Graduate Algebraic Geometry Seminar Fall 2017

When: Wednesdays 4:00pm

Where:Van Vleck TBD

Lizzie the OFFICIAL mascot of GAGS!!

Who: YOU!!

Why: The purpose of this seminar is to learn algebraic geometry 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.

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 DJ, 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.

Wish List

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.

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)

David Mumford "Picard Groups of Moduli Problems" (an early paper delving into the geometry of algebaric stacks)

Spring 2017

Date	Speaker	Title (click to see abstract)
January 18	Nathan Clement	TBD
January 25	Nathan Clement	TBD
February 1	TBD	TBD
February 8	TBD	TBD
February 15	TBD	TBD
February 22	TBD	TBD
March 1	TBD	TBD
March 8	TBD	TBD
March 15	TBD	TBD
March 22	Spring Break	No Seminar.
March 29	TBD	TBD
April 5	TBD	TBD
April 12	TBD	TBD
April 19	TBD	TBD
April 26	TBD	TBD

September 14

DJ Bruce

Title: Vignettes In Algebraic Geometry

Abstract:

Algebraic geometry is a massive forest, and it is often easy to become lost in the thicket of technical detail and seemingly endless abstraction. The goal of this talk is to take a step back out of these weeds, and return to our roots as algebraic geometers. By looking at three different classical problems we will explore various parts of algebraic geometry, and hopefully motivate the development of some of its larger machinery. Each problem will slowly build with no prerequisite assumed of the listener in the beginning.

September 21

Moisés Herradón Cueto

Title: Hilbert's 21 and The Riemann-Hilbert correspondence

Abstract: Enough with the algebra! Away with the schemes and categories! Consider a differential equation with some singularities, such as y'=1/x. Analysis tells us that its solutions can be extended along paths on the complex plane, but when a path loops around the singular point, 0 in this case, the solution might change. This phenomenon is called monodromy. Hilbert's twenty-first problem asks about the possible inverse of the monodromy construction: if some monodromy is prescribed on the plane with some points removed, is there a nice (Fuchsian), linear differential equation whose solutions have this monodromy? Attempting to solve this problem will quickly take us back to our cozy algebraic geometry world of sheaves and vector bundles. For those of us to whom the word sheaves produces a cold sweat running down our backs, this topic is a great way to motivate and introduce sheaves, and will ultimately give us a reason to care about nontrivial vector bundles.

No knowledge (or ignorance) of sheaves is required and the analysis in the talk will be contained in the tiny amount that I myself know.

September 28

Moisés Herradón Cueto

Title: Hilbert's 21 and The Riemann-Hilbert correspondence

Abstract: Enough with the algebra! Away with the schemes and categories! Consider a differential equation with some singularities, such as y'=1/x. Analysis tells us that its solutions can be extended along paths on the complex plane, but when a path loops around the singular point, 0 in this case, the solution might change. This phenomenon is called monodromy. Hilbert's twenty-first problem asks about the possible inverse of the monodromy construction: if some monodromy is prescribed on the plane with some points removed, is there a nice (Fuchsian), linear differential equation whose solutions have this monodromy? Attempting to solve this problem will quickly take us back to our cozy algebraic geometry world of sheaves and vector bundles. For those of us to whom the word sheaves produces a cold sweat running down our backs, this topic is a great way to motivate and introduce sheaves, and will ultimately give us a reason to care about nontrivial vector bundles.

No knowledge (or ignorance) of sheaves is required and the analysis in the talk will be contained in the tiny amount that I myself know.

October 5

No Talk This Week

Title: Research Computing in Algebra

Abstract: This weeks seminar conflicts with the "Research Computing in Algebra" workshop, and so instead we will not be having seminar this week. Instead we encourage everyone -- but especially those with little computational experience -- to go and learn how computation plays a major role in the research of your algebra peers, and how you can begin to integrate computation into your own research. Contact Steve Goldstein for more information.

October 12

Nathan Clement

Title: Spectral Curves and Higgs Bundles

Abstract: I will present some of the backround motivation for the study of Higgs Bundles, mainly pertaining to Nigel Hitchen's 1987 paper. I will then introduce the spectral curve associated to an operator and describe the relevant geometry.

October 19

Nathan Clement

Title: Spectral Curves and Blowups

Abstract: Continuing on from last time, I will now take a closer look at the geometry of the spectral curve. The main construction will be the lifting of a spectral curve to a blow up of the ambient surface, and the main tool for studying the geometry of this new spectral curve will be intersection theory in a surface.

October 26

Andrei Caldararu

Title: What is Mirror Symmetry?

Abstract: Mirror Symmetry is a surprising discovery made in physics around 1992. Its main initial statement was the conjecture that one can calculate certain enumerative invariants (curve counts) on a Calabi-Yau threefolds by carying out an apparently unrelated calculation (solving a differential equation) related to a very different Calabi-Yau threefold. Later, two mathematical explanations of mirror symmetry were proposed, one algebraic by Maxim Kontsevich (Homological Mirror Symmetry) and one geometric by Strominger-Yau-Zaslow.

November 2

Daniel Erman

Title: Deformation Theory

Abstract: Deformation Theory, What does it know? Does it know things? Let's find out!

November 9

Brandon Boggess

Title: Quasicoherent Sheaves and Saturation

Abstract: Given a module, one can form a quasicoherent sheaf on an affine scheme. In much the same way, we can get a quasicoherent sheaf on a projective scheme from any graded module. Unlike in the affine case, this construction fails to give an equivalence of categories. We will examine this construction and explore how saturation can fix this problem.

November 16

Wanlin Li

Title: Gonality of modular curves in characteristic p

Abstract: My talk is based on Bjorn Poonen's paper with this title. He gave a proof of given a bound on gonality, there are only finitely many modular curves in characteristic p. The same result for characteristic 0 was given by Abramovich in 1966. I will sketch the proof in this talk. This paper used Technics from both number theory and algebraic geometry.

November 23

No Seminar This Week

Title: Enjoy Thanksgiving!

Abstract: n/a

November 30

TBD

Title: TBD

Abstract: TBD

December 7

David Wagner

Title: Generic Freeness and the Dimension of Fibres

Abstract: The fact that the image of a projective variety is closed was known in some special cases as early as Newton, who gave ingenious methods for computing equations of the image (by hand!!). There is no need, though, to ask only about the set of positive-dimensional fibres; somewhat more generally, and under very modest assumptions about the schemes in question, the dimension of fibres is semi-continuous on the source (i.e. only jumps up). Guided carefully by David Eisenbud, we begin by proving the generic freeness lemma of Grothendieck and then pass on to the thoroughly lovely Chevalley's Theorem. After accepting a few basic facts about dimension (plus more theorems), our pastoral traipse through the domain of commutative algebra will be basically self-contained.