Department of

# Mathematics

Seminar Calendar
for Graduate Student Algebraic events the year of Tuesday, January 1, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    December 2018           January 2019          February 2019
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Wednesday, January 16, 2019

3:00 pm in 2 Illini Hall,Wednesday, January 16, 2019

#### Organizational Meeting

###### Sungwoo Nam (UIUC Math)

Wednesday, January 23, 2019

3:00 pm in 2 Illini Hall,Wednesday, January 23, 2019

#### Torelli Theorem for curves

###### Lutian Zhao   [email] (UIUC Math)

Abstract: Jacobians are parametrizing the degree 0 line bundles. By sending a curve to its Jacobian we can get a polarized Abelian variety. The Torelli Theorem states we can reverse this map, i.e. for a polarized Abelian variety we can reconstruct the same curve. In this talk, I’ll start from Jacobian and prove the theorem. If time permitted, I’ll define the Torelli map for nodal curves.

Wednesday, January 30, 2019

3:00 pm in 2 Illini Hall,Wednesday, January 30, 2019

#### Canceled

Wednesday, February 6, 2019

3:00 pm in Altgeld Hall,Wednesday, February 6, 2019

#### Murphy's law in Hilbert scheme

###### Sungwoo Nam (Illinois Math)

Abstract: One feature of moduli space is that although it parametrizes nice objects like smooth projective curves, it can be quite bad. In this talk, we will see lots of instances of these phenomena(mostly involving lots of cohomology computations) focusing on Hilbert scheme of curves in a projective space. I'll end with a discussion on Mumford's famous pathological example and Murphy's law formulated by Vakil.

Wednesday, February 13, 2019

3:00 pm in 2 Illini Hall,Wednesday, February 13, 2019

#### Equivariant Cohomology

###### Ciaran O'Neill (Illinois Math)

Abstract: I’ll define equivariant cohomology and give some basic examples. Then I’ll go into more detail for the case of a torus action on projective space.

Wednesday, February 20, 2019

3:00 pm in 2 Illini Hall,Wednesday, February 20, 2019

#### The Geometry of Spectral Curves

###### Matej Penciak (Illinois Math)

Abstract: One way of encoding the data of an integrable system is in terms of the spectral curves. From the curves, it is possible to obtain the constants of motion as integrals over cycles in the curves. In this talk, I will explain some of these classical aspects of integrable systems through some worked out examples. I will also introduce an action-coordinate (AC) duality for integrable systems. I will show how AC duality can be used to relate well-known integrable systems and even construct new integrable systems from old ones. Finally, I hope to describe what the action this AC duality has on spectral curves for some integrable systems of interest.

Wednesday, February 27, 2019

3:00 pm in 2 Illini Hall,Wednesday, February 27, 2019

#### Dieudonné crystals associated to formal groups

###### Ningchuan Zhang (Illinois Math)

Abstract: In this talk, I will introduce Dieudonné crystals associated to commutative formal group schemes. The focus of this talk will be on the construction of the contravariant Dieudonné crystal functor and explicit computation of some examples. I'll also mention its relation with extensions and deformations of formal groups if time allows.

Wednesday, March 6, 2019

3:00 pm in 2 Illini Hall,Wednesday, March 6, 2019

#### Abelian Varieties in Positive Characteristic

###### Ravi Donepudi (Illinois Math)

Abstract: This talk will be an introduction to the theory of abelian varieties over fields of positive characteristic. The presence of the non-separable Frobenius automorphism in this context gives the theory a flavor entirely different from over the complex numbers. An important question in this area is to characterize which abelian varieties (with extra data) arise as Jacobians of smooth curves. Much of the progress on this problem has been through studying some stratifications of moduli spaces of abelian varieties. We will introduce these moduli spaces and stratifications, and survey interesting results in this area.

Wednesday, March 13, 2019

3:00 pm in 2 Illini Hall,Wednesday, March 13, 2019

#### What are matrix factorizations?

###### Jesse Huang (Illinois Math)

Abstract: A matrix factorization is, roughly speaking, what looks like AB=fId where f is a polynomial and every square matrix in the equation takes value in the polynomial ring. This notion was originally introduced in the study of homological algebra on (singular) complete intersections and then generalized and made into a younger sibling of the derived category of coherent sheaves. The state-of-the-art consolidates the study of things like hypersurface singularities and (A to B) mirror symmetry for non-CYs. I will try to showcase some basics and survey through a handful of well-known results in this talk.

Wednesday, March 27, 2019

3:00 pm in 2 Illini Hall,Wednesday, March 27, 2019

#### Intersection Theory I - Rational Equivalence

###### Martino Fassina (Illinois Math)

Abstract: This is the first talk for our reading group on Intersection Theory. The material presented roughly corresponds to Chapter 1 of Fulton's book. I will introduce concepts such as cycles, rational equivalence, proper pushforwards and flat pullbacks. The focus will be on intuition and explicit examples.

Wednesday, April 3, 2019

3:00 pm in 2 Illini Hall,Wednesday, April 3, 2019

#### Intersection Theory II

###### Yidong Chen (Illinois Physics)

Abstract: In this talk, I'll follow chapter 2 of Fulton's book and talk about divisors, pseudo-divisors, and how to intersect with divisors. As an application, I'll discuss Chern class of line bundles. With time permitting, I'll move towards the definition of Chern class of vector bundles, but will most definitely leave the actual work to the next speaker.

Wednesday, April 10, 2019

3:00 pm in 2 Illini Hall,Wednesday, April 10, 2019

#### Intersection Theory III - Chern classes of vector bundles

Abstract: In this talk, based on chapter 3 of Fulton's "Intersection Theory", I will introduce Segre classes and Chern classes, and outline some of their basic properties. I will also discuss a few interesting examples and special cases.

Wednesday, April 17, 2019

3:00 pm in 2 Illini Hall,Wednesday, April 17, 2019

#### Intersection Theory IV

###### Jin Hyung To (Illinois Math)

Abstract: We study Section 4. We construct the Segre class of a closed subscheme which is a cycle class of the subscheme.

Wednesday, April 24, 2019

3:00 pm in 2 Illini Hall,Wednesday, April 24, 2019

#### Intersection Theory V-Intersection Products

###### Sungwoo Nam (Illinois Math)

Abstract: In this talk, we will see the important construction of deformation to the normal cone, which is an analog of the tubular neighborhood theorem in algebraic geometry. Using this, we will define intersection product with a regular codimension d subvariety, generalizing intersection with a divisor introduced in the second talk. Time permitting, we will see how to understand the number 3264 from the intersection theory point of view.

Wednesday, August 28, 2019

4:00 pm in Altgeld Hall 447,Wednesday, August 28, 2019

#### Organizational Meeting

Wednesday, September 4, 2019

4:00 pm in Altgeld Hall 447,Wednesday, September 4, 2019

#### Counting rational curves on K3 surfaces and modular forms

###### Sungwoo Nam (Illinois Math)

Abstract: Curve counting invariants on K3 surfaces turn out to have an interesting connection to modular forms via Yau-Zaslow formula. In this talk, starting from the basic properties of K3 surfaces, I’ll discuss two proofs of Yau-Zaslow formula due to Beauville which uses Euler characteristic of compactified Jacobian, and Bryan-Leung using Gromov-Witten technique. If time permits, I’ll describe generalizations of the formula such as Göttsche’s formula and Katz-Klemm-Vafa formula.

Wednesday, September 11, 2019

4:00 pm in 447 Altgeld Hall,Wednesday, September 11, 2019

#### Compactified Jacobian

###### Lutian Zhao (Illinois Math)

Abstract: In this talk, I will start by a brief review of the history of Jacobians. Then I will describe the definition of compactified Jacobian as they are crucial object in studying singular curves. The final goal is to understand a calculation on the compactified Jacobian of the curve $x^p-y^q=0$ for $p,q$ coprime, where the Euler characteristic of the Compactified Jacobian is exactly the Catalan number $C_{p,q}$.

Wednesday, September 18, 2019

4:00 pm in 447 Altgeld Hall,Wednesday, September 18, 2019

#### Intro to the Gorsky-Negut wall-crossing conjecture

###### Josh Wen (Illinois Math)

Abstract: The Hilbert scheme of points on the plane is a space that by now has been connected to many areas outside of algebraic geometry: e.g. algebraic combinatorics, representation theory, knot theory, etc. The equivariant K-theory of these spaces have a few distinguished bases important to making some of these connections. A new entrant to this list of bases is the Maulik-Okounkov K-theoretic stable bases. They depend in a piece-wise constant manner by a real number called the slope, and the numbers where the bases differ are called the walls. Gorsky and Negut have a conjecture relating the transition between bases when the slope crosses a wall to the combinatorics of q-Fock spaces for quantum affine algebras. I'll try to introduce as many of the characters of this story as I can as well as discuss a larger picture wherein these stable bases are geometric shadows of things coming from deformation quantization.

Wednesday, September 25, 2019

4:00 pm in 447 Altgeld Hall,Wednesday, September 25, 2019

#### The renormalized De Rham functor

###### Ciaran O'Neill (Illinois Math)

Abstract: I’ll start with some background, then give the definition of the renormalized De Rham functor (as defined by Drinfeld and Gaitsgory). This comes with a natural transformation to the ordinary De Rham functor. I’ll mention how this can potentially be used to prove Kirwan surjectivity in certain circumstances. There will also be an example or two.

Wednesday, October 2, 2019

4:00 pm in 447 Altgeld Hall,Wednesday, October 2, 2019

#### Intro to Gromov-Witten invariants

Abstract: Gromov-Witten invariants (often) count the number of curves (= Riemann surfaces) of a fixed genus in a projective variety (= nice complex manifold). I will introduce these invariants and compute a few examples.

Wednesday, October 23, 2019

4:00 pm in 447 Altgeld Hall,Wednesday, October 23, 2019

#### Deformations of the Frobenius

###### William Balderrama (Illinois Math)

Abstract: The deformation theory of the Frobenius homomorphism of a group scheme in positive characteristic is rich, and gives rise to various interesting algebraic structures. I'll introduce this problem, talk about the example of the multiplicative group, from which one naturally obtains Witt vectors and the notion of a delta ring, and talk about what is known about the case of elliptic curves.

Wednesday, October 30, 2019

4:00 pm in 447 Altgeld,Wednesday, October 30, 2019

#### A Case for Étale Cohomology

###### Justin Kelm (Illinois Math)

Abstract: This talk will serve as an introduction to étale cohomology. In particular, I will focus on the shortcomings of Zariski cohomology, why étale maps are a good algebro-geometric substitute for the notion of a "covering map" or "local" diffeomorphism, the technology that étale cohomlogy is built out of, and a basic computation or two that should illuminate why étale cohomology should give rise to a useful Weil cohomology theory.

Wednesday, November 20, 2019

4:00 pm in 447 Altgeld Hall,Wednesday, November 20, 2019

#### Geometry and arithmetic of curves over finite fields

###### Ravi Donepudi (Illinois Math)

Abstract: The theory of algebraic curves over a finite field runs entirely parallel to the classical theory of number fields (finite extensions of the rational numbers). Analogues of many results that are long standing open conjectures in the number field case are theorems in the case of curves over finite fields. We will introduce the key concepts in this area, survey important results and (time permitting) state some original results. This talk assumes only a passing familiarity with finite fields.