Seminar Calendar
for Graduate Algebraic Geometry Seminar events the year of Friday, April 21, 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.
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Thursday, January 19, 2017

Graduate Algebraic Geometry Seminar
3:00 pm   in 345 Altgeld Hall,  Thursday, January 19, 2017
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Submitted by jjwen2.
Organizational meeting

Friday, January 27, 2017

Graduate Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Friday, January 27, 2017
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Submitted by jjwen2.
Josh Wen (UIUC Math)
Raindrop. Droptop. Symmetric functions from DAHA.
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

Graduate Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Friday, February 3, 2017
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Submitted by jjwen2.
Eliana Duarte (UIUC Math)
Syzygies and Implicitization of tensor product surfaces
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

Graduate Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Friday, February 10, 2017
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Submitted by jjwen2.
Matej Penciak (UIUC Math)
The KP-CM correspondence
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

Graduate Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Friday, February 17, 2017
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Submitted by jjwen2.
Lutian Zhao (UIUC Math)
What is a Topological Quantum Field Theory?
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

Graduate Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Friday, February 24, 2017
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Submitted by jjwen2.
Sungwoo Nam (UIUC Math)
Quantum cohomology of Grassmannians and Gromov-Witten invariants
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

Graduate Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Friday, April 7, 2017
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Submitted by jjwen2.
Joseph Pruitt (UIUC Math)
An introduction to quantum cohomology and the quantum product
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

Graduate Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Friday, April 21, 2017
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Submitted by jjwen2.
Hadrian Quan (UIUC Math)
Maximal tori in the symplectomorphism groups of Hirzebruch surfaces
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

Graduate Algebraic Geometry Seminar
3:00 pm   in 243 Altgeld Hall,  Friday, May 5, 2017
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Submitted by jjwen2.
Jin Hyung To (UIUC Math)
Complete intersections in projective space
Abstract: We will go over complete intersection projective varieties (projective algebraic sets).