Difference between revisions of "Graduate Algebraic Geometry Seminar"

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(May 1)
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'''
 
'''
'''When:''' Wednesdays 4:25pm
+
'''When:''' Wednesdays 4:10pm
  
'''Where:''' Van Vleck B317 (Spring 2019)
+
'''Where:''' Van Vleck B215 (Fall 2018)
 
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]
 
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]
  
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== Give a talk! ==
 
== Give a talk! ==
 
We need volunteers to give talks this semester. If you're interested contact [mailto:cbooms@wisc.edu Caitlyn] or [mailto:drwagner@math.wisc.edu David], or just add yourself to the list (though in that case we might move your talk later without your permission). Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material.
 
We need volunteers to give talks this semester. If you're interested contact [mailto: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 ==
Line 105: Line 106:
 
__NOTOC__
 
__NOTOC__
  
== Spring 2019 ==
+
== Autumn 2018 ==
  
 
<center>
 
<center>
Line 114: Line 115:
 
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
 
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
 
|-
 
|-
| bgcolor="#E0E0E0"| February 6
+
| bgcolor="#E0E0E0"| September 12
| bgcolor="#C6D46E"| Vlad Sotirov
+
| bgcolor="#C6D46E"| Moisés Herradón Cueto
| bgcolor="#BCE2FE"|[[#February 6| Heisenberg Groups and the Fourier Transform]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#September 12| Hodge Theory: One hour closer to understanding what it's about]]
 
|-
 
|-
| bgcolor="#E0E0E0"| February 13
+
| bgcolor="#E0E0E0"| September 19
| bgcolor="#C6D46E"| David Wagner
+
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#BCE2FE"|[[#February 13| DG potpourri]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#September 12| Linear Resolutions of Edge Ideals]]
 +
|-
 +
| bgcolor="#E0E0E0"| September 26
 +
| bgcolor="#C6D46E"| Qiao He
 +
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#September 26| An Elementary Introduction to Geometric Langlands]]
 +
|-
 +
| bgcolor="#E0E0E0"| October 3
 +
| bgcolor="#C6D46E"| Wanlin Li
 +
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#October 3| Gonality of Curves and More]]
 
|-
 
|-
| bgcolor="#E0E0E0"| February 20
+
| bgcolor="#E0E0E0"| October 10
| bgcolor="#C6D46E"| Caitlyn Booms
+
| bgcolor="#C6D46E"| Ewan Dalby
| bgcolor="#BCE2FE"|[[#February 20| Completions of Noncatenary Local Domains and UFDs]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#October 10| A Gentle introduction to Grothendieck's Galois theory]]
 
|-
 
|-
| bgcolor="#E0E0E0"| February 27
+
| bgcolor="#E0E0E0"| October 17
| bgcolor="#C6D46E"| Sun Woo Park
+
| bgcolor="#C6D46E"| Johnnie Han
| bgcolor="#BCE2FE"|[[#February 27| Baker’s Theorem]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#October 17| Schubert Calculus]]
 
|-
 
|-
| bgcolor="#E0E0E0"| March 6
+
| bgcolor="#E0E0E0"| October 24
| bgcolor="#C6D46E"| Connor Simpson
+
| bgcolor="#C6D46E"| Solly Parenti
| bgcolor="#BCE2FE"|[[#March 6| Mason's Conjectures and Chow Rings of Matroids]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#October 24| Quadratic Polynomials]]
 
|-
 
|-
| bgcolor="#E0E0E0"| March 13
+
| bgcolor="#E0E0E0"| October 31
 
| bgcolor="#C6D46E"| Brandon Boggess
 
| bgcolor="#C6D46E"| Brandon Boggess
| bgcolor="#BCE2FE"|[[#March 13| Dial M_1,1 for moduli]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#October 31| How to Parameterize Elliptic Curves and Influence People]]
 
|-
 
|-
| bgcolor="#E0E0E0"| March 27
+
| bgcolor="#E0E0E0"| November 7
| bgcolor="#C6D46E"| Solly Parenti
+
| bgcolor="#C6D46E"| Vladimir Sotirov
| bgcolor="#BCE2FE"|[[#March 27| Quadratic Forms]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#November 7| Morita Duality and Local Duality]]
 +
|-
 +
| bgcolor="#E0E0E0"| November 14
 +
| bgcolor="#C6D46E"| David Wagner
 +
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#November 14| Homological Projective Duality]]
 
|-
 
|-
| bgcolor="#E0E0E0"| April 3
+
| bgcolor="#E0E0E0"| November 21
| bgcolor="#C6D46E"| Colin Crowley
+
| bgcolor="#C6D46E"| A turkey/Smallpox
| bgcolor="#BCE2FE"|[[#April 3| Riemann-Roch and Abel-Jacobi theory on a finite graph]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#November 21| TBD]]
 
|-
 
|-
| bgcolor="#E0E0E0"| April 10
+
| bgcolor="#E0E0E0"| November 28
| bgcolor="#C6D46E"| Alex Hof
+
| bgcolor="#C6D46E"| Asvin Gothandaraman
| bgcolor="#BCE2FE"|[[#April 10| Kindergarten GAGA]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#November 30| Deformation Theory]]
 
|-
 
|-
| bgcolor="#E0E0E0"| April 17
+
| bgcolor="#E0E0E0"| December 5
 
| bgcolor="#C6D46E"| Soumya Sankar
 
| bgcolor="#C6D46E"| Soumya Sankar
| bgcolor="#BCE2FE"|[[#April 17| Inseparable maps and quotients of varieties]]
+
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#December 5| One Step Closer to <math>B_{cris}</math>]]
 +
|-
 +
| bgcolor="#E0E0E0"| December 12
 +
| bgcolor="#C6D46E"| Sun Woo Park
 +
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar Fall 2018#December 12| A Survey of Newton Polygons]]
 +
|}
 +
</center>
 +
 
 +
== September 12 ==
 +
<center>
 +
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#E0E0E0"| April 24
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Moisés Herradón Cueto'''
| bgcolor="#C6D46E"| Wendy Cheng
 
| bgcolor="#BCE2FE"|[[#April 24| Introduction to Boij-S&#246;derberg Theory]]
 
 
|-
 
|-
| bgcolor="#E0E0E0"| May 1
+
| bgcolor="#BCD2EE" align="center" | Title: Hodge Theory: One hour closer to understanding what it's about
| bgcolor="#C6D46E"| Shengyuan Huang
+
|-
| bgcolor="#BCE2FE"|[[#May 1| Orbifold Singular Cohomology]]
+
| bgcolor="#BCD2EE"  | 
|}
+
Abstract:
 +
 
 +
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>
 +
 
 +
== September 19 ==
 +
<center>
 +
{| 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="#BCD2EE" align="center" | Title: Linear Resolutions of Edge Ideals
 +
|-
 +
| bgcolor="#BCD2EE"  |  
 +
Abstract:
 +
 
 +
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>
  
== February 6 ==
+
== September 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%" | '''Vladimir Sotirov'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Qiao He'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Heisenberg Groups and the Fourier Transform
+
| bgcolor="#BCD2EE"  align="center" | Title: An Elementary Introduction to Geometric Langlands
 
|-
 
|-
 
| bgcolor="#BCD2EE"  |   
 
| bgcolor="#BCD2EE"  |   
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.
+
Abstract:  
  
 +
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>
  
== February 13 ==
+
== October 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%" | '''David Wagner'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Wanlin Li'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: DG potpourri
+
| bgcolor="#BCD2EE"  align="center" | Title: Gonality of Curves and More
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: I will explain how differential graded categories made their way into AG as a way to solve some of the inadequacies of the ordinary derived category. We will then give examples of the utility of DG techniques.
+
| bgcolor="#BCD2EE"  |  
<br />
+
Abstract:  
[[File:Dg-meme.png|center]]
+
 
 +
I will introduce an invariant, gonality of curves, from the definition, properties to its applications on modular curves.  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== February 20 ==
+
== October 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%" | '''Caitlyn Booms'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Ewan Dalby'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Completions of Noncatenary Local Domains and UFDs
+
| bgcolor="#BCD2EE"  align="center" | Title: A Gentle introduction to Grothendiecks Galois theory
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: A ring is called catenary if for any pair of prime ideals P contained in Q, all saturated chains of prime ideals between P and Q have the same length.
+
| bgcolor="#BCD2EE"  |  
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.
+
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.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== February 27 ==
+
== October 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%" | '''Sun Woo Park'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Johnnie Han'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Baker's Theorem
+
| bgcolor="#BCD2EE"  align="center" | Title: Schubert Calculus
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: As a follow up talk to last semester, I will state and prove Baker’s theorem, a classical theorem which uses Newton Polygons to count the genus of a given curve. I will also briefly sketch how one can use Newton polygons to understand resolution of singularities of a given curve over valuation rings.
+
| bgcolor="#BCD2EE"  |  
<br />
+
Abstract:  
[[File:Sun_woo_baker.png|500px|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>
  
== March 6 ==
+
== October 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%" | '''Connor Simpson'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Solly Parenti'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Mason's Conjectures and Chow Rings of Matroids
+
| bgcolor="#BCD2EE"  align="center" | Title: Quadratic Polynomials
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: A matroid is a combinatorial structure that abstracts many different notions of independence, including linear independence in a vector space. Mason's Conjectures are a series of three increasingly strong inequalities on certain numerical invariants of matroids, the weakest of which resisted proof for over 40 years until its resolution in 2015. However, in the years since, all of Mason's conjectures have become theorems!
+
| bgcolor="#BCD2EE"  |  
 +
Abstract:  
  
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).
+
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>
 
</center>
  
== March 13 ==
+
== October 31 ==
 
<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 239: Line 285:
 
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Brandon Boggess'''
 
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Brandon Boggess'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Dial M_1,1 for moduli
+
| bgcolor="#BCD2EE"  align="center" | Title: How to Parameterize Elliptic Curves and Influence People
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: We will speculate wildly about what kind of object the moduli space of elliptic curves should be. I don't know what a stack is, and I promise not to try to define one.
+
| bgcolor="#BCD2EE"  |  
<br/>
+
Abstract:  
[[File:Dial-M-For-Elliptic.png|400px|center]]
+
 
 +
A classical guide to classifying curves for the geometrically minded grad student. I will assume basically zero AG background.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== March 27 ==
+
== November 7 ==
 
<center>
 
<center>
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
 
|-
 
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Solly Parenti'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Vladimir Sotirov'''
 +
|-
 +
| bgcolor="#BCD2EE"  align="center" | Title: Morita duality and local duality
 +
|-
 +
| bgcolor="#BCD2EE"  | 
 +
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>
 +
 
 +
== November 14 ==
 +
<center>
 +
{| 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="#BCD2EE" | Title: Quadratic Forms
+
| bgcolor="#BCD2EE"  align="center" | Title: Homological Projective Duality
 
|-
 
|-
| 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:  
 +
 
 +
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.  
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== April 3 ==
+
== November 21 ==
 
<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%" | '''TBD'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Riemann-Roch and Abel-Jacobi theory on a finite graph
+
| bgcolor="#BCD2EE"  align="center" | Title: TBD
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: A finite graph is like a Riemann surface, or a smooth projective curve. Following the paper by Baker and Norine, we will investigate linear equivalence of divisors on graphs, the Jacobian of a graph, and a combinatorial interpretation of these as a chip firing game.
+
| bgcolor="#BCD2EE"  |  
 +
Abstract:  
 +
 
 +
TBD
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== April 10 ==
+
== November 28 ==
 
<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%" | '''Asvin G'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Kindergarten GAGA
+
| bgcolor="#BCD2EE"  align="center" | Title: Deformation Theory
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: Join me in regressing to an infantile state as we discuss Serre's 1956 paper ''Algebraic geometry and analytic geometry'', widely considered to be the most influential work ever authored by a mathematician under the age of five. We will define the notion of an analytic space, construct the analytic space associated to any algebraic variety over <math>\mathbb{C}</math>, and examine the relationships between the two, including the equivalence between coherent algebraic sheaves and coherent analytic sheaves in the projective case.
+
| bgcolor="#BCD2EE"  |  
 +
Abstract:  
  
[[File:Badromancehof.png|500px|center]]
+
I will explain what deformation theory is and how to use it by doing a few examples.
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== April 17 ==
+
== November 7 ==
 
<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%" | '''TBD'''
 
|-
 
|-
| bgcolor="#BCD2EE" | Title: Inseparable maps and quotients of varieties
+
| bgcolor="#BCD2EE"  align="center" | Title: TBD
 
|-
 
|-
| bgcolor="#BCD2EE"  | Abstract: The theory of inseparable maps is inseparable from the study of varieties in positive characteristic, as are quotients of varieties by wonderfully non-reduced group schemes. I will talk about the theory of derivations and Lie algebras and how these are helpful in understanding both the structure of inseparable maps, as well as group-scheme actions on varieties.
+
| bgcolor="#BCD2EE"  |  
 +
Abstract:  
  
[[File:Prime_Characteristic.jpg|500px|center]]
+
TBD
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
== April 24 ==
+
== December 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%" | '''Wendy Cheng'''
+
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Soumya Sankar'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Introduction to Boij-S&#246;derberg Theory
+
| bgcolor="#BCD2EE"  align="center" | Title: One Step Closet to <math>B_{cris}</math>
 
|-
 
|-
| 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.
+
| 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>
 
</center>
  
== May 1 ==
+
== December 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%" | '''Sun Woo Park'''
 
|-
 
|-
| bgcolor="#BCD2EE"  | Title: Orbifold Singular Cohomology
+
| bgcolor="#BCD2EE"  align="center" | Title: A Survey of Newton Polygons
 
|-
 
|-
| 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.
+
| 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.)
 +
 
 
|}                                                                         
 
|}                                                                         
 
</center>
 
</center>
  
 
== Organizers' Contact Info ==
 
== Organizers' Contact Info ==
 +
  
 
[https://sites.google.com/wisc.edu/cbooms/ Caitlyn Booms]
 
[https://sites.google.com/wisc.edu/cbooms/ Caitlyn Booms]
  
 
[http://www.math.wisc.edu/~drwagner/ David Wagner]
 
[http://www.math.wisc.edu/~drwagner/ David Wagner]
 +
  
 
== Past Semesters ==
 
== Past Semesters ==
 +
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2019 Spring 2019]
 +
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]
 
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2018 Fall 2018]
  

Revision as of 12:48, 29 August 2019

When: Wednesdays 4:10pm

Where: Van Vleck B215 (Fall 2018)

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.)


Autumn 2018

Date Speaker Title (click to see abstract)
September 12 Moisés Herradón Cueto Hodge Theory: One hour closer to understanding what it's about
September 19 Caitlyn Booms Linear Resolutions of Edge Ideals
September 26 Qiao He An Elementary Introduction to Geometric Langlands
October 3 Wanlin Li Gonality of Curves and More
October 10 Ewan Dalby A Gentle introduction to Grothendieck's Galois theory
October 17 Johnnie Han Schubert Calculus
October 24 Solly Parenti Quadratic Polynomials
October 31 Brandon Boggess How to Parameterize Elliptic Curves and Influence People
November 7 Vladimir Sotirov Morita Duality and Local Duality
November 14 David Wagner Homological Projective Duality
November 21 A turkey/Smallpox TBD
November 28 Asvin Gothandaraman Deformation Theory
December 5 Soumya Sankar One Step Closer to [math]B_{cris}[/math]
December 12 Sun Woo Park A Survey of Newton Polygons

September 12

Moisés Herradón Cueto
Title: Hodge Theory: One hour closer to understanding what it's about

Abstract:

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!

September 19

Caitlyn Booms
Title: Linear Resolutions of Edge Ideals

Abstract:

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.

September 26

Qiao He
Title: An Elementary Introduction to Geometric Langlands

Abstract:

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.

October 3

Wanlin Li
Title: Gonality of Curves and More

Abstract:

I will introduce an invariant, gonality of curves, from the definition, properties to its applications on modular curves.

October 10

Ewan Dalby
Title: A Gentle introduction to Grothendiecks Galois theory

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.

October 17

Johnnie Han
Title: Schubert Calculus

Abstract:

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.

October 24

Solly Parenti
Title: Quadratic Polynomials

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.

October 31

Brandon Boggess
Title: How to Parameterize Elliptic Curves and Influence People

Abstract:

A classical guide to classifying curves for the geometrically minded grad student. I will assume basically zero AG background.

November 7

Vladimir Sotirov
Title: Morita duality and local duality

Abstract:

I will explain what it means for a ring to admit a dualizing module and how to construct such for nice local rings.

November 14

David Wagner
Title: Homological Projective Duality

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.

November 21

TBD
Title: TBD

Abstract:

TBD

November 28

Asvin G
Title: Deformation Theory

Abstract:

I will explain what deformation theory is and how to use it by doing a few examples.

November 7

TBD
Title: TBD

Abstract:

TBD

December 5

Soumya Sankar
Title: One Step Closet to [math]B_{cris}[/math]

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.

December 12

Sun Woo Park
Title: A Survey of Newton Polygons

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.)

Organizers' Contact Info

Caitlyn Booms

David Wagner


Past Semesters

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015