Department of

Mathematics


Seminar Calendar
for Topology Seminar events the year of Tuesday, February 19, 2019.

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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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Friday, January 18, 2019

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

Organizational Meeting

Nachiketa Adhikari (UIUC)

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.

Tuesday, February 19, 2019

11:00 am in 345 Altgeld Hall ,Tuesday, February 19, 2019

G-equivariant factorization algebras

Laura Wells (Notre Dame Math)

Abstract: Factorization algebras are a mathematical tool used to encode the data of the observables of a field theory. There are various notions of factorization algebra: one can define a factorization algebra on the open subsets of some fixed manifold; or alternatively, one can define a factorization algebra on the site of all manifolds of a given dimension with specified geometric structure. In this talk I will outline a comparison between two such notions: G-equivariant factorization algebras on a fixed model space M and factorization algebras on the site of all manifolds quipped with a (G, M)-structure (given by an atlas of charts in M and transition maps in G). I will introduce the definitions of these two concepts and then sketch the proof of their equivalence as (\infy,1)-categories.

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.

Tuesday, February 26, 2019

11:00 am in 345 Altgeld Hall,Tuesday, February 26, 2019

What we know so far about "topological Langlands Correspondence"

Andrew Salch (Wayne State University)

Abstract: I'll give a survey of some relationships between Galois representations and stable homotopy groups of finite CW-complexes which suggest the possibility of "topological Langlands correspondences." I'll explain what such correspondences ought to be, what their practical consequences are for number theory and for algebraic topology, and I'll explain the cases of such correspondences that are known to exist so far. As an application of one family of known cases, I'll give a topological proof of the Leopoldt conjecture for one particular family of number fields. Some of the results in this talk are joint work with M. Strauch.

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.

Tuesday, April 9, 2019

11:00 am in 345 Altgeld Hall,Tuesday, April 9, 2019

Classifying spectra of finite groups and chromatic homotopy theory

Nathan Stapleton (U Kentucky math)

Abstract: We will discuss a question about the functoriality of certain evaluation maps for classifying spectra of finite groups that arose when thinking about questions related to chromatic homotopy theory. I will describe a solution to this problem found in joint work with Reeh, Schlank.

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.

Tuesday, April 16, 2019

11:00 am in 345 Altgeld Hall,Tuesday, April 16, 2019

Iterated K-theory of the integers and higher Lichtenbaum-Quillen conjectures

Gabe Angelini-Knoll (Michigan State University)

Abstract: The Hurewicz image of the alpha family in the algebraic K-theory of the integers is know to correspond to special values of the Riemann zeta function, by work of Adams and Quillen. Lichtenbaum and Quillen conjectured that, more generally, there should be a relationship between special values of Dedekind zeta functions and algebraic K-theory. These conjectures have now largely been proven by work of Voevodsky and Rost. The red-shift conjectures of Ausoni-Rognes generalize the Lichtenbaum-Quillen conjecture to higher chromatic heights in a precise sense. In that same spirit, I conjecture that the n-th Greek letter family is detected in the Hurewicz image of the n-th iteration of algebraic K-theory of the integers. In my talk, I will sketch a proof of this conjecture in the case n=2 using the theory of trace methods. Specifically, I prove that the beta family is detected in the Hurewicz image of iterated algebraic K-theory of the integers. This is a higher chromatic height analogue of the result of Adams and Quillen. Consequently, by work of Behrens, Laures, and Larson iterated algebraic K-theory of the integers detects explicit information about certain modular forms.

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.