Department of

# Mathematics

Seminar Calendar
for Algebraic Geometry events the next 12 months of Sunday, January 1, 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

#### CANCELLED

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

Wednesday, September 27, 2017

4:00 pm in Altgeld Hall 141,Wednesday, September 27, 2017

#### Introduction to Higgs Bundles and Spectral Curves

###### Matej Penciak (Illinois Math)

Abstract: In this talk I will define Higgs bundles, and try to motivate them as "parametrized linear algebra". Along the way I will emphasize the spectral description of Higgs bundles. By the end, the hope is to define the Hitchin integrable system which plays a central role in algebraically integrable systems.

Wednesday, October 4, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 4, 2017

#### Parabolic Higgs bundles

###### Georgios Kydonakis (Illinois Math)

Abstract: The Narasimhan and Seshadri theorem, one of the seminal first results in the study of the moduli space of vector bundles over a Riemann surface, relates degree zero, stable vector bundles on a compact Riemann surface $X$ with unitary representations of ${{\pi }_{1}}\left( X \right)$. One direction to generalize this theorem is by allowing punctures in the Riemann surface and the correspondence, which now involves parabolic bundles, was carried out by Mehta and Seshadri. The version for fundamental group representations of the punctured Riemann surface into Lie groups other than $G=\text{U}\left( n \right)$ entails introducing the notion of parabolic Higgs bundles. We will describe these holomorphic objects and see examples of those corresponding to Fuchsian representations of the fundamental group of the punctured Riemann surface.

Tuesday, October 10, 2017

12:00 pm in 243 Altgeld Hall,Tuesday, October 10, 2017

#### Dynamics, geometry, and the moduli space of Riemann surfaces

###### Alex Wright (Stanford)

Abstract: The moduli space of Riemann surfaces of fixed genus is one of the hubs of modern mathematics and physics. We will tell the story of how simple sounding problems about polygons, some of which arose as toy models in physics, became intertwined with problems about the geometry of moduli space, and how the study of these problems in Teichmuller dynamics lead to connections with homogeneous spaces, algebraic geometry, dynamics, and other areas. The talk will mention joint works with Alex Eskin, Simion Filip, Curtis McMullen, Maryam Mirzakhani, and Ronen Mukamel.

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

#### Koszul duality and characters of tilting modules

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

Abstract: This talk is about the "Hecke category," a monoidal category that appears in various incarnations in geometric representation theory. I will explain some of these incarnations and their roles in solving classical problems, such as the celebrated Kazhdan-Lusztig conjectures on Lie algebra representations. These conjectures (proved in 1981) hinge on the fact that the derived category of constructible sheaves on a flag variety is equipped with an obvious monoidal action of the Hecke category on the right. It turns out that there is also a second, "hidden" action of the Hecke category on the left. The symmetry between the "hidden" left action and the "obvious" right action leads to the phenomenon known as Koszul duality. In the last part of the talk, I will discuss new results on Koszul duality with coefficients in a field of positive characteristic, with applications to characters of tilting modules for algebraic groups. This is joint work with S. Makisumi, S. Riche, and G. Williamson.

Wednesday, October 11, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 11, 2017

#### Hurwitz number

###### Hao Sun (Illinois Math)

Abstract: I will say something about Hurwitz number in algebraic geometry.

Tuesday, October 17, 2017

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

#### A Tate duality theorem for local Galois symbols

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

Abstract: Let $K$ be a $p$-adic field and $M$ a finite continuous $Gal(\overline{K}/K)$-module annihilated by a positive integer $n$. Local Tate duality is a perfect duality between the Galois cohomology of $M$ and the Galois cohomology of its dual module, $Hom(M,\mu_n)$. In the special case when $M=A[n]$ is the module of the $n$-torsion points of an abelian variety, Tate has a finer result. In this case the group $H^1(K,A[n])$ has a "significant subgroup", namely there is a map $A(K)/n\rightarrow H^1(K,A[n])$ induced by the Kummer sequence on $A$. Tate showed that under the perfect pairing for $H^1$, the orthogonal complement of $A(K)/n$ is the corresponding part, $A^\star(K)/n$, that comes from the points of the dual abelian variety $A^\star$ of $A$. The goal of this talk will be to present an analogue of this classical result for $H^2$. We will see that the "significant subgroup" in this case is given by the image of a cycle map from zero cycles on abelian varieties to Galois cohomology, while the orthogonal complement under Tate duality is given by an object of integral $p$-adic Hodge theory. We will then discuss how this computation fits with the expectations of the Bloch-Beilinson conjectures for abelian varieties defined over algebraic number fields.

Wednesday, October 18, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 18, 2017

#### Moment maps in Algebraic and Differential Geometry

Abstract: In geometry, group actions are both ubiquitous and convenient. In this talk, I’ll survey an interesting circle of ideas relating notions of stability for orbits of an action to the complex geometry of the space being acted on. Time permitting, I’ll mention how some of this story generalizes after passing from finite to infinite dimensional groups.

Tuesday, October 24, 2017

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

#### Degree bounds for invariant rings of quivers

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

Abstract: The ring of polynomial invariants for a rational representation of a reductive group is finitely generated. Nevertheless, it remains a difficult task to find a minimal set of generators, or even a bound on their degrees. Combining ideas originating from Hochster, Roberts and Kempf with the study of various ranks associated to linear matrices, we prove "polynomial" bounds for various invariant rings associated to quivers. The polynomiality of our bounds have strong consequences in algebraic complexity, notably a polynomial time algorithm for non-commutative rational identity testing. This is joint work with Derksen.

Wednesday, October 25, 2017

4:00 pm in Altgeld Hall 141,Wednesday, October 25, 2017

#### Zariski tagent space to the moduli space of vector bundles on an algebraic curve

###### Jin Hyung To   [email] (UIUC)

Abstract: Zariski tagent space to the moduli space of vector bundles on an algebraic curve Abs: We will show how to use deformations to find the Zariski tangent space. The moduli space of vector bundles is the GIT quotient of Hilbert scheme. Using this we find the Zariski tangent space of the moduli space of vector bundles.

Tuesday, October 31, 2017

11:00 am in 345 Altgeld Hall,Tuesday, October 31, 2017

#### Infinite Loop Spaces in Algebraic Geometry

###### Elden Elmanto (Northwestern)

Abstract: A classical theorem in modern homotopy theory states that functors from finite pointed sets to spaces satisfying certain conditions model infinite loop spaces (Segal 1974). This theorem offers a recognition principle for infinite loop spaces. An analogous theorem for Morel-Voevodsky's motivic homotopy theory has been sought for since its inception. In joint work with Marc Hoyois, Adeel Khan, Vladimir Sosnilo and Maria Yakerson, we provide such a theorem. The category of finite pointed sets is replaced by a category where the objects are smooth schemes and the maps are spans whose "left legs" are finite syntomic maps equipped with a K​-theoretic trivialization of its contangent complex. I will explain what this means, how it is not so different from finite pointed sets and why it was a natural guess. In particular, I will explain some of the requisite algebraic geometry. Time permitting, I will also provide 1) an explicit model for the motivic sphere spectrum as a torsor over a Hilbert scheme and, 2) a model for all motivic Eilenberg-Maclane spaces as simplicial ind-smooth schemes.

Wednesday, November 1, 2017

4:00 pm in Altgeld Hall 141,Wednesday, November 1, 2017

#### Zariski Tangent Space to the moduli of vector bundles on an algebraic curve II

###### Jin Hyung To   [email] (UIUC)

Abstract: We will find the Zariski tangent space to the moduli space of vector bundles. This is the quotient vector space of the Zariski tangent space to the Hilbert scheme. In particular, we find the Zariski tangent space to the Jacobian variety using deformations.

Tuesday, November 7, 2017

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

#### Irreducible components of exotic Springer fibers

###### Daniele Rosso (Indiana University Northwest)

Abstract: The Springer resolution is a resolution of singularities of the variety of nilpotent elements in a reductive Lie algebra. It is an important geometric construction in representation theory, but some of its features are not as nice if we are working in Type C (Symplectic group). To make the symplectic case look more like the Type A case, Kato introduced the exotic nilpotent cone and its resolution, whose fibers are called the exotic Springer fibers. We give a combinatorial description of the irreducible components of these fibers in terms of standard Young bitableaux and obtain an exotic Robinson-Schensted correspondence. This is joint work with Vinoth Nandakumar and Neil Saunders.

Wednesday, November 8, 2017

4:00 pm in Altgeld Hall 141,Wednesday, November 8, 2017

#### Local deformation theory of algebraic schemes

###### Sungwoo Nam (UIUC)

Abstract: Deformation theory is the study of variations of structure of a given object, which can be used to study the original object. It can also be regarded as local geometry of a moduli space, if it exists. In this talk, I will introduce (local) deformation theory, using the formalism of functor of artin rings and discuss automorphism-deformation-obstruction vector spaces and how these are realized as certain cohomologies. Along the way, I will give examples of application to concrete cases, including deformations of schemes, closed subschemes, nonsingular varieties, quasi-coherent sheaves.

Tuesday, November 14, 2017

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

#### Kodaira-Saito vanishing via Higgs bundles in positive characteristic

###### Donu Arapura (Purdue University)

Abstract: In 1990, Saito gave a strong generalization of Kodaira’s vanishing theorem using his theory of mixed Hodge modules. I want to explain the statement in the special case of a variation of Hodge structure on the complement of a divisor with normal crossings. Unlike Saito’s original proof, I will describe a proof using characteristic p methods.

Wednesday, November 15, 2017

4:00 pm in Altgeld Hall 141,Wednesday, November 15, 2017

#### Deformation theory of Galois representations

###### Ravi Donepudi   [email] (UIUC)

Abstract: The first systematic study of deformation theory in algebraic number theory, specifically its application to the theory of Galois representations, was done by Barry Mazur (1987). The goal of this talk is to motivate why this is a useful and interesting thing to do. We begin with discussing why one should study Galois representations in the first place, let alone deform them. Then, we define appropriate categories that serve as the domains of our deformation functors and discuss aspects of their representability. Finally, we give examples of Galois representations arising “naturally” from arithmetic objects (like elliptic curves and modular forms) and from algebraic geometry (via the étale cohomology of smooth projective varieties). Time permitting, we will discuss some conjectures in the theory of Galois representations and the role deformation theory plays in understanding them better. No scheme theory is assumed.

Thursday, November 16, 2017

11:00 am in 241 Altgeld Hall,Thursday, November 16, 2017

#### Kloosterman sums and Siegel zeros

###### James Maynard (Institute For Advanced Study)

Abstract: Kloosterman sums arise naturally in the study of the distribution of various arithmetic objects in analytic number theory. The 'vertical' Sato-Tate law of Katz describes their distribution over a fixed field $F_p$, but the equivalent 'horizontal' distribution as the base field varies over primes remains open. We describe work showing cancellation in the sum over primes if there are exceptional Siegel-Landau zeros. This is joint work with Sary Drappeau, relying on a blend of ideas from algebraic geometry, the spectral theory of automorphic forms and sieve theory.

Tuesday, November 28, 2017

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

#### Higher rank Clifford indices for curves on K3 surfaces

###### Chunyi Li (University of Warwick)

Abstract: The Clifford index $Cliff_1(C)$ of curve $C$ is the second most important invariant of $C$ after the genus, measuring the complexity of the curve in its moduli space. The celebrated work by Lazarsfeld indicates that $Cliff_1(C)$ is $g-1-[g/2]$ when $C\in |H|$ is on a polarized K3 surface $(X,H)$. Inspired by the work of Mercat, an adequate generalization $Cliff_r(C)$ for higher rank vector bundles has been defined by Lange and Newstead. Via the tool of Bridgeland stability condition, for curves on generic K3 surfaces we compute that $Cliff_r(C)=2(g-1-[g/r])/r$, when $g\geq r^2\geq 4$. In the talk, I will explain more details on this classical topic and how does stability condition help to solve them. This is a joint work with S. Feyzbakhsh.

Wednesday, November 29, 2017

4:00 pm in Altgeld Hall 141,Wednesday, November 29, 2017

#### Introduction to quiver varieties

###### Ciaran O'Neill (UIUC)

Abstract: I’ll cover all the necessary background to define quiver varieties. This will include giving a reminder of the GIT quotient, defining the twisted GIT quotient and the basics of quivers. I’ll end by stating a theorem guaranteeing when a certain canonical map between quiver varieties is a resolution of singularities.

Wednesday, December 6, 2017

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

#### A Counterexample to the Weitzenböck Conjecture in Characteristics $p > 2$

###### Stephen Maguire   [email] (UIUC)

Abstract: A variety $X$ is ruled if there is a bi-rational map $\mathbb{P}^1 × Y \rightarrow X$ for some variety Y . The study of ruled varieties is motivated by the desire to study rational varieties by weakening the condition. If a variety $X$ has a $\mathbb{G}_a$-action, then one sees that it is ruled. Also, simplicial toric varieties often have non-reductive automorphism groups. An example is $\mathbb{P}(1 : 1 : 2)$ whose automorphism group is isomorphic to $GL(2, k)\times \mathbb{G}_a^3$(semi-direct product). As a result one finds that constructing a type of non-reductive GIT is a worthy venture. If one wants to construct this type of non-reductive GIT one first needs to understand actions of $\mathbb{G}_a$, the simplest non-reductive group. The earliest theorem regarding Ga was Weitzenb¨ock’s Theorem, which states that a representation $\mu : \mathbb{G}_a → GL(V_n)$ has a finitely generated ring of invariants $k[X]^{\mathbb{G}_a}$ if the field $k$ is an algebraically close field of characteristic zero. Work then focused on whether the invariant ring of non-linear actions of Ga is finitely generated. The answer is no. At this point people wished to answer the Weitzenbock conjecture, namely whether the result of Weitzenb¨ock’s Theorem still holds if the characteristic of the field is positive. It is this question we wish to answer in the following talks. In this talk we begin the process of producing a representation $\mu : \mathbb{G}_a → GL(V_6)$ over an algebraically closed field k of characteristic $p > 2$ such that the ring of invariants $k[x_1,\ldots,x_6]^{\mathbb{G}_a}$ is not a finitely generated k-algebra. In order to do this, we reduce the problem of finding a counterexample to the Weitzenbock conjecture to a curve counting problem, and then use this reduction to further reduce this problem to a problem about the support of a bi-graded ring.

Wednesday, December 13, 2017

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

#### Representation and Character Variety of 3 Manifolds

###### Xinghua Gao   [email] (UIUC)

Abstract: I'll start with defining $SL_2(\mathbb{C})$ representation/character variety of 3 manifold groups. Then I'll talk about how twisted cohomology is used to determine smoothness of these varieties, followed by some applications.