Graduate Algebraic Geometry Seminar Spring 2022: Difference between revisions

Revision as of 00:26, 15 January 2022

When: TBD

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.

Spring 2022 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.

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

January 29

Colin Crowley

Title: Lefschetz hyperplane section theorem via Morse theory

Abstract: Morse theory allows you to learn about the topology of a manifold by studying the critical points of a nice function on the manifold. This perspective produces a nice proof of the theorem in the title, which concerns the homology of smooth projective varieties over C. I'll explain what the theorem says, say something about what Morse theory is and why it's related, and then finish with a neat example. I'm aiming to make this understandable to someone who's taken algebraic geometry 1 and topology 1.

February 5

Asvin Gothandaraman

Title: An introduction to unirationality

Abstract: I will introduce the notion of unirationality and show that cubic hypersurfaces are unirational (following Kollar). If time permits, I will also show that unirational varieties are simply connected.

February 12

Qiao He

Title:

Abstract:

February 19

Dima Arinkin

Title: Blowing down, blowing up: surface geometry

Abstract:A big question in algebraic geometry is how much one can change a variety without affecting it `generically'. More precisely, if two varieties are birational, how far can they be from being isomorphic?

The question is trivial for (smooth projective) curves: they are birational if and only if they are isomorphic. In higher dimension, the situation is much more interesting. The most fundamental operation are the `blowup', which is a kind of alteration of a variety within its birational isomorphism class, and its opposite, the blowdown.

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

February 26

Connor Simpson

Title: Intro to Toric Varieties

Abstract: A brief introduction to toric varieties: how to build them, formulas for computing topological data, toric blow-ups, and more.

March 4

Peter Wei

Title: An introduction to Grothendieck-Riemann-Roch Theorem

Abstract: The classical Riemann-Roch theorem tells you about how topological (genus) and analytical (through line bundle) properties on compact Riemann surface (i.e. smooth projective curve) relate to each other. Moreover, this theorem can be generalized to any vector bundles (or coherent sheaves) over any smooth projective varieties. Eventually, Grothendieck “relativized” this theorem as a property of a morphism between two projective varieties. In this talk I will introduce basic notions to formulate this theorem. If time permitting, enough examples will be given appropriately.

March 11

Caitlyn Booms

Title: Intro to Stanley-Reisner Theory

Abstract: Stanley-Reisner theory gives a dictionary between combinatorial objects (simplicial complexes) and algebraic objects (Stanley-Reisner rings). In this talk, I will introduce the main objects of study in this theory, describe this dictionary with several examples, and discuss how Stanley-Reisner theory can help us investigate algebra-geometric questions.

March 25

Steven He

Title: Braid group action on derived category

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.

April 1

Vlad Sotirov

Title:

Abstract:

April 8

Maya Banks

Title:

Abstract:

April 15

Alex Hof

Title: Embrace the Singularity: An Introduction to Stratified Morse Theory

Abstract: Early on in the semester, Colin told us a bit about Morse

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

April 22

Ruofan

Title: Birational geometry: existence of rational curves

Abstract: Rational curves on a variety control its birational geometry. It thus is important to determine whether they exist. People didn’t know how to do this systematically, before Mori discovered a deformation lemma which detect their existence, and bound their degree if they exist. I will briefly introduce Mori’s insight.