Department of

# Mathematics

Seminar Calendar
for Graduate Geometry/Topology Seminar events the year of Thursday, March 21, 2019.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    February 2019            March 2019             April 2019
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       1  2  3  4  5  6
3  4  5  6  7  8  9    3  4  5  6  7  8  9    7  8  9 10 11 12 13
10 11 12 13 14 15 16   10 11 12 13 14 15 16   14 15 16 17 18 19 20
17 18 19 20 21 22 23   17 18 19 20 21 22 23   21 22 23 24 25 26 27
24 25 26 27 28         24 25 26 27 28 29 30   28 29 30
31


Friday, January 18, 2019

4:00 pm in 141 Altgeld Hall,Friday, January 18, 2019

#### Organizational Meeting

Abstract: We will draft a schedule of the seminar talks this semester. Please join us and sign up if you want to speak (you don't have to decide on a topic or abstract now). As usual, there will be cookies. All are welcome!

Friday, January 25, 2019

4:00 pm in 141 Altgeld Hall,Friday, January 25, 2019

#### Symmetric functions and Hilbert schemes

###### Joshua Wen (UIUC)

Abstract: One source of applications of geometric and topological methods to combinatorics and representation theory is to proving various numbers are positive integers by showing that said numbers are dimensions of some vector space. A big example of this from more than a decade ago is Haiman’s proof of the Macdonald positivity conjecture, which further cemented an already tight connection between symmetric functions and the topology of Hilbert schemes of points in $\mathbb{C}^2$. I want to go through this story while highlighting two lessons that nobody taught me in grad school—that generating series are awesome for geometers and how to do geometry on a moduli space.

Friday, February 1, 2019

4:00 pm in 145 Altgeld Hall,Friday, February 1, 2019

#### Vector fields on Spheres

###### Brian Shin (UIUC)

Abstract: In this talk, I would like to tell the story of one of the classical problems in topology: how many pointwise linearly independent vector fields can you put on a sphere of dimension $n$. The famous Hairy Ball Theorem tells us that there are none if $n$ is even. On the other hand, if $n$ is one of 1, 3, or 7, we can construct $n$ such vector fields using the normed divison $\mathbb{R}$-algebra structures on complex numbers, quaternions, and octonions. In this talk, we'll discuss the complete resolution of this problem by Adams, using methods of geometry, algebra, and homotopy theory along the way.

Friday, February 8, 2019

4:00 pm in 145 Altgeld Hall,Friday, February 8, 2019

#### Hamiltonian Lie algebroids

###### Luka Zwaan (UIUC)

Abstract: Hamiltonian Lie algebroids were introduced quite recently by Blohmann and Weinstein, resulting from their work in general relativity. They are a generalisation of the usual notion of a Hamiltonian action of a Lie algebra on a presymplectic manifold to arbitrary Lie algebroids. In this talk, I will quickly recall this usual notion, and then discuss several ways Blohmann and Weinstein tried to generalise it. In the end, the most convenient method makes use of a choice of connection on the Lie algebroid.

Friday, February 15, 2019

4:00 pm in Altgeld Hall 145 ,Friday, February 15, 2019

#### Laplacian Operator and Hyperbolic Geometry

###### Xiaolong Han (Illinois Math)

Abstract: The Laplacian operator acting on functions on a Riemannian manifold is an analytic operator invariant under isometry of the manifold. Its spectrum encodes much geometric information of the manifold. In this talk, I will start with some basic properties of Laplacian operator and hyperbolic geometry. Then I will talk about how these two interact with each other. Time permitting, I will talk about some of my recent works. No background on Laplacian operator or hyperbolic geometry is assumed.

Friday, February 22, 2019

4:00 pm in 145 Altgeld Hall,Friday, February 22, 2019

#### 27 lines on smooth cubic surfaces

###### Ningchuan Zhang (UIUC)

Abstract: In this talk, I will show that there are $27$ projective lines on a smooth cubic surface in $\mathbb{CP}^3$ by a Chern class computation. This talk is based on a course project I did with Professor Sheldon Katz in Math 524 (now 514) in Spring 2015. No knowledge of algebraic geometry or characteristic classes is assumed.

Friday, March 1, 2019

4:00 pm in 145 Altgeld Hall,Friday, March 1, 2019

#### Exposition on motives

###### Tsutomu Okano (UIUC)

Abstract: The proof of Weil conjectures led Grothendieck to think about categories of motives. This is supposed to be an abelian category that contains all the arithmetic-geometric information of varieties. Such a category has not yet been proved to exist. However, there are convincing partial answers which I hope to communicate in this talk. I will describe Grothendieck's construction of pure Chow motives, then Voevodsky's construction of the conjectured derived category of motives. Towards the end, I will describe the connection with motivic homotopy theory.

Friday, March 8, 2019

4:00 pm in 145 Altgeld Hall,Friday, March 8, 2019

#### Basics of Chern Simons Theory

###### Yidong Chen (UIUC)

Abstract: In this talk I'll explain Atiyah's "axioms" for topological field theory and construct two examples: Chern Simons theory with finite group over any compact oriented manifold, and Chern Simons theory with compact simply connected Lie group over a compact connected 3-manifold. The latter (with SU(2)) is the quintessential example for Chern Simons theory in the physics literature.

Friday, March 15, 2019

4:00 pm in 145 Altgeld Hall,Friday, March 15, 2019

#### Some aspects of Foliations of 3-manifolds

###### Gayana Jayasinghe (UIUC)

Abstract: While foliations have proven to be a useful tool for studying the topology and geometry of manifolds, in lower dimensions, they allow one to create and admire extremely beautiful pictures. Renowned masters of this art such as William Thurston and David Gabai have developed a many-layered machinery to manipulate and construct "nice" foliations. I will assume very little knowledge and will introduce the basics, then talk about some things I found interesting. My props will be edible versions of these you can study at your leisure.

Friday, March 29, 2019

4:00 pm in 145 Altgeld Hall,Friday, March 29, 2019

#### Geometric ideas in number theory

###### Robert Dicks (UIUC)

Abstract: Jurgen Neukirch in 1992 wrote that Number Theory is Geometry. At first glance, it seems nothing could be further from the truth, but it turns out that tools such as vector bundles, cohomology, sheaves, and schemes have become indispensable for understanding certain chapters of number theory in recent times. The speaker aims to discuss an analogue in the context of number fields of the classical Riemann-Roch theorem, which computes dimensions of spaces of meromorphic functions on a Riemann surface in terms of its genus. The aim is for the talk to be accessible for any graduate student; we'll find out what happens.

Friday, April 5, 2019

4:00 pm in 145 Altgeld Hall,Friday, April 5, 2019

#### A pointless alternative to topological spaces

###### William Balderrama (UIUC)

Abstract: Fundamental to geometry and topology is the notion of a space. These are usually axiomatized as topological spaces, but there are alternative axiomatizations. In this talk, I will introduce one alternative, the locales, and describe some ways in which they can be better behaved than topological spaces.

Friday, April 12, 2019

4:00 pm in 145 Altgeld Hall,Friday, April 12, 2019

#### What is a Higgs bundle?

###### Matej Penciak (UIUC)

Abstract: In this talk I will introduce and try to motivate Higgs bundles as objects that naturally arise in algebra and geometry.

Friday, April 19, 2019

4:00 pm in 145 Altgeld Hall,Friday, April 19, 2019

#### Complex structures on real vector bundles

###### Abhra Abir Kundu (UIUC)

Abstract: In this talk, I will provide an interpretation of the question "Does a given real vector bundle admit a complex structure?" and offer an approach to understanding this question.

Friday, April 26, 2019

4:00 pm in 145 Altgeld Hall,Friday, April 26, 2019

#### Relatively hyperbolic groups and Dehn fillings

###### Heejoung Kim (UIUC)

Abstract: Geometric group theory has been studied extensively since Gromov introduced the notion of a hyperbolic group. For instance, the fundamental group of a hyperbolic surface is a hyperbolic group, but not the fundamental group of a cusped hyperbolic 3-manifold. From this motivating example, we consider a generalization of a hyperbolic group, called a relatively hyperbolic group. On the other hand, Thurston's Dehn filling theorem states that one can obtain further hyperbolic 3-manifolds from a given cusped hyperbolic 3-manifold. Groves and Manning extended Thurston's Dehn filling theorem to the context of relatively hyperbolic groups. In this talk, we will discuss hyperbolic groups, relatively hyperbolic groups, and the group-theoretic analog of Thurston's Dehn filling theorem in the context of relatively hyperbolic groups.