Department of

# Mathematics

Seminar Calendar
for Algebraic Geometry events the year of Wednesday, September 13, 2017.

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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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Thursday, January 19, 2017

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

#### Organizational meeting

Tuesday, January 24, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, January 24, 2017

#### On the Behrend function and its motivic version in Donaldson-Thomas theory

###### Yungfeng Jiang (U Kansas Math)

Abstract: The Behrend function, introduced by K. Behrend, is a fundamental tool in the study of Donaldson-Thomas invariants. In his foundational paper K. Behrend proves that the weighted Euler characteristic of the Donaldson-Thomas moduli space weighted by the Behrend function is the Donaldson-Thomas invariants defined by R. Thomas using virtual fundamental cycles. This makes the Donaldson-Thomas invariants motivic. In this talk I will talk about the basic notion of the Behrend function and apply it to several other interesting geometries. If time permits, I will also talk about the motivic version of the Behrend function and the famous Joyce-Song formula of the Behrend function identities.

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.

Tuesday, January 31, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, January 31, 2017

#### Kirwan surjectivity for quiver varieties

###### Tom Nevins (UIUC)

Abstract: Many interesting hyperkahler, or more generally holomorphic symplectic, manifolds are constructed via hyperkahler/holomorphic symplectic reduction. For such a manifold there is a “hyperkahler Kirwan map,” from the equivariant cohomology of the original manifold to the reduced space. It is a long-standing question when this map is surjective (in the Kahler rather than hyperkahler case, this has been known for decades thanks to work of Atiyah-Bott and Kirwan). I’ll describe a resolution of the question (joint work with K. McGerty) for Nakajima quiver varieties: their cohomology is generated by Chern classes of “tautological bundles.” If there is time, I will explain that this is a particular instance of a general story in noncommutative geometry. The talk will not assume prior familiarity with any of the notions above.

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.

Wednesday, February 8, 2017

3:00 pm in 243 Altgeld Hall,Wednesday, February 8, 2017

#### Stability and wall-crossing in algebraic geometry

###### Rebecca Tramel (UIUC)

Abstract: I will discuss two notions of stability in algebraic geometry: slope stability of vector bundles on curves, and Bridgeland stability for complexes of sheaves on smooth varieties. I will try and motivate both of these definitions with questions from algebraic geometry and from physics. I will then work through a few detailed examples to show how varying our notion of stability affects the set of stable objects, and how this relates to the geometry of the space we are studying.

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.

Monday, February 27, 2017

4:00 pm in 245 Altgeld Hall,Monday, February 27, 2017

#### Algebra, Combinatorics, Geometry

###### Hal Schenck (Department of Mathematics, University of Illinois)

Abstract: I'll give an overview of the spectacular success of algebraic methods in studying problems in discrete geometry and combinatorics. First we'll discuss the face vector (number of vertices, edges, etc.) of a convex polytope and recall Euler's famous formula for polytopes of dimension 3. Then we'll discuss graded rings, focusing on polynomial rings and quotients. Associated to a simplicial polytope P (every face is "like" a triangle) is a graded ring called the Stanley-Reisner ring, which "remembers" everything about P, and gives a beautiful algebra/combinatorics dictionary. I will sketch Stanley's solution to a famous conjecture using this machinery, and also touch on connections between P and toric varieties, which are objects arising in algebraic geometry.

Tuesday, February 28, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, February 28, 2017

#### BPS Counts on K3 surfaces and their products with elliptic curves

###### Sheldon Katz (UIUC)

Abstract: In this survey talk, I begin by reviewing the string theory-based BPS spectrum computations I wrote about with Klemm and Vafa in the late 1990s. These were presented to the algebraic geometry community as a prediction for Gromov-Witten invariants. But our calculations of the BPS spectrum contained much more information than could be interpreted via algebraic geometry at that time. During the intervening years, Donaldson-Thomas invariants were introduced, used by Pandharipande and Thomas in their 2014 proof of the original KKV conjecture. It has since become apparent that the full meaning of the KKV calculations, and more recent extensions, can be mathematically interpreted via motivic Donaldson-Thomas invariants. With this understanding, we arrive at precise and deep conjectures. I conclude by surveying the more recent work of myself and others in testing and extending these physics-inspired conjectures on motivic BPS invariants.

Tuesday, March 7, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, March 7, 2017

#### Bernstein-Sato polynomials for maximal minors

###### Andras Lorincz (Purdue University)

Abstract: Initially introduced for hypersurfaces, Bernstein-Sato polynomials have been recently defined for arbitrary varieties by N. Budur, M. Mustata and M. Saito. Nevertheless, they are notoriously difficult to compute with very few explicit cases known. In this talk, after giving the necessary background, I will discuss some techniques that allow the computation of the Bernstein-Sato polynomial of the ideal of maximal minors of a generic matrix. Time permitting, I will also talk about connections to topological zeta functions and show the monodromy conjecture for this case.Initially introduced for hypersurfaces, Bernstein-Sato polynomials have been recently defined for arbitrary varieties by N. Budur, M. Mustata and M. Saito. Nevertheless, they are notoriously difficult to compute with very few explicit cases known. In this talk, after giving the necessary background, I will discuss some techniques that allow the computation of the Bernstein-Sato polynomial of the ideal of maximal minors of a generic matrix. Time permitting, I will also talk about connections to topological zeta functions and show the monodromy conjecture for this case.

Tuesday, March 14, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, March 14, 2017

#### Categorical Gromov-Witten Invariants

###### Junwu Tu (University of Missouri )

Abstract: In this talk, following Costello and Kontsevich, we describe a construction of Gromov-Witten type invariants from cyclic A-infinity categories.

Tuesday, March 28, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, March 28, 2017

#### A variety with non-finitely generated automorphism group

###### John Lesieutre (UIC)

Abstract: If X is a projective variety, then Aut(X)/Aut^0(X) is a countable group, but little is known about what groups can occur. I will construct a six-dimensional variety for which this group is not finitely generated, and discuss how the construction can adapted to give an example of a complex variety with infinitely many non-isomorphic real forms.

Tuesday, April 4, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, April 4, 2017

#### Rational points of generic curves and the section conjecture

###### Tatsunari Watanabe (Purdue University)

Abstract: The section conjecture comes from Grothendieck's anabelian philosophy where he predicts that if a variety is "anabelian", then its arithmetic fundamental group should control its geometry. In this talk, I will introduce the section conjecture and the generic curve of genus g >=4 with no marked points as an example where the conjecture holds. The primary tool used is called weighted completion of profinite groups developed by R Hain and M Matsumoto. It linearizes a profinite group such as arithmetic mapping class groups and is relatively computable since it is controlled by cohomology groups.

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.

Tuesday, April 11, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, April 11, 2017

#### Enriched Hodge Structures

###### Deepam Patel (Purdue University)

Abstract: It is well known the the category of mixed Hodge structures does not give the right answer when studying cycles on possibly open/singular varieties. In this talk, we will discuss how the category of mixed Hodge structures can be `enriched’ to a category appropriate for studying algebraic cycles on infinitesimal thickenings of complex analytic varieties. This is based on joint work with Madhav Nori and Vasudevan Srinivas.

Thursday, April 13, 2017

2:00 pm in 241 Altgeld Hall,Thursday, April 13, 2017

#### L-values, Bessel moments and Mahler measures

###### Detchat Samart   [email] (UIUC)

Abstract: We will discuss some formulas and conjectures relating special values of L-functions associated to modular forms to moments of Bessel functions and Mahler measures. Bessel moments arise in the study of Feynman integrals, while Mahler measures have received a lot of attention from mathematicians over the past few decades due to their apparent connection with number theory, algebraic geometry, and algebraic K-theory. Though easy to verify numerically with high precision, most of these formulas turn out to be ridiculously hard to prove, and no machinery working in full generality is currently known. Some available techniques which have been used to tackle these problems will be demonstrated. Time permitting, we will present a meta conjecture of Konstevich and Zagier which gives a general framework of how one could verify these formulas using only elementary calculus.

Tuesday, April 18, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, April 18, 2017

#### Stable quotients and the B-model

###### Rahul Pandharipande (ETH Zurich)

Abstract: I will give an account of recent progress on stable quotient invariants, especially from the point of view of the B-model and present a geometrical derivation of the holomorphic anomaly equation for local CY cases (joint work with Hyenho Lho).

Friday, April 21, 2017

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

#### 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.

4:00 pm in 241 Altgeld Hall,Friday, April 21, 2017

#### It’s hard being positive: symmetric functions and Hilbert schemes

###### Joshua Wen (UIUC Math)

Abstract: Macdonald polynomials are a remarkable basis of $q,t$-deformed symmetric functions that have a tendency to show up various places in mathematics. One difficult problem in the theory was the Macdonald positivity conjecture, which roughly states that when the Macdonald polynomials are expanded in terms of the Schur function basis, the corresponding coefficients lie in $\mathbb{N}[q,t]$. This conjecture was proved by Haiman by studying the geometry of the Hilbert scheme of points on the plane. I’ll give some motivations and origins to Macdonald theory and the positivity conjecture and highlight how various structures in symmetric function theory are manifested in the algebraic geometry and topology of the Hilbert scheme. Also, if you like equivariant localization computations, then you’re in luck!

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).

Thursday, August 31, 2017

11:00 am in 241 Altgeld Hall,Thursday, August 31, 2017

#### Polynomial Roth type theorems in Finite Fields

###### Dong Dong (Illinois Math)

Abstract: Recently, Bourgain and Chang established a nonlinear Roth theorem in finite fields: any set (in a finite field) with not-too-small density contains many nontrivial triplets $x$, $x+y$, $x+y^2$. The key step in Bourgain-Chang's proof is a $1/10$-decay estimate of some bilinear form. We slightly improve the estimate to a $1/8$-decay (and thus a better lower bound for the density is obtained). Our method is also valid for 3-term polynomial progressions $x$, $x+P(y)$, $x+Q(y)$. Besides discrete Fourier analysis, algebraic geometry (theorems of Deligne and Katz) is used. This is a joint work with Xiaochun Li and Will Sawin.

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. I’ll 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, I’ll 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.).

Thursday, September 7, 2017

1:00 pm in Altgeld Hall 243,Thursday, September 7, 2017

#### Character values, combinatorics, and some p-adic representations of finite groups

###### Michael Geline (Northern Illinois University)

Abstract: There are many questions which remain open from Brauer's modular representation theory of finite groups, and little agreement exists over the extent to which they ought to ultimately depend on the classification of finite simple groups. In studying a classification-free approach to one such question, involving lattices over the p-adics, I was led to an elementary combinatorial problem which is interesting in its own right and somewhat amenable to analysis by means of character values. I will present this problem, the original conjecture of Brauer which gave rise to it, and my own progress in the area. If it is necessary to mention algebraic geometry, there is something of a long-shot analogy between the character values and the Weil conjectures.

Tuesday, September 12, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, September 12, 2017

#### To Be Announced

###### Chris Dodd (UIUC)

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.

Tuesday, September 19, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, September 19, 2017

#### Character values, combinatorics, and some p-adic representations of finite groups

###### Michael Geline (Northern Illinois University)

Abstract: There are many questions which remain open from Brauer's modular representation theory of finite groups, and little agreement exists over the extent to which they ought to ultimately depend on the classification of finite simple groups. In studying a classification-free approach to one such question, involving lattices over the p-adics, I was led to an elementary combinatorial problem which is interesting in its own right and somewhat amenable to analysis by means of character values. I will present this problem, the original conjecture of Brauer which gave rise to it, and my own progress in the area. If it is necessary to mention algebraic geometry, there is something of a long-shot analogy between the character values and the Weil conjectures.

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.

Tuesday, September 26, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, September 26, 2017

#### To Be Announced

###### Mark Penney (University of Oxford / Max Planck Institute (Bonn))

Tuesday, October 10, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, October 10, 2017

#### To Be Announced

###### Pramod Achar (Louisiana State University)

Tuesday, October 17, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, October 17, 2017

#### To Be Announced

###### Evangelia Gazaki (University of Michigan)

Tuesday, October 24, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, October 24, 2017

#### To Be Announced

###### Visu Makam (University of Michigan)

Tuesday, November 7, 2017

3:00 pm in 243 Altgeld Hall,Tuesday, November 7, 2017