Department of

Mathematics


Seminar Calendar
for Algebraic Geometry Seminar events the year of Friday, September 14, 2018.

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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 2018           September 2018          October 2018    
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                        30                                         

Tuesday, January 16, 2018

4:00 pm in Illini Hall 1,Tuesday, January 16, 2018

Organizational meeting

Abstract: We'll have a cookies party while deciding what'll be in the seminar this semester.

Tuesday, January 23, 2018

4:00 pm in Illini Hall 1,Tuesday, January 23, 2018

Moduli space of compact Riemann surfaces

Jin Hyung To (UIUC)

Abstract: We will overview the moduli space of compact Riemann surfaces.

Tuesday, January 30, 2018

4:00 pm in Illini Hall 1,Tuesday, January 30, 2018

Riemann-Roch Formula and The Dimension of Our Universe

Lutian Zhao   [email] (UIUC)

Abstract: In this talk we'll introduce the classical Riemann-Roch formula, which appears as a vast generalization of the Euler-Maclaurin formula for the integrals. As an interesting application, the critical dimension for the bosonic string theory can be calculated by these formula to be d=26, which matches with the physical prediction using light-cone quantization. No basic knowledge on string theory and Riemann-Roch will be assumed.

Tuesday, February 6, 2018

4:00 pmTuesday, February 6, 2018

Cancelled

Abstract: Cancelled

Tuesday, February 13, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, February 13, 2018

A noncommutative McKay correspondence

Chelsea Walton (UIUC)

Abstract: The aim of this talk is two-fold-- (1) to recall the classic McKay correspondence in the commutative/ classic setting for group actions on polynomial rings, and (2) to present a generalization of the McKay correspondence in the noncommutative/ quantum setting of Hopf algebra actions on noncommutative analogues of polynomial rings. Many notions will be defined from scratch during the talk and pre-talk, and useful examples will be provided during the pre-talk.

4:00 pm in Illini Hall 1,Tuesday, February 13, 2018

Introduction to Cohomological Field Theory

Sungwoo Nam (UIUC)

Abstract: Cohomological field theory(CohFT) was first introduced by Kontsevich and Manin to organize the data of Gromov-Witten theory and quantum cohomology into a list of axioms. Although its main model is Gromov-Witten theory, it has been also successful dealing with problems outside of Gromov-Witten theory. In this talk, I will introduce the notion of CohFT, Givental-Teleman’s classification of semisimple CohFTs and some concrete examples. Basic knowledge of Gromov-Witten theory will be helpful, but it is not assumed in this talk.

Tuesday, February 20, 2018

4:00 pm in Illini Hall 1,Tuesday, February 20, 2018

A Bird's-Eye View of Seiberg Witten Integrable Systems

Matej Penciak (UIUC)

Abstract: In this talk I will give a rudimentary description of supersymmetric gauge theories, and focus on the particular case of $N=2$ supersymmetry in dimension $4$ with gauge group $SU(2)$. In this setting, originally noticed and explained by Seiberg and Witten in 1994, the moduli of vacua exhibits the structure of an algebraic integrable system. I will explain how this structure manifests itself, and the give a sketch of the calculation that Seiberg and Witten made in their original paper. If time permits, I will explain the generalization of this story to more general gauge groups, and with possible additional matter fields included in the theory.

Tuesday, February 27, 2018

4:00 pm in 1 Illini Hall,Tuesday, February 27, 2018

Introduction to GIT

Itziar Ochoa de Alaiza Gracia (UIUC)

Abstract: The aim of this talk is to give the motivation for the GIT quotient. We will do so by introducing different notions of quotients, illustrated by examples. Finally we will define the Affine and projective GIT quotients.

Tuesday, March 6, 2018

3:00 pm in Illini Hall 2,Tuesday, March 6, 2018

(normally ordered) Tensor product of Tate objects and decomposition of higher adeles

Aron Heleodoro (Northwestern University)

Abstract: In this talk I will introduce the different tensor products that exist on Tate objects over vector spaces (or more generally coherent sheaves on a given scheme). As an application, I will explain how these can be used to describe higher adeles on an n-dimensional smooth scheme. Both Tate objects and higher adeles would be introduced in the talk. (This is based on joint work with Braunling, Groechenig and Wolfson.)

4:00 pm in 1 Illini Hall,Tuesday, March 6, 2018

Introduction to GIT, II

Itziar Ochoa de Alaiza Gracia

Abstract: The aim of this talk is to give the motivation for the GIT quotient. We will do so by introducing different notions of quotients, illustrated by examples. Finally we will define the Affine and projective GIT quotients.

Tuesday, March 13, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, March 13, 2018

Constructible 1-motives

Simon Pepin Lehalleur (Freie Universität Berlin)

Abstract: Thanks to the work of Voevodsky, Morel, Ayoub, Cisinski and Déglise, we have at our disposal a mature theory of triangulated categories of mixed motivic sheaves with rational coefficients over general base schemes, with a "six operations" formalism and the expected relationship with algebraic cycles and algebraic K-theory. A parallel development has taken place in the study of Voevodsky's category of mixed motives over a perfect field, where the subcategory of 1-motives (i.e., generated by motives of curves) has been completely described by work of Orgogozo, Barbieri-Viale, Kahn and Ayoub. We explain how to combine these two sets of ideas to study the triangulated category of 1-motivic sheaves over a base. Our main results are the definition of the motivic t-structure for constructible 1-motivic sheaves, a precise relation with Deligne 1-motives, and the extraction of the "1-motivic part" of a general motivic sheaves via a "motivic Picard functor".

4:00 pm in 1 Illini Hall,Tuesday, March 13, 2018

Algebraic Morse theory from GIT

Jesse Huang (UIUC)

Abstract: Birational geometry is closely tied to GIT quotients and variations. In this episode of GIT series, I will apply the machinery to a countable set of basic examples, through which we shall see how the change of linearization produces elementary birational transformations.

Tuesday, March 27, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, March 27, 2018

A gentle approach to the de Rham-Witt complex

Akhil Mathew (The University of Chicago)

Abstract: The de Rham-Witt complex of a smooth algebra over a perfect field provides a chain complex representative of its crystalline cohomology, a canonical characteristic zero lift of its algebraic de Rham cohomology. We describe a simple approach to the construction of the de Rham-Witt complex. This relates to a homological operation L\eta_p on the derived category, introduced by Berthelot and Ogus, and can be viewed as a toy analog of a cyclotomic structure. This is joint work with Bhargav Bhatt and Jacob Lurie.

Tuesday, April 3, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, April 3, 2018

Equal sums of higher powers of binary quadratic forms, I

Bruce Reznick (UIUC)

Abstract: We will describe all non-trivial solutions to the equation $f_1^d(x,y) + f_2^d(x,y) = f_3^d(x,y) + f_4^d(x,y)$ for quadratic forms $f_j \in \mathbb C[x,y]$. No particular prerequisites are needed and tools will be derived during the talk. Lots of fun stuff. The content of the second talk, next week, will be shaped by the reaction to this one.

Tuesday, April 10, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, April 10, 2018

Equal sums of higher powers of binary quadratic forms, II

Bruce Reznick (UIUC)

Abstract: A continuation of last week, but newcomers are welcome and will be brought up to speed. I promise to completely satisfy any curiosity you might have about the representation of binary sextic forms as a sum of two cubes of binary forms and as a sum of three cubes of binary forms. A theorem about universal representations as a sum of three cubes resolves the first non-trivial case of a conjecture of Boris Shapiro.

4:00 pm in 1 Illini Hall,Tuesday, April 10, 2018

GIT quotients of flag varieties

Joshua Wen (UIUC)

Abstract: As we’ve seen, GIT quotients depend on a choice of line equivariant line bundle, and varying this choice can lead to drastic or subtle changes between quotients. After introducing a framework for ‘variation of GIT’ by Dolgachev and Hu, I want to consider a case of the flag variety and its action either by a torus or semisimple group. Here, one already knows many equivariant line bundles, and studying dimensions of invariant sections leads to results of representation-theoretic significance.

Tuesday, April 24, 2018

4:00 pm in 1 Illini Hall,Tuesday, April 24, 2018

Moduli of Twisted Curve

Hao Sun (UIUC)

Abstract: I'll give an introductory talk about the twisted curves. Twisted curves are related to the study of r-spin Witten classes and r-spin geometry of the moduli space of curves.

Tuesday, August 28, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, August 28, 2018

Organizational Meeting

The ghost of Arthur Coble (?)

Wednesday, August 29, 2018

4:00 pm in 2 Illini Hall,Wednesday, August 29, 2018

Organizational Meeting

Tuesday, September 4, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, September 4, 2018

Tamely ramified geometric Langlands correspondence in positive characteristic

Shiyu Shen (UIUC Math)

Abstract: I will describe a generic version of tamely ramified geometric Langlands correspondence (GLC) in positive characteristic for $GL_n$, generalizing the work of Bezrukavnikov-Braverman on the unramified case. Let $X$ be a smooth projective curve over an algebraically closed field $k$ of characteristic $p>n$. I will give a spectral description of the parabolic Hitchin fiber over an open subset of the Hitchin base, and describe a correspondence between flat connections with regular singularities on $X$ and twisted Higgs bundles on the Frobenius twist $X^{(1)}$. Then I will explain how to use a twisted version of Fourier-Mukai transform to establish the GLC.

Wednesday, September 5, 2018

4:00 pm in 2 Illini Hall,Wednesday, September 5, 2018

(Crystalline) differential operators in positive characteristic

Shiyu Shen (UIUC Math)

Abstract: I will talk about several features of (Crystalline) differential operators in characteristic $p$, including the Azumaya property and two theorems by Cartier.

Tuesday, September 11, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, September 11, 2018

A geometric model for complex analytic equivariant elliptic cohomology

Arnav Tripathy (Harvard)

Abstract: Elliptic cohomology has always been a natural big brother to ordinary cohomology and K-theory and is often implicated in the trichotomies of integrable systems or geometric representation theory. However, computations with elliptic cohomology are often made difficult by the fact that we do not know geometric representatives for elliptic cohomology classes. I will explain in this talk a step forward, in joint work with D. Berwick-Evans, for the case of equivariant elliptic cohomology over the complex numbers by using geometric constructions inspired by supersymmetric field theory. This talk will need no prior knowledge of either elliptic cohomology or field theories.

Wednesday, September 12, 2018

4:00 pm in 2 Illini Hall,Wednesday, September 12, 2018

Differential Equations from Hodge Theory

Lutian Zhao   [email] (UIUC Math)

Abstract: The classical theory of elliptic integrals is the milestone in the history of various fields in math: algebraic geometry, differential equations, number theory,.. etc. In this talk, I'll use this as the motivating example for the theory of periods. I'll talk about how we get some equations for the periods and interpret these equations in terms of Hodge theory. As an interesting application, I'll calculate the number of rational curves on quintic threefold by these differential equations. Only complex analysis is assumed.

Wednesday, September 19, 2018

4:00 pm in 2 Illini Hall,Wednesday, September 19, 2018

Window equivalences and spherical functors

Jesse Huang   [email] (UIUC Math)

Abstract: We will appreciate some recent results on the derived categories of GIT quotients through the basic example of a flip/flop.

Tuesday, September 25, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, September 25, 2018

Kasteleyn Operators from Mirror Symmetry

Eric Zaslow (Northwestern University)

Abstract: Kenyon-Okounkov-Sheffied showed that the statistical properties of dimer configurations on bipartite graphs on a two-torus are determined by a spectral curve. Goncharov-Kenyon showed how edge weights of the graph define a cluster integrable system. I will show how both of these results follow from sheaf quantization in the context of homological mirror symmetry (HMS): the spaces involved are moduli spaces of objects in two categories related by HMS. This talk is based on joint work with David Treumann and Harold Williams.

Wednesday, September 26, 2018

4:00 pm in 2 Illini Hall,Wednesday, September 26, 2018

Line bundles on abelian varieties

Matej Penciak   [email] (UIUC Math)

Abstract: In this talk I want to give classical results on line bundles on abelian varieties. I’ll begin with the Appel-Humbert theorem. Then I will give an introduction to theta functions and show that they can be identified with sections of line bundles. This will connect the older approach with the more well-known description in terms of divisors. Finally, I’ll consider more general problem of classifying vector bundles on elliptic curves, and describe Atiyah’s solution to the classification problem.

Tuesday, October 2, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, October 2, 2018

S(ymplectic) duality

Justin Hilburn

Abstract: In this talk I would like to briefly sketch how one can use the tools of derived symplectic geometry and holomorphically twisted gauge theories to derive a relationship between symplectic duality and local Langlands. Our starting point will be an observation due to Gaiotto-Witten that a 3d N=4 theory with a G-flavor symmetry is a boundary condition for 4d N=4 SYM with gauge group G. By examining the relationship between boundary observables and bulk lines we will be able to derive constructions originally due to Braverman, Finkelberg, Nakajima. By examine the relationship between boundary lines and bulk surface operators one can derive new connections to local geometric Langlands. This is based on joint work with PhilsangYoo, Tudor Dimofte, and Davide Gaiotto.

Wednesday, October 3, 2018

4:00 pm in 2 Illini Hall,Wednesday, October 3, 2018

Flops and derived categories of threefolds, Part 1

Sungwoo Nam (UIUC Math)

Abstract: In his paper, Bridgeland showed that derived categories of threefolds, which are related by flopping operations, are equivalent. Besides its own interest, this result can be used to study behavior of invariants of threefolds under birational morphisms. In this talk, we will present Bridgeland's work for two weeks. As the main idea involves constructing flop as a moduli space of perverse point sheaves, I'll introduce some notions such as derived categories and t-structures and their properties relevant to the proof. After that, I will give application of the theorem on birational Calabi-Yau threefolds and curve counting invariants.

Tuesday, October 9, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, October 9, 2018

On the Hitchin fibration for algebraic surfaces

Tsao-Hsien Chen (University of Chicago)

Abstract: In his work on non-abelian Hodge theory, Simpson constructs the Hitchin map from the moduli spaces of Higgs bundles over an arbitrary smooth algebraic variety X to an affine space, generalizing Hitchin's construction in the case of when X is a Riemann surface. Very little is known about the geometry of Simpson's Hitchin map except in the case when X is one-dimensional. In the talk I will report on some recent developments on the structure of Hitchin map for higher dimensional varieties with emphasis on the case of algebraic surfaces. Joint work with B.C. Ngo.

Wednesday, October 10, 2018

4:00 pm in 2 Illini Hall,Wednesday, October 10, 2018

Flops and derived categories of threefolds, part 2

Ciaran O'Neill (UIUC Math)

Abstract: We will give the proof of Bridgelands theorem, stated last time. We will introduce Fourier-Mukai transforms, an important part of the proof.

Tuesday, October 16, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, October 16, 2018

A Spectral Description of the Ruijsenaars-Schneider System

Matej Penciak (UIUC Math)

Abstract: The Ruijsenaars-Schneider (RS) integrable hierarchy is a many-particle system which can be viewed as a relativistic analogue of the Calogero-Moser system. The integrability and Lax form of the system has been known since it was introduced by Ruijsenaars and Schneider. In this talk I will give background on the RS system, and some classical results on elliptic functions. Then I will explain work in preparation that identifies the RS system and its Lax matrix in terms of spectral sheaves living in the total space of projective bundles on cubic curves. This work provides input to a larger project (some of it joint with David Ben-Zvi and Tom Nevins), and I will give an outline for why this spectral description will be useful in the larger project.

Wednesday, October 17, 2018

4:00 pm in 2 Illini Hall,Wednesday, October 17, 2018

Derived categories of abelian categories and applications

Jin Hyung To

Abstract: TBD

Wednesday, October 24, 2018

4:00 pm in 2 Illini Hall,Wednesday, October 24, 2018

Derived categories of abelian categories, II

Jin Hyung To

Abstract: TBA

Tuesday, October 30, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, October 30, 2018

Orientation data for coherent sheaves on local $\mathbb{P}^2$

Yun Shi (UIUC Math)

Abstract: Orientation data is an ingredient in the definition of Motivic Donaldson-Thomas (DT) invariant. Roughly speaking, it is a square root of the virtual canonical bundle on a moduli space. It has been shown that there is a canonical orientation data for the stack of quiver representations for a quiver with potential. In this talk, I will briefly introduce Motivic DT invariant, and the role of orientation data in its definition. I will then give a construction of orientation data for the stack of coherent sheaves on local $\mathbb{P}^2$ based on the canonical orientation data from quiver representations.

Wednesday, October 31, 2018

4:00 pm in 2 Illini Hall,Wednesday, October 31, 2018

Congruences of modular forms from an algebro-geometric perspective

Ningchuan Zhang (UIUC Math)

Abstract: In this talk, I’ll give an algebro-geometric explanation of congruences of normalized Eisenstein series following chapter 4 of Nicholas Katz’s paper “$p$-adic properties of modular schemes and modular forms”. The key idea in Katz’s paper is to establish a $p$-adic Riemann-Hilbert correspondence that can translate congruences of normalized Eisenstein series to that of continuous $\mathbb{Z}_p^\times$-representations in rank $1$ free $\mathbb{Z}_p$-modules. The latter is very easy to compute given that $\mathbb{Z}_p^\times$ is topologically cyclic when $p\neq 2$.

Wednesday, November 7, 2018

4:00 pm in 2 Illini Hall,Wednesday, November 7, 2018

Model theory and ideas from algebraic geometry

Chieu Minh Tran (UIUC Math)

Abstract: In this (very soft) talk, I will give a model theory crash course and explain how ideas from classical (or ancient?) algebraic geometry have played an important role in shaping the field as we know it today.

Tuesday, November 13, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, November 13, 2018

Severi degrees via representation theory

Yaim Cooper (IAS)

Abstract: The Severi degrees of $\mathbb{P}^1 \times \mathbb{P}^1$ can be computed in terms of an explicit operator on the Fock space $F[\mathbb{P}^1]$. We will discuss this and variations on this theme. We will explain how to use this approach to compute the relative Gromov-Witten theory of other surfaces, such as Hirzebruch surfaces and $E \times\mathbb{P}^1$. We will also discuss operators for calculating descendants. Joint with R. Pandharipande.

Friday, November 16, 2018

4:00 pm in 243 Altgeld Hall,Friday, November 16, 2018

De Rham epsilon lines and the epsilon connection

Michael Groechenig (University of Toronto)

Abstract: De Rham epsilon lines for holonomic D-modules on curves were introduced by Deligne and Beilinson-Bloch-Esnault. This formalism includes a product formula, expressing the determinant of cohomology of a holonomic D-module as a tensor product of the epsilon lines computed with respect to a non-zero rational 1-form. Patel generalised the theory of de Rham epsilon factors to arbitrary dimensions. A curious feature of BBE’s 1-dimensional theory, is the epsilon connection which appears when studying the variation of the epsilon lines on the space of non-zero 1-forms. In this talk I will explain how properties of algebraic K-theory yield a conjectural candidate for the epsilon connection in arbitrary dimensions.

Tuesday, November 27, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, November 27, 2018

To Be Announced

Yajnaseni Dutta (Northwestern University)

Tuesday, December 11, 2018

3:00 pm in 243 Altgeld Hall,Tuesday, December 11, 2018

To Be Announced

Yaim Cooper (IAS)