Department of

Mathematics


Seminar Calendar
for Topology Seminar events the year of Saturday, November 9, 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.

Friday, August 30, 2019

4:00 pm in 141 Altgeld Hall,Friday, August 30, 2019

Organizational Meeting

Nachiketa Adhikari (UIUC)

Abstract: This is a seminar for graduate students interested in geometry and/or topology. We will draft a schedule of the seminar talks this semester. Please join us and sign up if you would like to speak. You don't need to have a topic or abstract to sign up; you can speak about your own research, or give an expository talk about something you find interesting. As usual, there will be cookies. All are welcome!

Friday, September 6, 2019

4:00 pm in 141 Altgeld Hall,Friday, September 6, 2019

Geometry by example: the projective plane

Nachiketa Adhikari (UIUC)

Abstract: In this expository talk, I will introduce the projective plane, and use it to explore a range of ideas including moment polytopes, localization formulas and intersection theory.

Tuesday, September 10, 2019

11:00 am in 347 Altgeld Hall,Tuesday, September 10, 2019

Dirichlet character twisted Eisenstein series and $J$-spectra

Ningchuan Zhang

Abstract: Bernoulli numbers show up in both the $q$-expansions of normalized Eisenstein series and the image of the $J$-homomorphism in the stable homotopy groups of spheres. Number theorists have defined generalized Bernoulli numbers and twisted Eisenstein series associated to Dirichlet characters. The goal of this talk is to construct a family of Dirichlet character twisted $J$-spectra and explain the relations between their homotopy groups and congruences of the twisted Eisenstein series. In the course of that, we will generalize Nicholas Katz’s algebro-geometric explanation of congruences of the (untwisted) normalized Eisenstein series in his Antwerp notes.

Friday, September 13, 2019

4:00 pm in 141 Altgeld Hall,Friday, September 13, 2019

Construction of a Poisson manifold of strong compact type

Luka Zwaan (UIUC)

Abstract: I will start the talk with a short introduction to Poisson geometry, going over several equivalent ways of defining a Poisson structure and giving some basic properties and examples. After that I will focus on a specific class of Poisson manifolds, namely those we call ''of compact type''. There are several compactness types, and finding non-trivial examples for the strongest type turns out to be quite difficult. I will sketch a construction which makes use of the many strong properties of K3 surfaces.

Friday, September 20, 2019

4:00 pm in 141 Altgeld Hall,Friday, September 20, 2019

A Geometric Proof of Lie's Third Theorem

Shuyu Xiao (UIUC)

Abstract: There are three basic results in Lie theory known as Lie's three theorems. These theorems together tell us that: up to isomorphism, there is a one-to-one correspondence between finite-dimensional Lie algebras and simply connected Lie groups. While the first two theorems are easy to prove with the most basic differential geometry knowledge, the third one is somehow a deeper result which needs relatively advanced tools. In this talk, I will go over the proof given by Van Est, in which he identifies any finite-dimensional Lie algebra with a semi-direct product of its center and its adjoint Lie algebra. I will introduce Lie group cohomology, Lie algebra cohomology and how they classify the abelian extensions of Lie groups and Lie algebras and thus determine the Lie algebra structure on the semi-direct product.

Friday, September 27, 2019

4:00 pm in 141 Altgeld Hall,Friday, September 27, 2019

Thurston’s Construction of pseudo-Anosovs

Christopher Loa (UIUC)

Abstract: In the 1970’s, Thurston classified Mod(S) for higher genus surfaces in a widely circulated preprint, “remarkable for its brevity and richness.” This classification turns out to be a trichotomy (finite order, reducible, or pseudo-Anosov), just like the classification of automorphisms of the torus (finite order, reducible, or Anosov). The aim of this talk is to spell out his construction “for a large class of examples of diffeomorphisms in canonical form.” The real treasure of this construction is that it allows us to easily get our hands on pseudo-Anosov maps, a seemingly difficult task. As Thurston himself wrote “. . . it is pleasant to see something of this abstract origin made very concrete.” We motivate the construction by first classifying the automorphisms of the torus. Knowledge of basic linear algebra and hyperbolic geometry is assumed, and familiarity with mapping class groups will be helpful for following along.

Friday, October 18, 2019

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

Lines in Space

Brian Shin (UIUC)

Abstract: Consider four lines in three-dimensional space. How many lines intersect these given lines? In this expository talk, I'd like to discuss this classical problem of enumerative geometry. Resolving this problem will give us a chance to see some interesting algebraic geometry and algebraic topology. If time permits, I'll discuss connections to motivic homotopy theory.

Friday, October 25, 2019

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

Geometry + Topology + Analysis + Algebra

Cameron Rudd (UIUC)

Abstract: Barring a last minute change of heart, this talk will be about analytic analogues of typical algebraic invariants of manifolds that have proven to be useful in understanding how geometric and topological features of aspherical Riemannian manifolds influence one another.

Tuesday, October 29, 2019

11:00 am in 347 Altgeld Hall,Tuesday, October 29, 2019

Topological Quillen localization of structured ring spectra

Yu Zhang (Ohio State University)

Abstract: Homotopy groups and stable homotopy groups of spaces are main invariants in algebraic topology. Homotopy groups are very powerful but difficult to compute in practice. Stable homotopy groups, on the other hand, are easier to work with, at the expense of losing unstable information. Structured ring spectra are spectra with certain algebraic structure encoded by the action of an operad O. For such O-algebras, the analog of stable homotopy groups is played by Topological Quillen (TQ) homology groups. In this talk, we will discuss the following question: What information can be seen by TQ-homology? In particular, we will discuss TQ-localization of O-algebras and show the TQ-Whitehead theorem for homotopy pro-nilpotent O-algebras.

Friday, November 1, 2019

4:00 pm in 141 Altgeld Hall,Friday, November 1, 2019

Flexibility vs Rigidity in hyperbolic geometry

Xiaolong Han (UIUC)

Abstract: "Most" closed surfaces have a hyperbolic structure. We can ask similar questions in higher dimensions. In this talk, I will talk about some interesting phenomena and duality by looking at examples and theorems in hyperbolic geometry. I will also talk about rigidity, like how having isomorphic first fundamental group implies existence of diffeomorphism / isometry. The talk will mostly use intuition and requires no prior knowledge of hyperbolic geometry.

Tuesday, November 5, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 5, 2019

Loop space constructions of elliptic cohomology

Matthew Spong

Abstract: Elliptic cohomology is a type of generalised cohomology theory related to elliptic curves which was introduced in the late 1980s. An important motivation for its introduction, which came from physics, was to help understand index theory for families of differential operators over free loop spaces. Yet for a long time, the only known constructions of elliptic cohomology were purely algebraic, and the precise connection to free loop spaces remained obscure. In this talk, I will summarise two constructions of complex analytic, equivariant elliptic cohomology: one from the K-theory of free loop spaces, and one from the ordinary cohomology of double free loop spaces. If time permits, I will also describe the construction of a Chern character-type map from the former to the latter.

Friday, November 8, 2019

4:00 pm in 141 Altgeld Hall,Friday, November 8, 2019

Finite Element Exterior Calculus

Nikolas Wojtalewicz (UIUC)

Abstract: In this talk, we will begin by discussing a basic example of a finite element method. We will state the basic formulation of this method, and then briefly discuss some of its limitations. We will follow up by talking about Hilbert complexes (such as the De Rahm complex), discretizing such complexes, and then about Finite Element Exterior Calculus. If time permits, we will show some examples where FEEC has been particularly successful.

Tuesday, November 12, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 12, 2019

Equivariant symmetric monoidal categories and K-theory

Peter Bonventre (University of Kentucky)

Abstract: Symmetric monoidal categories have (at least) two allures to homotopy theorists: they describe the algebraic structure appearing in many categories of interest, and they effectively model all connective spectra. Equivariantly, both of these tasks become more interesting, as the category of (connective) genuine equivariant spectra is significantly more subtle. In this talk, I introduce a new model for equivariant symmetric monoidal categories which both describes the algebraic structure in equivariant categories and has a K-theory functor to genuine G-spectra. I will give several examples and applications, including comparisons to many previously proposed models of genuine symmetric monoidal categories and equivariant algebra.

11:00 am in 347 Altgeld Hall,Tuesday, November 12, 2019

Equivariant symmetric monoidal categories and K-theory

Peter Bonventre (University of Kentucky)

Abstract: Symmetric monoidal categories have (at least) two allures to homotopy theorists: they describe the algebraic structure appearing in many categories of interest, and they effectively model all connective spectra. Equivariantly, both of these tasks become more interesting, as the category of (connective) genuine equivariant spectra is significantly more subtle. In this talk, I introduce a new model for equivariant symmetric monoidal categories which both describes the algebraic structure in equivariant categories and has a K-theory functor to genuine G-spectra. I will give several examples and applications, including comparisons to many previously proposed models of genuine symmetric monoidal categories and equivariant algebra.

Friday, November 15, 2019

4:00 pm in 141 Altgeld Hall,Friday, November 15, 2019

A Small Introduction to Big Concepts in Hyperbolic Manifolds

Joseph Malionek (UIUC)

Abstract: Hyperbolic Manifolds appear in some surprising places throughout geometry and topology, but unless you are in a field that requires thorough knowledge of them, you probably don't have the time or energy to learn about them on your own. I want to save you some work by introducing a couple of the bigger concepts and using them to sketch out some of the bigger theorems. An intuitive (but not necessarily formal) knowledge of Riemannian Geometry is assumed.

Tuesday, November 19, 2019

11:00 am in 347 Altgeld Hall,Tuesday, November 19, 2019

Two theories of real cyclotomic spectra

Jay Shah

Abstract: The topological Hochschild homology $THH(R)$ constitutes a powerful and well-studied invariant of an associative ring $R$. As originally shown by Bokstedt, Hsiang and Madsen, $THH(R)$ admits the elaborate structure of a cyclotomic spectrum, whose formulation depends upon equivariant stable homotopy theory. More recently, inspired by considerations in p-adic Hodge theory, Nikolaus and Scholze demonstrated (under a bounded-below assumption) that the data of a cyclotomic spectrum is entirely captured by a system of circle-equivariant Frobenius maps, one for each prime p. They also give a formula for the topological cyclic homology $TC(R)$ directly from these maps. The purpose of this talk is to extend the work of Nikolaus and Scholze in order to accommodate the study of real topological Hochschild homology $THR$, which is a $C_2$-equivariant refinement of $THH$ defined for an associative ring with an anti-involution, or more generally an $E_\sigma$-algebra in $C_2$-spectra. The key idea is to make use of the $C_2$-parametrized Tate construction. This is joint work with J.D. Quigley and is based on the arXiv preprint 1909.03920.

Tuesday, December 10, 2019

11:00 am in 347 Altgeld Hall,Tuesday, December 10, 2019

To Be Announced

Foling Zou (U Chicago)