Department of

Mathematics


Seminar Calendar
for Algebraic Geometry Seminar events the year of Wednesday, September 11, 2019.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2019           September 2019          October 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.

Tuesday, January 29, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, January 29, 2019

Non-reduced Parabolic Group Schemes

William Haboush (UIUC Math)

Abstract: In the 90’s I and my student N. Lauritzen described all possible non reduced parabolic subgroup schemes of a semisimple algebraic group. These lead to complete homogeneous spaces with very interesting properties. Among other things they provide counterexamples which were crucial to the Mori program. Now that the Lusztig conjecture has been shown to be completely false I am revisiting this material hoping to make some interesting contribution to the decomposition problem for Weyl modules.

Wednesday, January 30, 2019

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

Canceled

Tuesday, February 5, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, February 5, 2019

Non-reduced Parabolic Group Schemes, II

William Haboush (UIUC Math)

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.

Tuesday, February 19, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, February 19, 2019

Symplectic Springer theory

Kevin McGerty (University of Oxford and UIUC)

Abstract: One of the classical results of geometric representation theory is Springer's realization of representations of a Weyl group in the cohomology of the vanishing locus of nilpotent vector fields on the associated flag variety. A rich strain of current research focuses on attempting to extend aspects of Lie theory to the more general context of ``conical symplectic resolutions''. We will discuss, based on the discovery of Markman and Namikawa that such varieties have a natural analogue of a Weyl group, to what extent one can build an analogue of Springer's theory in this context, recovering for example a construction of Weyl group actions on the cohomology of quiver varieties, first discovered by Nakajima, which unlike previous construction does not require painful explicit verification of the braid relation.

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.

Tuesday, February 26, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, February 26, 2019

Pure cohomology of multiplicative quiver varieties

Thomas Nevins (UIUC)

Abstract: Multiplicative quiver varieties are certain quasiprojective algebraic varieties, defined by Crawley-Boevey and Shaw, associated to quivers. Examples include many moduli spaces of surface group representations (with punctures), a.k.a. moduli spaces of connections on punctured surfaces. I will introduce the basics of these varieties and explain joint work with McGerty that describes generators of the Hodge-theoretically "pure" part of their cohomology rings.

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.

Tuesday, April 9, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, April 9, 2019

Quantization of algebraic exact Lagrangians in cotangent bundles

Christopher Dodd (UIUC Math)

Abstract: Exact Lagrangians play an important role in symplectic topology; in algebraic geometry they seem to be almost unstudied. In this talk I’ll explain some recent results about their structure and in particular I’ll show that, in the affine case, they admit certain canonical noncommutative deformations. Time permitting I’ll explain how this implies the vanishing of certain invariants in their de Rham cohomology.

Wednesday, April 10, 2019

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

Intersection Theory III - Chern classes of vector bundles

Nachiketa Adhikari (Illinois Math)

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.

Tuesday, April 16, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, April 16, 2019

Stokes decompositions and wild monodromy

Philip Boalch (Orsay)

Abstract: Just like a Hodge structure can be described equivalently in terms of the Hodge filtration or the Hodge decomposition, a Stokes structure has several equivalent descriptions. The best known are the Stokes filtrations and the Stokes local systems (or wild monodromy representations). In this talk I will explain how to formalise the notion of {\em Stokes decompositions}, to intermediate between them. This is part of an attempt (the Lax project) to understand the bestiary of complete hyperkahler manifolds that occur as moduli spaces of algebraic Higgs bundles on the affine line.

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.

Tuesday, April 23, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, April 23, 2019

Virtual Euler characteristics of Quot scheme of surfaces

Rahul Pandharipande (ETH Zurich)

Abstract: Let S be a nonsingular projective surface. Quot schemes of quotients on S with supports of dimensions 0 and 1 always have 2-term obstruction theories (and therefore also have natural virtual fundamental classes). I will explain what we know about the virtual Euler characteristics in this theory: theorems, conjectures, and a lot of examples. Joint work with Dragos Oprea.

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.

Tuesday, May 7, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, May 7, 2019

Construction of the Poincare sheaf on the stack of Higgs bundles

Mao Li (University of Wisconsin)

Abstract: An important part of the Langlands program is to construct the Hecke eignsheaf for irreducible local systems. Conjecturally, the classical limit of the Hecke eignsheaves should correspond to the Poincare sheaf on the stack of Higgs bundles. The Poincare sheaf for the compactified Jacobian of reduced planar curves have been constructed in the pioneering work of Dima Arinkin. In this talk I will present the construction of the Poincare sheaf on the stack of rank two Higgs bundles for any smooth projective curve over the entire Hitchin base, and it turns out to be a maximal Cohen-Macaulay sheaf. This includes the case of nonreduced spectral curves, and thus provides the first example of the existence of the Poincare sheaf for nonreduced planar curves.

Monday, May 13, 2019

3:00 pm in 243 Altgeld Hall,Monday, May 13, 2019

Kirwan-Ness stratifications in algebraic geometry

Itziar Ochoa (Yale University)

Abstract: Given an algebraic variety $X$ with an action of a reductive group $G$, geometric invariant theory splits $X$ as the disjoint union $X=X^{ss}\sqcup X^{un}$ of the semistable and unstable locus. The Kirwan-Ness stratification refines $X$ even more by describing $X^{un}$ as a disjoint union of strata $X^{un}=\displaystyle\sqcup_{\beta\in\textsf{KN}} S_\beta$ determined by 1-parameter subgroups $\beta$. In this talk we will describe an algorithm that finds the $\beta$'s and show that such algorithm can be simplified when our space is of the form $T^*V$ where $V$ is a vector space.

Tuesday, August 27, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, August 27, 2019

Organizational Meeting

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.

Tuesday, September 10, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, September 10, 2019

Deformation theory and partition Lie algebras

Akhil Mathew (U Chicago)

Abstract: A theorem of Lurie and Pridham states that over a field of characteristic zero, derived "formal moduli problems" (i.e., deformation functors defined on derived Artinian commutative rings), correspond precisely to differential graded Lie algebras. This formalizes a well-known philosophy in deformation theory, and arises from Koszul duality between Lie algebras and commutative algebras. I will report on joint work with Lukas Brantner, which studies the analogous situation for arbitrary fields. The main result is that formal moduli problems are equivalent to a category of "partition Lie algebras"; these are algebraic structures (which agree with DG Lie algebras in characteristic zero) which arise from a monad built from the partition complex.

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 $C_{p,q}$.

Tuesday, September 17, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, September 17, 2019

P=W, a strange identity for Hitchin systems

Zili Zhang (U Michigan)

Abstract: Start with a compact Riemann surface X with marked points and a complex reductive group G. According to Hitchin-Simpson’s nonabelian Hodge theory, the pair (X,G) comes with two new complex varieties: the character variety M_B and the Higgs moduli M_D. I will present some aspects of this story and discuss an identity P=W indexed by affine Dynkin diagrams – occurring in the singular cohomology groups of M_D and M_B, where P and W dwell. Based on joint work with Junliang Shen.

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.

Tuesday, September 24, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, September 24, 2019

Motivic Chern classes and Iwahori invariants of principal series

Changjian Su (University of Toronto)

Abstract: Let G be a split reductive p-adic group. In the Iwahori invariants of an unramified principal series representation of G, there are two bases, one of which is the so-called Casselman basis. In this talk, we will prove conjectures of Bump, Nakasuji and Naruse about certain transition matrix coefficients between these two bases. The ingredients of the proof involve Maulik and Okounkov's stable envelopes and Brasselet--Schurmann--Yokura's motivic Chern classes for the complex Langlands dual groups. This is based on joint work with P. Aluffi, L. Mihalcea and J. Schurmann.

Tuesday, October 1, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, October 1, 2019

To Be Announced

Michael Brown (UW-Madison)

Abstract: TBA

Tuesday, October 8, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, October 8, 2019

To Be Announced

Nick Rozenblyum (U Chicago)

Abstract: TBA

Tuesday, October 15, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, October 15, 2019

To Be Announced

Claudiu Raicu (University of Notre Dame)

Abstract: TBA

Tuesday, October 22, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, October 22, 2019

To Be Announced

Wenliang Zhang (UIC)

Abstract: TBA

Tuesday, November 5, 2019

3:00 pm in 243 Altgeld Hall,Tuesday, November 5, 2019

To Be Announced

Ben Antieau (UIC)

Abstract: TBA

Tuesday, December 3, 2019

1:00 pm in 243 Altgeld Hall,Tuesday, December 3, 2019

To Be Announced

Andrei Negut (MIT)

Abstract: TBA