Department of

Mathematics


Seminar Calendar
for Topology Seminar events the year of Wednesday, September 16, 2020.

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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 24, 2020

4:00 pm in 141 Altgeld Hall,Friday, January 24, 2020

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 31, 2020

4:00 pm in 141 Altgeld Hall,Friday, January 31, 2020

Introduction to Orbifolds

Brannon Basilio (UIUC)

Abstract: We can generalize the notion of a manifold to include singularities; thus we can define a new object called orbifolds. In this talk, we will give an introduction to the notion of orbifolds, including examples, covering orbifold, Euler number of an orbifold, and the classification theorem of 2-orbifolds.

Friday, February 7, 2020

4:00 pm in 141 Altgeld Hall,Friday, February 7, 2020

A probabilistic approach to quantizing Yang-Mills theory

Kesav Krishnan (UIUC)

Abstract: I will introduce the problem of rigorously quantizing Yang Mills Theory, and how probability theory can be used to that end. If time permits, I will talk about the discrete gauge-string duality as introduced by Sourav Chatterjee

Tuesday, February 11, 2020

11:00 am in 243 Altgeld Hall,Tuesday, February 11, 2020

Vanishing and Realizability

Shane Clark (University of Kentucky)

Abstract: The Reidemeister trace of an endomorphism of a CW complex is a lower bound for the number of fixed points (up to homotopy) of that endomorphism. For an endomorphism $f$, the Reidemeister trace of $f^n$ is a lower bound for the number of fixed points of $f^n$, however it can be a far from an optimal lower bound. One method of addressing this discrepancy constructs an equivariant map, the n^{th} Fuller trace $f$, which carries information about the periodic points of a map $f$. However, we must ask how much information is retained by this equivariant construction? In this talk we show that the n^{th} Fuller trace of $f$ is a complete invariant for describing a minimum set of periodic points for maps of tori.

Friday, February 14, 2020

4:00 pm in 141 Altgeld Hall,Friday, February 14, 2020

Bounds on volumes of mapping tori

Heejoung Kim (UIUC)

Abstract: For a surface $S$ and a homeomorphism $f: S\to S$, the mapping torus of $S$ by $f$ is defined by $M_f=(S\times [0,1])/((x,0)\sim (f(x), 1))$. In particular, for a closed surface $S$ of genus at least 2 and a pseudo-Anosov element $f$ of the mapping class group of $S$, $M_f$ is a hyperbolic manifold. Brock provided bounds of the hyperbolic volume of $M_f$ from a hyperbolic structure on $M_f$ by using its Weil-Petersson metric. And then Agol gave a sharp upper bound for the volume in terms of the translation distance on the pants graph $P(S)$ which is associated with pants decomposition on $S$. In this talk, we will discuss mapping class groups and Agol's proof on the upper bound.

Friday, February 21, 2020

4:00 pm in 141 Altgeld Hall,Friday, February 21, 2020

Unifying Galois Theories with Categorification

Robert (Joseph) Rennie (UIUC)

Abstract: Since its inception nearly two centuries ago, what we call "Galois Theory" (say in an undergraduate algebra course) has led to many analogous results, and thus attained the status of a sort of metatheorem. In Galois' case, this concept was applied to fields, yielding an equivalence between some lattice of field extensions and a lattice of subgroups of a corresponding "galois group" ... under certain conditions. Later on, the same concept was shown to be present in Topology, with extensions being replaced by their dual notion of covering spaces, and the galois group being replaced by the fundamental group... again, under certain conditions. Even later, Galois' results for fields were generalized to arbitrary rings, introducing new associated data along the way. In this talk, we explore the process of formally unifying all of these "Galois Theories" into one Galois Principle, with the aim of developing an intuition for identifying some of its infinite use-cases in the wilds of Math (e.g. Algebra, Topology, and Logic). Along the way, I aim to discuss explicitly and to motivate categorification to the working mathematician using the results of this talk as concrete examples.

Tuesday, February 25, 2020

11:00 am in 243 Altgeld Hall,Tuesday, February 25, 2020

\'Etale K-theory

Akhil Mathew (U Chicago)

Abstract: I will explain some general structural results about algebraic K-theory and its \'etale sheafification, in particular its approximation by Selmer K-theory. This is based on some recent advances in topological cyclic homology. Joint work with Dustin Clausen.

Friday, February 28, 2020

4:00 pm in 141 Altgeld Hall,Friday, February 28, 2020

Arnold-Liouville Theorem

Jungsoo (Ben) Park (UIUC)

Abstract: This talk will be an introduction to fundamental concepts of symplectic geometry. Furthermore, we will delve into the proof of Arnold-Liouville theorem: https://en.wikipedia.org/wiki/Liouville–Arnold_theorem.

Tuesday, March 3, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 3, 2020

Approximating higher algebra by derived algebra

William Balderrama

Abstract: Obstruction theories and spectral sequences from the basic computational tools for accessing complex homotopical structure by means of pure algebra. Many of these are constructed via a careful examination of the relevant notion of ``free resolution''; the difficulties in their construction are then in maintaining sufficient control over these resolutions, as well as in the identification of the relevant obstruction groups. I will describe a general conceptual framework for producing these tools, based on a higher categorical variation on the notion of an algebraic theory, which is easily applicable to a wide variety of situations and provides a direct bridge between homotopical structure and algebraic structure.

Tuesday, March 10, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 10, 2020

$C_2$-equivariant homotopy groups of spheres

Mark Behrens (Notre Dame Math)

Abstract: I will explain how $RO(C_2)$-graded $C_2$-equivariant homotopy groups of spheres can be deduced from non-equivariant stable homotopy groups of stunted projective spaces, and the computation of Mahowald invariants.

Friday, March 13, 2020

4:00 pm in 141 Altgeld Hall,Friday, March 13, 2020

Poincare duality for singular spaces

Gayana Jayasinghe (UIUC)

Abstract: Poincare duality of manifolds is a classical theorem which can be phrased in terms of the homology and cohomology groups of manifolds. However, when we look at singular spaces, this fails to hold for the usual homology and cohomology groups. In the setting of a certain class of singular spaces know as topological pseudomanifolds, which include orbifolds, algebraic varieties, moduli spaces and many other natural objects, one can extend these groups in order to recover some form of Poincare duality. I'll present how this was achieved by Goresky and MacPherson with their Intersection homology, and by Cheeger using L^2 cohomology and explain how they are related to each other, in similar spirit to the equivalence in the smooth setting. I'll only assume a basic knowledge of homology and cohomology.

Tuesday, March 24, 2020

11:00 am in 243 Altgeld Hall,Tuesday, March 24, 2020

CANCELLED

Christina Osborne (OSU Math)

Tuesday, April 7, 2020

11:00 am in 243 Altgeld Hall,Tuesday, April 7, 2020

To Be Announced

Ben Antieau (UIC )

Friday, August 28, 2020

4:00 pm in Zoom,Friday, August 28, 2020

Organizational Meeting

Brannon Basilio (UIUC)

Abstract: Organizational meeting to discuss who wants to give talks and when. The link will be sent to gradgeomtopo mailing list, but please email Brannon basilio3 at illinois for edu to join the list.

Friday, September 11, 2020

4:00 pm in Zoom,Friday, September 11, 2020

Introduction to Seifert surfaces and Knot Invariants

Brannon Basilio (UIUC)

Abstract: Seifert surfaces are used to study invariants of knots. In this talk, we introduce the notion of Seifert surfaces and how they give rise to the Alexander invariant and ultimately the Alexander polynomial through the universal abelian covering space.

Tuesday, September 15, 2020

11:00 am in Zoom,Tuesday, September 15, 2020

How do field theories detect the torsion in topological modular forms?

Dan Berwick-Evans   [email] (UIUC)

Abstract: Since the mid 1980s, there have been hints of a connection between 2-dimensional field theories and elliptic cohomology. This lead to Stolz and Teichner's conjectured geometric model for the universal elliptic cohomology theory of topological modular forms (TMF) for which cocycles are 2-dimensional (supersymmetric, Euclidean) field theories. Properties of these field theories lead naturally to the expected integrality and modularity properties of classes in TMF. However, the abundant torsion in TMF has always been mysterious from for the supersymmetric sigma model with target determines a cocycle representative of the generator of pi_3(TMF) = Z/24. (Contact Jeremiah Heller for Zoom info: jbheller@illinois.edu)

Friday, September 18, 2020

4:00 pm in Zoom,Friday, September 18, 2020

Floer theory and low-dimensional topology

Nachiketa Adhikari (UIUC)

Abstract: This talk will be an introduction to Floer theory and the key ideas behind it. If time permits, I will also sketch its connections to low-dimensional topology and, in particular, knots. Please email Brannon at basilio3 AT illinois DOT edu for the Zoom details.

Friday, September 25, 2020

4:00 pm in Zoom,Friday, September 25, 2020

Framed Cobordism and the Homotopy Groups of Spheres

Brian Shin

Abstract: The computation of the homotopy groups of spheres has dominated the minds of homotopy theorists since the 1930s. To this day, much is still not known about these groups. In this expository talk, I'd like to discuss one of the earliest approaches to understanding these groups: the geometry of framed cobordisms. Please contact basilio3 (at) illinois (dot) edu for Zoom details.