Department of

# Mathematics

Seminar Calendar
for Algebraic Geometry 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.
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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.

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.

Thursday, September 5, 2019

3:00 pm in 347 Altgeld Hall,Thursday, September 5, 2019

#### Polytopes, polynomials and recent results in 1989 mathematics

###### Bruce Reznick   [email] (University of Illinois at Urbana-Champaign)

Abstract: Hilbert’s 17th Problem discusses the possibility of writing polynomials in several variables which only take non-negative values as a sum of squares of polynomials. One approach is to substitute squared monomials into the arithmetic-geometric inequality. Sometimes this is a sum of squares, sometimes it isn’t, and I proved 30 years ago that this depends on a property of the polytope whose vertices are the exponents of the monomials in the substitution. What’s new here is an additional then-unproved claim in that paper and its elementary, but non-obvious proof. This talk lies somewhere in the intersection of combinatorics, computational algebraic geometry and number theory and is designed to be accessible to first year graduate students.

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.

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.

Friday, September 27, 2019

4:00 pm in 345 Altgeld Hall,Friday, September 27, 2019

#### O-minimal complex analysis according to Peterzil–Starchenko (Part 1)

###### Lou van den Dries (UIUC)

Abstract: This is the first of two survey talks on the subject of the title. Neer (and others?) will follow up with a more detailed treatment in later talks. O-minimal complex analysis is one way that ideas from o-minimality have been used in recent work in arithmetic algebraic geometry (Pila, Zannier, Tsimerman, Klingler,…), the other one being the Pila–Wilkie theorem. The two topics relate because important objects like the family of Weierstrass p-functions turn out to be "o-minimal".

Tuesday, October 1, 2019

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

#### Standard Conjecture D for matrix factorizations

Abstract: In 1968, Grothendieck posed a family of conjectures concerning algebraic cycles called the Standard Conjectures. The conjectures have been proven in some special cases, but they remain open in general. In 2011, Marcolli-Tabuada realized two of these conjectures as special cases of more general conjectures, involving differential graded categories, which they call Noncommutative Standard Conjectures C and D. The goal of this talk is to discuss a proof, joint with Mark Walker, of Noncommutative Standard Conjecture D in a special case which does not fall under the purview of Grothendieck's original conjectures: namely, in the setting of matrix factorizations.

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.

Friday, October 4, 2019

4:00 pm in 345 Altgeld Hall,Friday, October 4, 2019

#### O-minimal complex analysis according to Peterzil–Starchenko (Part 2)

###### Lou van den Dries (UIUC)

Abstract: This is the first of two survey talks on the subject of the title. Neer (and others?) will follow up with a more detailed treatment in later talks. O-minimal complex analysis is one way that ideas from o-minimality have been used in recent work in arithmetic algebraic geometry (Pila, Zannier, Tsimerman, Klingler,…), the other one being the Pila–Wilkie theorem. The two topics relate because important objects like the family of Weierstrass p-functions turn out to be "o-minimal".

Tuesday, October 8, 2019

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

#### Character stacks and shtukas in the topological setting

###### Nick Rozenblyum (U Chicago)

Abstract: I will describe a general categorical framework leading to shtukas (in the sense of Drinfeld) and excursion operators (in the sense of V. Lafforgue) on moduli spaces. In particular, I will give a concrete description of the space of functions on (derived) character varieties. I will explain how this leads to the spectral action in the context of Betti geometric Langlands and (conjecturally) to the spectral decomposition in geometric Langlands over finite fields via a categorification of Grothendieck's function-sheaf correspondence. This is joint work with Gaitsgory, Kazhdan, and Varshavsky.

Tuesday, October 15, 2019

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

#### Koszul Modules and Green’s Conjecture

###### Claudiu Raicu (University of Notre Dame)

Abstract: Formulated in 1984, Green’s Conjecture predicts that one can recognize the intrinsic complexity of an algebraic curve from the syzygies of its canonical embedding. Green’s Conjecture for a general curve has been resolved in two landmark papers by Voisin in the early 00s. I will explain how the recent theory of Koszul modules provides more elementary solutions to this problem, by relating it to the study of the syzygies of some very concrete surfaces. Joint work with M. Aprodu, G. Farkas, S. Papadima, S. Sam and J. Weyman.

Friday, October 18, 2019

4:00 pm in 141 Altgeld Hall,Friday, October 18, 2019

#### Lines in Space

###### Brian Shin (UIUC)

Abstract: Consider four lines in three-dimensional space. How many lines intersect these given lines? In this expository talk, I'd like to discuss this classical problem of enumerative geometry. Resolving this problem will give us a chance to see some interesting algebraic geometry and algebraic topology. If time permits, I'll discuss connections to motivic homotopy theory.

Tuesday, October 22, 2019

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

#### Structure of local cohomology modules associated with projective varieties

###### Wenliang Zhang (UIC)

Abstract: Let R be a polynomial ring over a field and I be an ideal of R. The local cohomology modules H^j_I(R) are rarely finitely generated as R-modules. However, they have finite length when viewed as objects in the category of D-modules in characteristic 0 (or in the category of F-modules in characteristic p). Computing the actual length (in the appropriate category) has been an open problem; it is also an open problem just to determine whether they are simple objects (in the appropriate category). In this talk, I will explain a solution to this problem when the ideal I is a homogeneous prime ideal and the projective variety Proj(R/I) has mild singularities in characteristic p. This is a joint work with Nicholas Switala.

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.

Friday, October 25, 2019

2:00 pm in 347 Altgeld Hall,Friday, October 25, 2019

#### Noncommutative algebra from a geometric point of view

###### Xingting Wang (Howard University)

Abstract: In this talk, I will discuss how to use algebro-geometric and Poisson geometric methods to study the representation theory of 3-dimensional Sklyanin algebras, which are noncommutative analogues of polynomial algebras of three variables. The fundamental tools we are employing in this work include the noncommutative projective algebraic geometry developed by Artin-Schelter-Tate-Van den Bergh in 1990s and the theory of Poisson order axiomatized by Brown and Gordon in 2002, which is based on De Concini-Kac-Priocesi’s earlier work on the applications of Poisson geometry in the representation theory of quantum groups at roots of unity. This is joint work with Milen Yakimov and Chelsea Walton.

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.

Tuesday, November 5, 2019

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

#### Rational points and derived equivalence

###### Ben Antieau (UIC)

Abstract: Suppose that X and Y are smooth projective varieties over a field k and suppose that X and Y have equivalent derived categories of sheaves. If X has a rational point, does Y have a rational point? This question was asked 10 years ago by Esnault. I will report on joint work with Addington, Frei, and Honigs which shows that, in general, the answer is ‘no’, in contrast to what happens for curves (Antieau—Krashen—Ward) or in dimension at most 3 over finite fields (Honigs).

Thursday, November 7, 2019

4:00 pm in 245 Altgeld Hall,Thursday, November 7, 2019

#### A survey of Sperner theory

###### Richard Stanley   [email] (MIT and University of Miami)

Abstract: Let $X$ be a collection of subsets of an $n$-element set $S$ such that no element of $X$ is a subset of another. In 1927 Emanuel Sperner showed that the number of elements of $X$ is maximized by taking $X$ to consist of all subsets of $S$ with $\lfloor n/2\rfloor$ elements. This result started the subject of \emph{Sperner theory}, which is concerned with the largest subset $A$ of a finite partially ordered set $P$ that forms an \emph{antichain}, that is, no two elements of $A$ are comparable in $P$. We will give a survey of Sperner theory, focusing on some connections with linear algebra and algebraic geometry.

Monday, November 11, 2019

3:00 pm in 441 Altgeld Hall,Monday, November 11, 2019

#### Introduction to Picard groups

###### Venkata Sai Narayana Bavisetty (UIUC Math)

Abstract: Picard groups have been classically studied in commutative algebra and algebraic geometry. These can be suitably generalised to define picard groups of $\mathbb{E}_{\infty}$ ring spectra. In this situation a natural question to ask is "How is the picard group of $R$ related to the picard group of $R_*$?". In this expository talk, I will explain some methods which have been used to answer the above question.

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.

Monday, December 2, 2019

3:00 pm in 441 Altgeld Hall,Monday, December 2, 2019

#### The Gross-Hopkins duality

###### Ningchuan Zhang (UIUC Math)

Abstract: In this talk, we introduce the Gross-Hopkins duality in chromatic homotopy theory, which relates the the Spanier-Whitehead duality and the Brown-Comenetz duality in the $K(n)$-local category. We will mainly focus on the underlying algebraic geometry of the duality phenomena and work out some explicit examples at height 1.

Tuesday, December 3, 2019

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

#### The Beauville-Voisin conjecture for Hilb(K3) and the Virasoro algebra

###### Andrei Negut (MIT)

Abstract: We give a geometric representation theory proof of a mild version of the Beauville-Voisin Conjecture for Hilbert schemes of K3 surfaces, namely the injectivity of the cycle map restricted to the subring of Chow generated by tautological classes. Our approach involves lifting formulas of Lehn and Li-Qin-Wang from cohomology to Chow groups, and using them to solve the problem by invoking the irreducibility criteria of Virasoro algebra modules, due to Feigin-Fuchs. Joint work with Davesh Maulik.

Wednesday, December 4, 2019

4:00 pm in 245 Altgeld Hall,Wednesday, December 4, 2019

#### Geometry and arithmetic over finite fields

###### Ravi Donepudi

Abstract: We will discuss algebraic, geometric and combinatorial approaches in the study of curves over finite fields. Our goal will be to give the audience a survey of the basic tools, questions and results in this area. We will discuss topics such as: Counting points on curves, the Weil conjectures, Jacobians and the Torelli problem. We will attempt to demonstrate that there are a variety of problems in this field ranging from those requiring heavy tools from algebraic geometry to those that can be investigated by writing a few lines of computer code. No prior knowledge is assumed beyond a passing acquaintance with the field of integers modulo a prime.