Graduate Algebraic Geometry Seminar Fall 2021: Difference between revisions

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'''
'''
'''When:''' Thursday 5:00-6:00 PM CST
'''When:''' 5:00-6:00 PM Thursdays


'''Where:''' https://uwmadison.zoom.us/j/92877740706?pwd=OVo0QmxRVEdUQ3RnUWpoWmFRRUI3dz09
'''Where:''' TBD
[[Image:cat.jpg|thumb|220px| | Lizzie the OFFICIAL mascot of GAGS!!]]
[[Image:cat.jpg|thumb|220px| | 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.
'''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.
'''Why:''' The purpose of this seminar is to learn algebraic geometry and commutative algebra by giving and listening to talks in a informal setting. Sometimes people present an interesting paper they find. Other times people give a prep talk for the [https://hilbert.math.wisc.edu/wiki/index.php?title=Algebra_and_Algebraic_Geometry_Seminar&redirect=yes 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 [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@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is [https://groups.google.com/u/2/a/g-groups.wisc.edu/g/gags here].
'''
 
''' Organizers: ''' [https://johndcobb.github.io John Cobb], [https://sites.google.com/view/colincrowley/home Colin Crowley].


== Give a talk! ==
== Give a talk! ==
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.
We need volunteers to give talks this semester. If you're interested, please fill out [https://forms.gle/sa3ARndYSkBhT6LR9 this form]. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the [https://hilbert.math.wisc.edu/wiki/index.php?title=Graduate_Algebraic_Geometry_Seminar main page].
 
=== Fall 2021 Topic Wish List ===
This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.
* Stacks for Kindergarteners
* Motives for Kindergarteners
* Applications of Beilinson resolution of the diagonal, Fourier Mukai transforms in general
* Wth did June Huh do and what is combinatorial hodge theory?
* Computing things about Toric varieties
* Reductive groups and flag varieties
* Introduction to arithmetic geometry -- what are some big picture ideas of what "goes wrong" when not over an algebraically closed field?
* Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
* Going from line bundles and divisors to vector bundles and chern classes
* A History of the Weil Conjectures
* Mumford & Bayer, "What can be computed in Algebraic Geometry?"
* A pre talk for any other upcoming talk


== Being an audience member ==
== 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:
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:
* Do Not Speak For/Over the Speaker
* Ask Questions Appropriately:
* Ask Questions Appropriately
 
== Spring 2021 ==


==Talks==
<center>
<center>
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"
|-
|-
| bgcolor="#E0E0E0"| February 4
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''
| bgcolor="#C6D46E"| Asvin Gothandaraman
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 4| A Bertini type theorem via probability]]
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
|-
| bgcolor="#E0E0E0"| February 25
| bgcolor="#C6D46E"| Colin Crowley
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 25| TBD]]
|-
| bgcolor="#E0E0E0"| March 11
| bgcolor="#C6D46E"| Roufan Jiang
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| TBD]]
|-
| bgcolor="#E0E0E0"| March 18
| bgcolor="#C6D46E"| Alex Hof
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 18| An Introduction to the Deformation Theory of Complete Intersection Singularities]]
|-
| bgcolor="#E0E0E0"| March 25
| bgcolor="#C6D46E"| Chiahui (Wendy) Cheng
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Explicit Bound on Collective Strength of Regular Sequences of Three Homogeneous Polynomials]]
|-
| bgcolor="#E0E0E0"| April 1
| bgcolor="#C6D46E"| Erika Pirnes
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Reconstruction conjecture in graph theory (Note: special time at noon!)]]
|-
| bgcolor="#E0E0E0"| April 8
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| TBD]]
|}
</center>
 
== February 4 ==
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: A Bertini type theorem via probability
|-
| bgcolor="#BCD2EE"  | Abstract:  I will prove that most hyperplane slices are irreducible over any field by reducing to finite fields and applying probabilistic arguments. The talk will be very elementary!
|}                                                                       
</center>
== February 25 ==
<center>
{| 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="#BCD2EE"  align="center" | Title: TBD
|-
| bgcolor="#BCD2EE"  | Abstract:  TDB
|}                                                                       
</center>
== March 11 ==
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Roufan Jiang'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
| bgcolor="#BCD2EE"  | Abstract:  TBD
|}                                                                       
</center>
== March 18 ==
<center>
{| 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="#BCD2EE"  align="center" | Title: An Introduction to the Deformation Theory of Complete Intersection Singularities
|-
| bgcolor="#BCD2EE"  | Abstract: Essentially what it says in the title; I'll give a fairly laid-back overview of some of the basic definitions and results about deformations of complete intersection singularities, including the Kodaira-Spencer map and the existence of versal deformations in the isolated case. If time permits, I'll discuss Morsification of isolated singularities. Very little background will be assumed.
|}                                                                       
</center>
 
== March 25 ==
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Chiahui (Wendy) Cheng'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: Explicit Bound on Collective Strength of Regular Sequences of Three Homogeneous Polynomials
|-
| bgcolor="#BCD2EE"  | Abstract: Let f_1,...,f_r in k[x_1,...,x_n] be homogeneous polynomial of degree d. Ananyan and Hochster (2016) proved that there exists a bound N=N(r,d) where if collective strength of f_1,...,f_r is greater than or equal to N, then f_1,...,f_r are regular sequence. In this paper, we study the explicit bound N(r,d) when $r=3$ and d=2,3 and show that N(3,2)=2 and N(3,3)>2.
|}                                                                       
</center>
== April 1 ==
<center>
{| 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="#BCD2EE"  align="center" | Title: Reconstruction conjecture in graph theory (Note: special time at noon!)
|-
| bgcolor="#BCD2EE"  | Abstract: The deck of a graph with n vertices is a multiset of n unlabeled graphs, each obtained from the original graph by deleting a vertex (and the edges incident to it). The reconstruction conjecture says that if two finite simple graphs with at least three vertices have the same deck, then they are isomorphic. The talk is going to focus on examples, and does not assume previous knowledge about graph theory.
|}                                                                       
</center>
 
== April 8 ==
<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: TBD
|-
| bgcolor="#BCD2EE"  | Abstract:  TBD
|}                                                                       
</center>
 
== April 29 ==
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Owen Goff'''
|-
| bgcolor="#BCD2EE"  align="center" | Title: TBD
|-
| bgcolor="#BCD2EE"  | Abstract:  TBD
|}                                                                       
</center>
 
== Fall 2020 ==
 
<center>
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"
|-
|-
| bgcolor="#E0E0E0"| September 30
| bgcolor="#E0E0E0"| September 30
| bgcolor="#C6D46E"| Asvin Gothandaraman
| bgcolor="#C6D46E"| Yifan Wei
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#September 30| Title]]
| bgcolor="#BCE2FE"|[[#September 30| On Chow groups and K groups]]
|-
|-
| bgcolor="#E0E0E0"| October 5
| bgcolor="#E0E0E0"| October 7
| bgcolor="#C6D46E"| Yifan Wei
| bgcolor="#C6D46E"| Owen Goff
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 5| On the Analytic Side (GAGA)]]
| bgcolor="#BCE2FE"|[[#October 7 | Roguish Noncommutativity and the Onsager Algebra]]
|-
|-
| bgcolor="#E0E0E0"| October 14
| bgcolor="#E0E0E0"| October 14
| bgcolor="#C6D46E"| Owen Goff
| bgcolor="#C6D46E"| Peter YI WEI
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 14| The Magic and Comagic of Hopf Algebras]]
| bgcolor="#BCE2FE"|[[#October 14 | Pathologies in Algebraic Geometry]]
|-
|-
| bgcolor="#E0E0E0"| October 21
| bgcolor="#E0E0E0"| October 21
| bgcolor="#C6D46E"| Roufan Jiang
| bgcolor="#C6D46E"| Asvin G
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 21| TBD]]
| bgcolor="#BCE2FE"|[[#October 21 | Introduction to Arithmetic Schemes]]  
|-
|-
| bgcolor="#E0E0E0"| October 28
| bgcolor="#E0E0E0"| October 28
| bgcolor="#C6D46E"| Erika Pirnes
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#October 28| Introduction to representation theory via an example]]
| bgcolor="#BCE2FE"|[[#October 28 | Classifying Varieties of Minimal Degree]]  
|-
|-
| bgcolor="#E0E0E0"| November 4
| bgcolor="#E0E0E0"| November 4
| bgcolor="#C6D46E"| Alex Mine
| bgcolor="#C6D46E"| John Cobb
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 4| The Internal Language of the Category of Sheaves]]
| bgcolor="#BCE2FE"|[[#November 4 | Syzygies and Koszul Cohomology]]  
|-
|-
| bgcolor="#E0E0E0"| November 11
| bgcolor="#E0E0E0"| November 11
| bgcolor="#C6D46E"| John Cobb
| bgcolor="#C6D46E"| Colin Crowley
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 11| Introduction to Boij-Söderberg Theory]]
| bgcolor="#BCE2FE"| [[#November 11 | Introduction to Geometric Invariant Theory]]  
|-
|-
| bgcolor="#E0E0E0"| November 18
| bgcolor="#E0E0E0"| November 23
| bgcolor="#C6D46E"| Yunfan He
| bgcolor="#C6D46E"| Connor Simpson
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 18| Introduction to mixed Hodge structure]]
| bgcolor="#BCE2FE"|[[#November 18 | Combinatorial Hodge Theory]]  
|-
| bgcolor="#E0E0E0"| November November 25
| bgcolor="#C6D46E"| Maya Banks
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#November 25| TBD]]
|-
|-
| bgcolor="#E0E0E0"| December 2
| bgcolor="#E0E0E0"| December 2
| bgcolor="#C6D46E"| Peter Wei
| bgcolor="#C6D46E"| Alex Mine
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 2| K3 Surfaces and Their Moduli]]
| bgcolor="#BCE2FE"|[[#December 2 | Fourier-Mukai Transforms]]  
|-
|-
| bgcolor="#E0E0E0"| December 9
| bgcolor="#E0E0E0"| December 9
| bgcolor="#C6D46E"| Wendy Cheng
| bgcolor="#C6D46E"| Yu Luo
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 9| TBD]]
| bgcolor="#BCE2FE"|[[#December 9 | Stacks for Kindergarteners]]  
|-
| bgcolor="#E0E0E0"| December 16
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#December 16| Characteristic Dependence of Syzygies of Random Monomial Ideals]]
|}
|}
</center>
</center>


== Spring 2020 ==
=== September 30 ===
 
<center>
{| style="color:black; font-size:120%" border="0" cellpadding="14" cellspacing="5"
|-
| bgcolor="#D0D0D0" width="300" align="center"|'''Date'''
| bgcolor="#A6B658" width="300" align="center"|'''Speaker'''
| bgcolor="#BCD2EE" width="300" align="center"|'''Title (click to see abstract)'''
|-
| bgcolor="#E0E0E0"| January 29
| bgcolor="#C6D46E"| Colin Crowley
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#January 29| Lefschetz hyperplane section theorem via Morse theory]]
|-
| bgcolor="#E0E0E0"| February 5
| bgcolor="#C6D46E"| Asvin Gothandaraman
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 5| An Introduction to Unirationality]]
|-
| bgcolor="#E0E0E0"| February 12
| bgcolor="#C6D46E"| Qiao He
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 12| Title]]
|-
| bgcolor="#E0E0E0"| February 19
| bgcolor="#C6D46E"| Dima Arinkin
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 19| Blowing down, blowing up: surface geometry]]
|-
| bgcolor="#E0E0E0"| February 26
| bgcolor="#C6D46E"| Connor Simpson
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#February 26| Intro to toric varieties]]
|-
| bgcolor="#E0E0E0"| March 4
| bgcolor="#C6D46E"| Peter
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 4| An introduction to Grothendieck-Riemann-Roch Theorem]]
|-
| bgcolor="#E0E0E0"| March 11
| bgcolor="#C6D46E"| Caitlyn Booms
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 11| Intro to Stanley-Reisner Theory]]
|-
| bgcolor="#E0E0E0"| March 25
| bgcolor="#C6D46E"| Steven He
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#March 25| Braid group action on derived categories]]
|-
| bgcolor="#E0E0E0"| April 1
| bgcolor="#C6D46E"| Vlad Sotirov
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 1| Title]]
|-
| bgcolor="#E0E0E0"| April 8
| bgcolor="#C6D46E"| Maya Banks
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 8| Title]]
|-
| bgcolor="#E0E0E0"| April 15
| bgcolor="#C6D46E"| Alex Hof
| bgcolor="#BCE2FE"|[[Graduate Algebraic Geometry Seminar#April 15| Embrace the Singularity: An Introduction to Stratified Morse Theory]]
|-
| bgcolor="#E0E0E0"| April 22
| bgcolor="#C6D46E"| Ruofan
| 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]]
|}
</center>
 
== January 29 ==
<center>
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Colin Crowley'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yifan Wei'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Lefschetz hyperplane section theorem via Morse theory
| bgcolor="#BCD2EE"  align="center" | Title: On Chow groups and K groups
|-
|-
| 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="#BCD2EE"  |  
|}                                                                       
Abstract:  
</center>


== February 5 ==
We define Chow groups and K groups for non-singular varieties, illustrate some basic properties, and explain how intersection theory is done using K groups (on a smooth surface). Then we proceed to compute the K group of a non-singular curve. On higher dimensions there might be some issues, if time permits we will show how these issues can be mitigated, and why Grothendieck-Riemann-Roch is one of the greatest theorems in algebraic geometry (in my humble opinion).
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin Gothandaraman '''
|-
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to unirationality
|-
| 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.  
|}                                                                       
</center>


== February 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%" | '''Qiao He'''
|-
| bgcolor="#BCD2EE"  align="center" | Title:
|-
| bgcolor="#BCD2EE"  | Abstract:
|}                                                                         
|}                                                                         
</center>
</center>


== February 19 ==
=== October 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%" | '''Dima Arinkin'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Owen Goff'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Blowing down, blowing up: surface geometry
| bgcolor="#BCD2EE"  align="center" | Title: Roguish Noncommutativity and the Onsager Algebra
|-
|-
| 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?
| bgcolor="#BCD2EE"  |  
Abstract:  


The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the
While throughout algebraic geometry and many other fields we like commutative rings, we often wonder what happens if our ring is not commutative. Say, for instance, you have A^2, but instead of xy=yx you have a relation xy = qyx for some constant q. In this talk I will discuss the consequences of this relation and how it relates to an object of combinatorial nature called the q-Onsager algebra.
situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.


In my talk, I will introduce blowups and discuss their properties. Then (time permitting) I would like to look deeper at the case
of surfaces, where the combination of blowups and intersection theory provides a complete and beautiful picture. (If we do get to this point, I won't assume any knowledge of intersection theory: to an extent, this talk is my excuse to introduce it.)
|}                                                                         
|}                                                                         
</center>
</center>


== February 26 ==
=== October 14 ===
<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%" | '''Peter YI WEI'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Intro to Toric Varieties
| bgcolor="#BCD2EE"  align="center" | Title: Pathologies in Algebraic Geometry
|-
|-
| bgcolor="#BCD2EE"  | Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.
| bgcolor="#BCD2EE"  |  
Abstract: This talk serves as a brief discussion on pathologies in algebraic geometry, inspired by a short thread of Daniel Litt’s twitter. No hard preliminaries! :)
 
|}                                                                         
|}                                                                         
</center>
</center>


== March 4 ==
=== October 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%" | '''Peter Wei'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Asvin G'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: An introduction to Grothendieck-Riemann-Roch Theorem
| bgcolor="#BCD2EE"  align="center" | Title: Introduction to Arithmetic Schemes
|-
|-
| 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.
| bgcolor="#BCD2EE"  |  
Abstract: Many of us are comfortable working with varieties over the complex numbers (or other fields) but part of the magic is that it's almost as easy to consider varieties over more exotic rings like the integers or the p-adics.
 
I'll explain how to think about such varieties and then use them to prove the birational invariance of Hodge numbers for Calabi-Yau's over the complex numbers using results from finite fields and p-adic analysis!
 
TBD
 
|}                                                                         
|}                                                                         
</center>
</center>


== March 11 ==
=== October 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"
Line 349: Line 156:
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Caitlyn Booms'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Intro to Stanley-Reisner Theory
| bgcolor="#BCD2EE"  align="center" | Title: Classifying Varieties of Minimal Degree
|-
|-
| 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.
| bgcolor="#BCD2EE"  |  
Abstract:  
 
The degree of a variety embedded in projective space is a well-defined invariant, and there is a sense in which some varieties have minimal degree. Long ago, Del Pezzo and Bertini classified geometrically all possible projective varieties of minimal degree. More recently, Eisenbud and Goto gave an algebraic notion that classifies such varieties. In this talk, we will introduce the necessary background and explore these two theorems and the ways they are connected.
 
|}                                                                         
|}                                                                         
</center>
</center>


== March 25 ==
=== November 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%" | '''Steven He'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Braid group action on derived category
| bgcolor="#BCD2EE"  align="center" | Title: Syzygies and Koszul Cohomology
|-
|-
| bgcolor="#BCD2EE"  | Abstract: In this talk, I will define spherical object and A_m-configuration in derived category of coherent sheaves, and say a few words about the motivation coming from the homological mirror symmetry.
| bgcolor="#BCD2EE"  |  
Abstract:  
 
Early on in the history of algebraic geometry it was recognized that many properties/invariants of projective varieties could be deduced by looking at their hyperplane sections. Starting in the 1950s, this classical picture was gradually refined into general theory by people like Serre and Kodaira — many hard-earned numbers could now be obtained by more brainless methods. I hope to motivate a few ideas introduced in the 1980’s as a continuation of this story beginning from Serre’s vanishing theorem.
|}                                                                         
|}                                                                         
</center>
</center>


== April 1 ==
=== November 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%" | '''Colin Crowley'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title:  
| bgcolor="#BCD2EE"  align="center" | Title: Introduction to Geometric Invariant Theory
|-
|-
| bgcolor="#BCD2EE"  | Abstract:  
| bgcolor="#BCD2EE"  |  
Abstract:  
 
Given a group action on a variety, is there a quotient variety?  How do you construct it? Geometric invariant theory gives partial answers to these questions for projective varieties and a particular class of groups (reductive groups). I’ll give an overview of how GIT quotients work, which will be in the language of Hartshorne chapter one and does not require any knowledge of schemes. (Although I may need to talk a little about ample line bundles. I haven't decided yet.)
 
With the remaining time I'll sketch how these ideas are used in constructing (coarse) moduli spaces of semistable vector bundles, and mention which areas of math use these ideas today.
|}                                                                         
|}                                                                         
</center>
</center>


== April 8 ==
=== November 23 ===
<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%" | '''Maya Banks'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Connor Simpson'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title:  
| bgcolor="#BCD2EE"  align="center" | Title: Combinatorial Hodge Theory
|-
|-
| bgcolor="#BCD2EE"  | Abstract:  
| bgcolor="#BCD2EE"  |  
Abstract: We'll talk about log-concave sequences of natural numbers and the
evolution of methods for proving log-concavity over the past decade.
In particular, we'll talk about "what June Huh did", then talk about
alternative roads to similar results, including recent work by Chan & Pak.
 
|}                                                                         
|}                                                                         
</center>
</center>


== April 15 ==
=== December 2 ===
<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%" | '''Alex Mine'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Embrace the Singularity: An Introduction to Stratified Morse Theory
| bgcolor="#BCD2EE"  align="center" | Title: Fourier-Mukai Transforms
|-
|-
| bgcolor="#BCD2EE"  | Abstract: Early on in the semester, Colin told us a bit about Morse
| bgcolor="#BCD2EE"  |  
Theory, and how it lets us get a handle on the (classical) topology of
Abstract:  
smooth complex varieties. As we all know, however, not everything in
 
life goes smoothly, and so too in algebraic geometry. Singular
I'll say a few things about derived category of sheaves and talk about Fourier-Mukai transforms, which are certain functors between the derived categories of sheaves on two schemes. In particular, I will try to elucidate what is so "Fourier" about them.
varieties, when given the classical topology, are not manifolds, but
 
they can be described in terms of manifolds by means of something called
a Whitney stratification. This allows us to develop a version of Morse
Theory that applies to singular spaces (and also, with a bit of work, to
smooth spaces that fail to be nice in other ways, like non-compact
manifolds!), called Stratified Morse Theory. After going through the
appropriate definitions and briefly reviewing the results of classical
Morse Theory, we'll discuss the so-called Main Theorem of Stratified
Morse Theory and survey some of its consequences.
|}                                                                         
|}                                                                         
</center>
</center>


== April 22 ==
=== December 9 ===
<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%" | '''Ruofan'''
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''Yu Luo'''
|-
|-
| bgcolor="#BCD2EE"  align="center" | Title: Birational geometry: existence of rational curves
| bgcolor="#BCD2EE"  align="center" | Title: Stacks for Kindergarteners
|-
|-
| 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.
| bgcolor="#BCD2EE"  |  
|}                                                                       
Abstract:  
</center>


== April 29 ==
After struggling for a while, a kindergartner manages to build a LEGO Death Star. One day, while our kindergartner is at school, their father manages to break it. He hurriedly buys a new set, builds it, and secretly replaces the broken Death Star. Even though our kindergartner does not know it, we know that two Death Stars are not the same. That is, even though the two Death Stars are isomorphic, they are not canonically isomorphic. Motivated by this, we define the stacks in algebraic geometry to study the moduli problem.
<center>
{| style="color:black; font-size:100%" table border="2" cellpadding="10" width="700" cellspacing="20"
|-
| bgcolor="#A6B658" align="center" style="font-size:125%" | '''John Cobb'''
|-
| bgcolor="#BCD2EE"  align="center" | Title:
|-
| bgcolor="#BCD2EE"  | Abstract:
|}                                                                         
|}                                                                         
</center>
</center>


== Past Semesters ==
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2021 Spring 2021]


== Organizers' Contact Info ==
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Fall_2020 Fall 2020]
 
[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?
[https://www.math.wisc.edu/wiki/index.php/Graduate_Algebraic_Geometry_Seminar_Spring_2020 Spring 2020]


* DG Schemes.
==Ed Dewey's Wish List Of Olde==__NOTOC__
Back in the day Ed and Nathan made this list of topics they wanted to hear. They all sound super duper cool, but it's also true that they had many years of AG behind their backs, so this list might not be very representative of what the GAGS audience wants to hear bout.
Here are the topics we're '''DYING''' to learn about!  Please consider looking into one of these topics and giving one or two GAGS talks.
===Specifically Vague Topics===
* D-modules 101: basics of D-modules, equivalence between left and right D-modules, pullbacks, pushforwards, maybe the Gauss-Manin Connection. Claude Sabbah's introduction to the subject could be a good place to start.
* Sheaf operations on D-modules (the point is that then you can get a Fourier-Mukai transform between certain O-modules and certain D-modules, which is more or less how geometric Langlands is supposed to work)
===Interesting Papers & Books===
* ''Symplectic structure of the moduli space of sheaves on an abelian or K3 surface'' - Shigeru Mukai.
* ''Residues and Duality'' - Robin Hatshorne.
** Have you heard of Serre Duality? Would you like to really understand the nuts and bolts of it and its generalizations? If so this book is for you. (You wouldn't need to read the whole book to give a talk ;).)
* ''Coherent sheaves on P^n and problems in linear algebra'' - A. A. Beilinson.
** In this two page paper constructs the semi-orthogonal decomposition of the derived category of coherent sheaves on projective space. (This topic is very important, and there are a ton of other resources for this result and the general theory of derived categories.)
* ''Frobenius splitting and cohomology vanishing for Schubert varieties'' - V.B. Mehta and A. Ramanathan.
** In characteristic p the fact that (x+y)^p=x^p+y^p means that one has the Frobenius morphism, which sends f to f^p. In this paper the authors introduce the notion of what it means for a variety to be Frobenius split, and use this to prove certain cohomologcal vanishing results for Schubert varieties. Since then Frobenius splitting -- and its related cousins (F-regularity, strong F-regularity, F-purity, etc.) have played large roles in geometry and algebra in characteristic p. This is a good place to get a sense for what kicked all this stuff off!
* ''Schubert Calculus'' - S. L. Kleiman and Dan Laksov.
** An introduction to Schubert calculus suitable for those of all ages. I am told the paper essentially only uses linear algebra!
* ''Rational Isogenies of Prime Degree'' - Barry Mazur.
** In this paper Mazur classifies all isogenies of rational elliptic curves of prime order. As a result of this he deduces his famous result that the torsion subgroup of an elliptic curve (over Q) is one of 15 abelian groups. This definitely stares into the land of number theory, but certainly would still be of interest to many.
* ''Esquisse d’une programme'' - Alexander Grothendieck.
** Originating from a grant proposal in the mid 1980's this famous paper outlines a tantalizing research program, which seeks to tie numerous different areas of math (algebraic geometry, Teichmuller theory, Galois theory, etc.) together. This is where Grothendieck introduced his famous Lego game and dessin d'enfant. While just a research proposal this paper has seemingly inspired a ton of cool math, and will allow you to "blow peoples’ minds". (The original paper is in French, but there are English translations out there.)
* ''Géométrie algébraique et géométrie analytique'' - J.P. Serre.
** A projective variety X over the complex numbers has two lives, an algebraic and an analytic, depending on which topology one wishes to work with. That is one can think about X as a complex manifold and work with holomorphic functions or as an algebraic variety and work with regular functions. Hence to any complex projective variety we have two sheaf theories and as a result two cohomology theories. In this famous paper Serre compares these two and shows they are in fact the same. (''Note: This is a super fundamental result that is used all the time; normally in the following way: Uhh... What do you mean by cohomology? Well by GAGA or something it doesn't really mater.) (The original paper is in French, but there are English translations out there.)
* ''Limit linear series: Basic theory''- David Eisenbud and Joe Harris.
** One of the more profitable tools -- especially when studying moduli spaces -- in a geometers tool box is the theory of degenerations. However, sometimes we care about more than just the variety we are degenerating and want to keep track of things like vector/line bundles. In this paper Eisenbud and Harris develop the theory of degenerating a curve together with a linear series. From this they prove a ton of cool results: M_g is of general type for g>24, Brill-Noether theory, etc.
* ''Picard Groups of Moduli Problems'' - David Mumford.
** This paper is essentially the origin of algebraic stacks.
* ''The Structure of Algebraic Threefolds: An Introduction to Mori's Program'' - Janos Kollar
** This paper is an introduction to Mori's famous ``minimal model'' program, which is a far reaching program seeking to understand the birational geometry of higher dimensional varieties.
* ''Cayley-Bacharach Formulas'' - Qingchun Ren, Jürgen Richter-Gebert, Bernd Sturmfels.
** A classical result we all learn in a first semester of algebraic geometry is that 5 points in the plane (in general position) determine a unique plane conic. One can similarly show that 9 (general) points in the plane determine a unique plane cubic curve. This paper tries to answer the question: ``What is equation for this cubic curve?''.
* ''On Varieties of Minimal Degree (A Centennial Approach)'' - David Eisenbud and Joe Harris.
** Suppose X is a projective variety embedded in projective space so that X is not contained in any hyperplane. By projecting from general points one can see that the degree of X is at least codim(X)+1. This paper discusses the classification of varieties that achieve this lower degree bound i.e. varieties of minimal degree. This topic is quite classical and the paper seems to contain a nice mixture of classical and modern geometry.
* ''The Gromov-Witten potential associated to a TCFT'' - Kevin J. Costello.
** This seems incredibly interesting, but fairing warning this paper has been described as ''highly technical'', which considering it uses A-infinity algebras and the derived category of a  Calabi-Yau seems like a reasonable description. (This paper may be covered in Caldararu's Spring 2017 topics course.)
== Past Semesters ==
[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_Fall_2019 Fall 2019]


Latest revision as of 22:57, 21 November 2021

When: 5:00-6:00 PM Thursdays

Where: TBD

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. Sometimes people present an interesting paper they find. Other times people give a prep talk for the 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@g-groups.wisc.edu by sending an email to gags+subscribe@g-groups.wisc.edu. If you prefer (and are logged in under your wisc google account) the list registration page is here.

Organizers: John Cobb, Colin Crowley.

Give a talk!

We need volunteers to give talks this semester. If you're interested, please fill out this form. Beginning graduate students are particularly encouraged to give a talk, since it's a great way to get your feet wet with the material. If you would like some talk ideas, see the list on the main page.

Fall 2021 Topic Wish List

This was assembled using input from an interest form at the beginning of the semester. Choose one and you will have the rare guarantee of having one interested audience member. Feel free to add your own.

  • Stacks for Kindergarteners
  • Motives for Kindergarteners
  • Applications of Beilinson resolution of the diagonal, Fourier Mukai transforms in general
  • Wth did June Huh do and what is combinatorial hodge theory?
  • Computing things about Toric varieties
  • Reductive groups and flag varieties
  • Introduction to arithmetic geometry -- what are some big picture ideas of what "goes wrong" when not over an algebraically closed field?
  • Geothendieck '66, "On the de Rham Cohomology of Algebraic Varieties"
  • Going from line bundles and divisors to vector bundles and chern classes
  • A History of the Weil Conjectures
  • Mumford & Bayer, "What can be computed in Algebraic Geometry?"
  • A pre talk for any other upcoming talk

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

Talks

Date Speaker Title (click to see abstract)
September 30 Yifan Wei On Chow groups and K groups
October 7 Owen Goff Roguish Noncommutativity and the Onsager Algebra
October 14 Peter YI WEI Pathologies in Algebraic Geometry
October 21 Asvin G Introduction to Arithmetic Schemes
October 28 Caitlyn Booms Classifying Varieties of Minimal Degree
November 4 John Cobb Syzygies and Koszul Cohomology
November 11 Colin Crowley Introduction to Geometric Invariant Theory
November 23 Connor Simpson Combinatorial Hodge Theory
December 2 Alex Mine Fourier-Mukai Transforms
December 9 Yu Luo Stacks for Kindergarteners

September 30

Yifan Wei
Title: On Chow groups and K groups

Abstract:

We define Chow groups and K groups for non-singular varieties, illustrate some basic properties, and explain how intersection theory is done using K groups (on a smooth surface). Then we proceed to compute the K group of a non-singular curve. On higher dimensions there might be some issues, if time permits we will show how these issues can be mitigated, and why Grothendieck-Riemann-Roch is one of the greatest theorems in algebraic geometry (in my humble opinion).

October 7

Owen Goff
Title: Roguish Noncommutativity and the Onsager Algebra

Abstract:

While throughout algebraic geometry and many other fields we like commutative rings, we often wonder what happens if our ring is not commutative. Say, for instance, you have A^2, but instead of xy=yx you have a relation xy = qyx for some constant q. In this talk I will discuss the consequences of this relation and how it relates to an object of combinatorial nature called the q-Onsager algebra.

October 14

Peter YI WEI
Title: Pathologies in Algebraic Geometry

Abstract: This talk serves as a brief discussion on pathologies in algebraic geometry, inspired by a short thread of Daniel Litt’s twitter. No hard preliminaries! :)

October 21

Asvin G
Title: Introduction to Arithmetic Schemes

Abstract: Many of us are comfortable working with varieties over the complex numbers (or other fields) but part of the magic is that it's almost as easy to consider varieties over more exotic rings like the integers or the p-adics.

I'll explain how to think about such varieties and then use them to prove the birational invariance of Hodge numbers for Calabi-Yau's over the complex numbers using results from finite fields and p-adic analysis!

TBD

October 28

Caitlyn Booms
Title: Classifying Varieties of Minimal Degree

Abstract:

The degree of a variety embedded in projective space is a well-defined invariant, and there is a sense in which some varieties have minimal degree. Long ago, Del Pezzo and Bertini classified geometrically all possible projective varieties of minimal degree. More recently, Eisenbud and Goto gave an algebraic notion that classifies such varieties. In this talk, we will introduce the necessary background and explore these two theorems and the ways they are connected.

November 4

John Cobb
Title: Syzygies and Koszul Cohomology

Abstract:

Early on in the history of algebraic geometry it was recognized that many properties/invariants of projective varieties could be deduced by looking at their hyperplane sections. Starting in the 1950s, this classical picture was gradually refined into general theory by people like Serre and Kodaira — many hard-earned numbers could now be obtained by more brainless methods. I hope to motivate a few ideas introduced in the 1980’s as a continuation of this story beginning from Serre’s vanishing theorem.

November 11

Colin Crowley
Title: Introduction to Geometric Invariant Theory

Abstract:

Given a group action on a variety, is there a quotient variety? How do you construct it? Geometric invariant theory gives partial answers to these questions for projective varieties and a particular class of groups (reductive groups). I’ll give an overview of how GIT quotients work, which will be in the language of Hartshorne chapter one and does not require any knowledge of schemes. (Although I may need to talk a little about ample line bundles. I haven't decided yet.)

With the remaining time I'll sketch how these ideas are used in constructing (coarse) moduli spaces of semistable vector bundles, and mention which areas of math use these ideas today.

November 23

Connor Simpson
Title: Combinatorial Hodge Theory

Abstract: We'll talk about log-concave sequences of natural numbers and the evolution of methods for proving log-concavity over the past decade. In particular, we'll talk about "what June Huh did", then talk about alternative roads to similar results, including recent work by Chan & Pak.

December 2

Alex Mine
Title: Fourier-Mukai Transforms

Abstract:

I'll say a few things about derived category of sheaves and talk about Fourier-Mukai transforms, which are certain functors between the derived categories of sheaves on two schemes. In particular, I will try to elucidate what is so "Fourier" about them.

December 9

Yu Luo
Title: Stacks for Kindergarteners

Abstract:

After struggling for a while, a kindergartner manages to build a LEGO Death Star. One day, while our kindergartner is at school, their father manages to break it. He hurriedly buys a new set, builds it, and secretly replaces the broken Death Star. Even though our kindergartner does not know it, we know that two Death Stars are not the same. That is, even though the two Death Stars are isomorphic, they are not canonically isomorphic. Motivated by this, we define the stacks in algebraic geometry to study the moduli problem.

Past Semesters

Spring 2021

Fall 2020

Spring 2020

Fall 2019

Spring 2019

Fall 2018

Spring 2018

Fall 2017

Spring 2017

Fall 2016

Spring 2016

Fall 2015

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