Department of

# Mathematics

Seminar Calendar
for Algebraic Geometry events the year of Monday, April 22, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
      March 2019             April 2019              May 2019
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2       1  2  3  4  5  6             1  2  3  4
3  4  5  6  7  8  9    7  8  9 10 11 12 13    5  6  7  8  9 10 11
10 11 12 13 14 15 16   14 15 16 17 18 19 20   12 13 14 15 16 17 18
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24 25 26 27 28 29 30   28 29 30               26 27 28 29 30 31
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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.

4:00 pm in 245 Altgeld Hall,Wednesday, February 6, 2019

#### Systems of Calogero-Moser Type

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

Abstract: It is well known that many-particle systems are in general not solvable analytically. For some specific choices of interactions between particles though, a lot can be said. In this talk I aim to give an introduction to systems of Calogero-Moser type and the surprising role of algebraic geometry in their solvability. I will also give a perspective on how this subject plays a role in some hot topics in mathematics in general: Hitchin integrable systems, geometric representation theory, and the geometric Langlands philosophy.

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.

Friday, February 22, 2019

4:00 pm in 145 Altgeld Hall,Friday, February 22, 2019

#### 27 lines on smooth cubic surfaces

###### Ningchuan Zhang (UIUC)

Abstract: In this talk, I will show that there are $27$ projective lines on a smooth cubic surface in $\mathbb{CP}^3$ by a Chern class computation. This talk is based on a course project I did with Professor Sheldon Katz in Math 524 (now 514) in Spring 2015. No knowledge of algebraic geometry or characteristic classes is assumed.

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.

Thursday, February 28, 2019

4:00 pm in 245 Altgeld Hall,Thursday, February 28, 2019

#### Quivers, representation theory and geometry

###### Kevin McGerty (University of Oxford and Visiting Fisher Professor, University of Illinois)

Abstract: A quiver is an oriented graph. It has a natural algebra associated to it called the path algebra, which as the name suggests has a basis given by paths in the quiver with multiplication given by concatenation. The representation theory of these algebras encompasses a number of classical problems in linear algebra, for example subspace arrangements and Jordan canonical form. A remarkable discovery of Gabriel however in the 1970s revealed a deep connection between these algebras and Lie theory, which has subsequently lead to a rich interaction between quivers, Lie theory and algebraic geometry. This talk will begin by outlining the elementary theory of representations of path algebras, explain Gabriel's result and survey some of the wonderful results which it has led to in Lie theory: the discovery of the canonical bases of quantum groups, the geometric realization of representations of affine quantum groups by Nakajima, and most recently deep connections between representations of symplectic reflection algebras and affine Lie algebras.

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.

Thursday, March 7, 2019

11:00 am in 241 Altgeld Hall,Thursday, March 7, 2019

#### Diophantine problems and a p-adic period map

###### Brian Lawrence (University of Chicago)

Abstract: I will outline a proof of Mordell's conjecture / Faltings's theorem using p-adic Hodge theory. I'll start with a discussion of cohomology theories in algebraic geometry, and build from there. The paper is joint with Akshay Venkatesh.

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

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.