 ==
+
== March 27 ==
 
<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 275: Line 253:
 
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Solly Parenti'''
 
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Solly Parenti'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Quadratic Polynomials
+
| bgcolor="#BCD2EE" | Title: Quadratic Forms
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract:
 
 
 
Quadratic polynomials have been studied forever. You can't just like play around with them and expect cool exciting math things like modular forms or special values of L-functions to show up, that would be ridiculous.
 
|}                                                                       
 
</center>
 
 
 
== October 31 ==
 
<center>
 
{| 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="#BCD2EE"  align="center" | Title: How to Parameterize Elliptic Curves and Influence People
+
| 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:  
 
 
 
A classical guide to classifying curves for the geometrically minded grad student. I will assume basically zero AG background.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== November 7 ==
+
== April 3 ==
 
<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"  align="center" | Title: Morita duality and local duality
+
| bgcolor="#BCD2EE" | Title: Riemann-Roch and Abel-Jacobi theory on a finite graph
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
Abstract:
 
 
 
I will explain what it means for a ring to admit a dualizing module and how to construct such for nice local rings.
 
 
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== November 14 ==
+
== April 10 ==
 
<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%" | '''Alex Hof'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Homological Projective Duality
+
| bgcolor="#BCD2EE"  | Title: Kindergarten GAGA
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
Abstract:  
 
  
I will introduce the derived category with the goal of undestanding Kuznetsov's HPD, a mysterious phenomenon which has produced a great number of examples and theorems in AG. We will give a demonstration of the duality in the case of an intersection of quadrics.  
+
[[File:Badromancehof.png|500px|center]]
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== November 21 ==
+
== April 17 ==
 
<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%" | '''TBD'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Soumya Sankar'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: TBD
+
| bgcolor="#BCD2EE" | Title: Inseparable maps and quotients of varieties
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| 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.
Abstract:  
 
  
TBD
+
[[File:Prime_Characteristic.jpg|500px|center]]
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== November 28 ==
+
== April 24 ==
 
<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%" | '''Asvin G'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Wendy Cheng'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: Deformation Theory
+
| bgcolor="#BCD2EE"  | Title: Introduction to Boij-S&#246;derberg Theory
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| bgcolor="#BCD2EE"  | Abstract: Boij-S&#246;derberg theory describes the Betti diagrams of graded modules over the polynomial ring, up to multiplication by a rational number. Analog Eisenbud-Schreyer theory describes the cohomology tables of vector bundles on projective spaces up to rational multiple. We give an introduction and survey of these newly developed areas.
Abstract:  
 
 
 
I will explain what deformation theory is and how to use it by doing a few examples.
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== November 7 ==
+
== May 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%" | '''TBD'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Shengyuan Huang'''
 
|-
 
|-
| bgcolor="#BCD2EE"  align="center" | Title: TBD
+
| bgcolor="#BCD2EE"  | Title: Orbifold Singular Cohomology
 
|-
 
|-
| bgcolor="#BCD2EE"  |  
+
| bgcolor="#BCD2EE"  | Abstract: Let [X/G] be a stack which is a global quotient of a manifold X by a finite group G.  There is a way to construct an orbifold singular cohomology ring.  It is the correct generalization of singular cohomology of a topological space, because it coincides with the singular cohomology of a crepant resolution of the quotient space X/G.  I will compute several example to explain this.  Moreover, (orbifold) singular cohomology ring of a space should corresponds to the (orbifold) Hochschild cohomolgy ring of its mirror if you believe Homological Mirror Symmetry.  I will briefly compare these two sides of Homological Mirror Symmetry by computing concrete examples.
Abstract:  
 
 
 
TBD
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== December 5 ==
+
== Organizers' Contact Info ==
<center>
 
{| 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="#BCD2EE"  align="center" | Title: One Step Closet to <math>B_{cris}</math>
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: I will talk about various comparison theorems in <math>p</math>-adic cohomology, and time permitting, describe the crystalline side of things in greater detail.
 
|}                                                                       
 
</center>
 
 
 
== December 12 ==
 
<center>
 
{| 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="#BCD2EE"  align="center" | Title: A Survey of Newton Polygons
 
|-
 
| bgcolor="#BCD2EE"  | 
 
Abstract: I will give a survey on how understanding newton polygons can be useful in solving many different problems in algebraic geometry: from the proof of p-adic Weierstrass Formula to the re-formulization of Tate's Algorithm for elliptic curves. (Since I will focus on providing various applications of newton polygons, I will not be able to present rigorous proofs to most of the statements I will make in this talk.)
 
  
TBD
+
[https://sites.google.com/wisc.edu/cbooms/ Caitlyn Booms]
|}                                                                       
 
</center>
 
  
== Organizers' Contact Info ==
+
[http://www.math.wisc.edu/~drwagner/ David Wagner]
  
[https://www.math.wisc.edu/~moises Moisés Herradón Cueto]
+
== Past Semesters ==
 +
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]
  
== Past Semesters ==
 
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2018 Spring 2018]
  

Latest revision as of 13:22, 1 May 2019

When: Wednesdays 4:25pm

Where: Van Vleck B317 (Spring 2019)

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 Caitlyn 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:

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)

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.)


Spring 2019

Date Speaker Title (click to see abstract)
February 6 Vlad Sotirov Heisenberg Groups and the Fourier Transform
February 13 David Wagner DG potpourri
February 20 Caitlyn Booms Completions of Noncatenary Local Domains and UFDs
February 27 Sun Woo Park Baker’s Theorem
March 6 Connor Simpson Mason's Conjectures and Chow Rings of Matroids
March 13 Brandon Boggess Dial M_1,1 for moduli
March 27 Solly Parenti Quadratic Forms
April 3 Colin Crowley Riemann-Roch and Abel-Jacobi theory on a finite graph
April 10 Alex Hof Kindergarten GAGA
April 17 Soumya Sankar Inseparable maps and quotients of varieties
April 24 Wendy Cheng Introduction to Boij-Söderberg Theory
May 1 Shengyuan Huang Orbifold Singular Cohomology

February 6

Vladimir Sotirov
Title: Heisenberg Groups and the Fourier Transform

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.

February 13

David Wagner
Title: DG potpourri
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.


Dg-meme.png

February 20

Caitlyn Booms
Title: Completions of Noncatenary Local Domains and UFDs
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.

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.

February 27

Sun Woo Park
Title: Baker's Theorem
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.


Sun woo baker.png

March 6

Connor Simpson
Title: Mason's Conjectures and Chow Rings of Matroids
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!

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 13

Brandon Boggess
Title: Dial M_1,1 for moduli
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.


Dial-M-For-Elliptic.png

March 27

Solly Parenti
Title: Quadratic Forms
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.

April 3

Colin Crowley
Title: Riemann-Roch and Abel-Jacobi theory on a finite graph
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.

April 10

Alex Hof
Title: Kindergarten GAGA
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.
Badromancehof.png

April 17

Soumya Sankar
Title: Inseparable maps and quotients of varieties
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.
Prime Characteristic.jpg

April 24

Wendy Cheng
Title: Introduction to Boij-Söderberg Theory
Abstract: Boij-Söderberg theory describes the Betti diagrams of graded modules over the polynomial ring, up to multiplication by a rational number. Analog Eisenbud-Schreyer theory describes the cohomology tables of vector bundles on projective spaces up to rational multiple. We give an introduction and survey of these newly developed areas.

May 1

Shengyuan Huang
Title: Orbifold Singular Cohomology
Abstract: Let [X/G] be a stack which is a global quotient of a manifold X by a finite group G. There is a way to construct an orbifold singular cohomology ring. It is the correct generalization of singular cohomology of a topological space, because it coincides with the singular cohomology of a crepant resolution of the quotient space X/G. I will compute several example to explain this. Moreover, (orbifold) singular cohomology ring of a space should corresponds to the (orbifold) Hochschild cohomolgy ring of its mirror if you believe Homological Mirror Symmetry. I will briefly compare these two sides of Homological Mirror Symmetry by computing concrete examples.

Organizers' Contact Info

Caitlyn Booms

David Wagner

Past Semesters

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015