Department of

# Mathematics

Seminar Calendar
for Graduate Algebraic Geometry events the next 12 months of Monday, August 22, 2016.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
      July 2016             August 2016           September 2016
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1  2       1  2  3  4  5  6                1  2  3
3  4  5  6  7  8  9    7  8  9 10 11 12 13    4  5  6  7  8  9 10
10 11 12 13 14 15 16   14 15 16 17 18 19 20   11 12 13 14 15 16 17
17 18 19 20 21 22 23   21 22 23 24 25 26 27   18 19 20 21 22 23 24
24 25 26 27 28 29 30   28 29 30 31            25 26 27 28 29 30
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Thursday, August 25, 2016

3:00 pm in 441 Altgeld Hall,Thursday, August 25, 2016

#### Organizational meeting

Abstract: Besides scheduling this semesters talks, we'll need to discuss the possibility of grad students giving pretalks at the AG lunch and organizing a summer minicourse program like the do Michigan ( http://www-personal.umich.edu/~takumim/minicourses2016.html ). It'll be lit and stocked with cookies!

Thursday, September 1, 2016

3:00 pm in 441 Altgeld Hall,Thursday, September 1, 2016

#### ADE classifications in Slodowy slices

###### Josh Wen (UIUC Math)

Abstract: Among other things, the ADE Dynkin diagrams classify: finite subgroups of $SL_2(\mathbb{C})$ (McKay correspondence); isolated surface singularities with multiplicity 2 that can be resolved by successive blowups (Kleinian/du Val singularities); and some complex simple Lie algebras (Killing-Cartan/Dynkin classfication). I'll tell you a story about how these three families are tied together in the geometry of the Grothendieck-Springer resolution and Slodowy slices.

Thursday, September 15, 2016

3:00 pm in 441 Altgeld Hall,Thursday, September 15, 2016

#### An introduction to D-modules and crystals

###### Emily Cliff (UIUC Math)

Abstract: We will introduce the notion of a D-module on a variety X, a generalization of the concept of a vector bundle with flat connection. We know that over a smooth manifold, a vector bundle with flat connection is equivalent to a local system, a family of vector spaces over the manifold related to each other by parallel transport along paths in the manifold. It seems hard to translate this picture back to algebraic geometry, for example because we have no good notion of paths in a variety, but luckily Grothendieck introduced the idea of crystals of sheaves: these are sheaves with parallel transport between infinitesimally close points. We will explain this definition, and sketch the proof of the equivalence between crystals and D-modules. We will consider advantages of each approach, and along the way will see naturally occurring (and not-too-scary) examples of stacks and their categories of sheaves.

Thursday, September 22, 2016

3:00 pm in 441 Altgeld Hall,Thursday, September 22, 2016

#### Intersection Homology and $L^2$ Cohomology

Abstract: Despite its name, singular homology is perhaps not the best topological tool for studying singular spaces. For a non-singular complex projective variety $X$, one has access to a host of classical results: Poincare duality, the de Rham theorem, the Hodge-Dolbeault isomorphism. For a singular variety many of these results no longer hold. One solution is intersection homology, which was developed by Goresky-MacPherson to modify singular homology in order to recover Poincare duality. In this talk we will (with lots of pictures!) motivate and introduce intersection homology and $L^2$ cohomology. Time permitting, we will discuss some open problems concerning what the de Rham theorem might look like for these new invariants.

Thursday, October 13, 2016

3:00 pm in 441 Altgeld Hall,Thursday, October 13, 2016

#### Construction of virtual fundamental class

###### Yun Shi (UIUC Math)

Abstract: Virtual fundamental class is an important construction for modern enumerative geometry. In this talk, I will explain the construction given by Behrend and Fantechi. I will also talk a little about its application to Donaldson-Thomas theory.

Thursday, October 27, 2016

3:00 pm in 441 Altgeld Hall,Thursday, October 27, 2016

#### The Gopakumar-Vafa Invariants

###### Lutian Zhao (UIUC Math)

Abstract: In 1998, Gopakumar and Vafa argued from M-theory that BPS counts (now known as Gopakumar-Vafa invariants) have the same "generating function" as the Gromov-Witten invariants. In particular, these invariants are integral, and they agree with naive curve counting in many cases. Also, it explains the contribution of multicovering and bubbling phenomena. The basic idea of this counting is to use Lefschetz decomposition on the moduli space of D-Branes to "virtually count" the number of abelian varieties. In this talk, I will discuss why it is a promising counting invariant and give some easy cases of this counting. The serious difficulty of this counting is the definition of moduli of D-Branes, which only have a satisfactory description at g=0. If time permits, I will describe some attempts by Hosono-Saito-Takahashi, Kiem-Li and Maulik-Toda on this theory.

Friday, November 4, 2016

3:00 pm in 241 Altgeld Hall,Friday, November 4, 2016

#### KN 1-parameter subgroups for representations of quivers

###### Itziar Ochoa (UIUC Math)

Abstract: Given a projective variety X with an action of a complex reductive group G, the quotient space $X/ G$ may not exist in the category of algebraic varieties. In order to fix this problem, Geometric Invariant Theory gives a construction of a $G$-invariant open subset $X^{ss}$ of X for which the algebraic quotient exists. The Kirwan-Ness (KN) stratification refines $X$ and its unique open stratum coincides with the set $X^{ss}$. When $X$ is a linear representation $G$, the semistable locus $X^{ss}\subset X$ and a KN stratification of $X \backslash X^{ss}$ are associated to a choice of a homomorphism $G\rightarrow \mathbb{G}_m$ . That is, we can write $X=X^{ss}\sqcup \bigsqcup_{\alpha\in \text{KN}}S_\alpha,$ where $S_{\alpha}$ are locally closed smooth pieces and KN indexes the 1-parameter subgroups that determine the stratification. In this situation, Nevins and McGerty give an algorithm to find the KN 1-parameter subgroups. I will ilustrate the algorithm with some examples and at the end I will focus on the case when $X$ is a representation of the cyclic quiver.

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

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).