Department of

Mathematics


Seminar Calendar
for Algebraic Geometry seminar events the year of Tuesday, February 19, 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.
     January 2019          February 2019            March 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.

Tuesday, April 23, 2019

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

Double ramification cycles for target varieties

Rahul Pandharipande (ETH Zurich)

Abstract: A basic question in the theory of algebraic curves is whether a divisor represents the zeros and poles of a rational function. An explicit solution in terms of periods was given by the work of Abel and Jacobi in the 19th century. In the past few years, a different approach to the question has been pursued: what is the class in the moduli of pointed curves of the locus of such divisors? The answer in Gromov-Witten theory is given by Pixton's formula for the double ramification cycle. I will discuss recent work with F. Janda, A. Pixton, and D. Zvonkine which considers double ramification cycles for target varieties X (where Pixton's original question is viewed as the X=point case). I will also discuss the associated relations studied by Y. Bae.