Department of

Mathematics


Seminar Calendar
for Graduate Algebraic Geometry Seminar events the year of Thursday, September 14, 2017.

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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 2017           September 2017          October 2017    
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        1  2  3  4  5                   1  2    1  2  3  4  5  6  7
  6  7  8  9 10 11 12    3  4  5  6  7  8  9    8  9 10 11 12 13 14
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 20 21 22 23 24 25 26   17 18 19 20 21 22 23   22 23 24 25 26 27 28
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Thursday, January 19, 2017

3:00 pm in 345 Altgeld Hall,Thursday, January 19, 2017

Organizational meeting

Friday, January 27, 2017

3:00 pm in 243 Altgeld Hall,Friday, January 27, 2017

Raindrop. Droptop. Symmetric functions from DAHA.

Josh Wen (UIUC Math)

Abstract: In symmetric function theory, various distinguished bases for the ring of (deformed) symmetric functions come from specifying an inner product on said ring and then performing Gram-Schmidt on the monomial symmetric functions. In the case of Jack polynomials, there is an alternative characterization as eigenfunctions for the Calogero-Sutherland operator. This operator gives a completely integrable system, hinting at some additional algebraic structure, and an investigation of this structure digs up the affine Hecke algebra. Work of Cherednik and Matsuo formalize this in terms of an isomorphism between the affine Knizhnik-Zamolodichikov (KZ) equation and the quantum many body problem. Looking at q-analogues yields a connection between the affine Hecke algebra and Macdonald polynomials by relating the quantum affine KZ equation and the Macdonald eigenvalue problem. All of this can be streamlined by circumventing the KZ equations via Cherednik's double affine Hecke algebra (DAHA). I hope to introduce various characters in this story and give a sense of why having a collection of commuting operators can be a great thing.

Friday, February 3, 2017

3:00 pm in 243 Altgeld Hall,Friday, February 3, 2017

Syzygies and Implicitization of tensor product surfaces

Eliana Duarte (UIUC Math)

Abstract: A tensor product surface is the closure of the image of a map $\lambda:\mathbb{P}^1\times \mathbb{P}^1\to \mathbb{P}^3$. These surfaces arise in geometric modeling and in this context it is useful to know the implicit equation of $\lambda$ in $\mathbb{P}^{3}$. Currently, syzygies and Rees algebras provide the fastest and most versatile method to find implicit equations of parameterized surfaces. Knowing the structure of the syzygies of the polynomials that define the map $\lambda$ allows us to formulate faster algorithms for implicitization of these surfaces and also to understand their singularities. We show that for tensor product surfaces without basepoints, the existence of a linear syzygy imposes strong conditions on the structure of the syzygies that determine the implicit equation. For tensor product surfaces with basepoints we show that the syzygies that determine the implicit equation of $\lambda$ are closely related to the geometry of the set of points at which $\lambda$ is undefined.

Friday, February 10, 2017

3:00 pm in 243 Altgeld Hall,Friday, February 10, 2017

The KP-CM correspondence

Matej Penciak (UIUC Math)

Abstract: In this talk I will describe how two seemingly unrelated integrable systems have an unexpected connection. I will begin with the classical story first worked out by Airault, McKean, and Moser. I will then describe a more modern interpretation of the relation due to Ben-Zvi and Nevins.

Friday, February 17, 2017

3:00 pm in 243 Altgeld Hall,Friday, February 17, 2017

What is a Topological Quantum Field Theory?

Lutian Zhao (UIUC Math)

Abstract: In this talk we will introduce the physicists' definition of topological quantum field theory, mainly focusing on cohomological quantum field theory introduced by Witten. We will discuss topological twisting and see what topological invariant is actually computed. If time permits, we will see how Gromov-Witten invariants are constructed by physics.

Friday, February 24, 2017

3:00 pm in 243 Altgeld Hall,Friday, February 24, 2017

Quantum cohomology of Grassmannians and Gromov-Witten invariants

Sungwoo Nam (UIUC Math)

Abstract: As a deformation of classical cohomology ring, (small) quantum cohomology ring of Grassmannians has a nice description in terms of quantum Schubert classes and it has (3 point, genus 0) Gromov-Witten invariants as its structure constants. In this talk, we will describe how 'quantum corrections' can be made to obtain quantum Schubert calculus from classical Schubert calculus. After studying its structure, we will see that the Gromov-Witten invariants, which define ring structure of quantum cohomology of Grassmannians, are equal to the classical intersection number of two-step flag varieties. If time permits, we will discuss classical and quantum Littlewood-Richardson rule using triangular puzzles.

Friday, April 7, 2017

3:00 pm in 243 Altgeld Hall,Friday, April 7, 2017

An introduction to quantum cohomology and the quantum product

Joseph Pruitt (UIUC Math)

Abstract: The quantum cohomology ring of a variety is a q-deformation of the ordinary cohomology ring. In this talk I will define the quantum cohomology ring, discuss attempts to describe the quantum cohomology rings of toric varieties via generators and relations, and I will close with some methods to actually work with the quantum product.

Friday, April 21, 2017

3:00 pm in 243 Altgeld Hall,Friday, April 21, 2017

Maximal tori in the symplectomorphism groups of Hirzebruch surfaces

Hadrian Quan (UIUC Math)

Abstract: In this talk, I'll discuss some beautiful results of Yael Karshon. After introducing the family of Hirzebruch surfaces, I'll highlight how certain toric actions identify these spaces with trapezoids in the complex plane. Finally, I'll describe the necessary and sufficient conditions she finds to determine when any two such surfaces are symplectomorphic. No knowledge of symplectic manifolds or toric varieties will be assumed.

Friday, May 5, 2017

3:00 pm in 243 Altgeld Hall,Friday, May 5, 2017

Complete intersections in projective space

Jin Hyung To (UIUC Math)

Abstract: We will go over complete intersection projective varieties (projective algebraic sets).

Wednesday, September 6, 2017

4:00 pm in 141 Altgeld Hall,Wednesday, September 6, 2017

Some fun with Hilbert schemes of points on surfaces

Joshua Wen (Illinois Math)

Abstract: The Hilbert scheme of points of $X$, parametrizes zero-dimensional subschemes of $X$. These spaces can be messy in general, but in the case that $X$ is a smooth surface, the Hilbert scheme is smooth as well. Rather than being some esoteric moduli space, the Hilbert scheme in this case is something one can get to know. Iíll introduce the Nakajima-Grojnowski construction of a Heisenberg algebra action that can be used to compute its Borel-Moore homology in a somewhat surprising way. Focusing on the case where $X$ is the plane, Iíll highlight connections with symmetric function theory. The goal is to give an overview of the surprising and useful structures the Hilbert scheme has, and perhaps this semester we can either study those structures in more detail or study larger-picture explanations for why those structures exist in the first place (moduli of sheaves on surfaces, 4d-gauge theory, etc.).

Wednesday, September 13, 2017

4:00 pm in Altgeld Hall 141,Wednesday, September 13, 2017

Factorization Algebra and Spaces

Matej Penciak (Illinois Math)

Abstract: In this talk I will introduce the notions of factorization algebras and spaces, and give an idea of where they fit into modern representation theory.

Wednesday, September 20, 2017

4:00 pm in Altgeld Hall 141,Wednesday, September 20, 2017

Higgs bundle and related "physics"

Lutian Zhao (Illinois Math)

Abstract: Higgs bundle was a math term introduced by Nigel Hitchin as a rough analogue of Higgs boson in standard model of physics. It turns out that it is deeply rooted in the N=4 super Yang-Mills world, where Kapustin and Witten realize the geometric Langlands correspondence as a special case of S-duality. In this talk. I'll introduce the history and notion of the Higgs bundle, and try to talk about some of the idea in their "proof". No knowledge of physics is assumed.