Department of

Mathematics


Seminar Calendar
for Graduate Geometry and Topology Seminar events the year of Monday, May 17, 2021.

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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 29, 2021

4:00 pm in Zoom,Friday, January 29, 2021

Organizaitonal Meeting

Brannon (UIUC)

Abstract: We will be having our first organizational meeting. Please email basilio3(at)illinois(dot)edu for the Zoom information.

Friday, February 12, 2021

4:00 pm in Zoom,Friday, February 12, 2021

Geometry of Knots

Brannon Basilio (UIUC)

Abstract: In this talk, we give an introduction to the geometry of knots. We first start with an example of how to decompose a knot complement into ideal tetrahedron and the conditions needed in order to obtain a hyperbolic structure on the tetrahedron. We then talk briefly of recent work in knot theory that uses this decomposition to obtain bounds on the hyperbolic volume of the knot complement. For Zoom information, please email basilio3 (at) illinois (dot) edu.

Friday, February 26, 2021

4:00 pm in Zoom,Friday, February 26, 2021

A Length and an Area Walk Into a Bar Complex

Cameron Rudd (UIUC)

Abstract: I will discuss some occurrences of lengths and area in geometry. Please contact basilio3 (at) illinois (dot) edu for Zoom information.

Friday, March 5, 2021

4:00 pm in Zoom,Friday, March 5, 2021

To (Conformal) Infinity and Beyond!

Hadrian Quan (UIUC)

Abstract: In this talk I'll describe a few geometric and analytic questions in the context of asymptotically hyperbolic spaces: manifolds which look like hyperbolic space at infinity. I'll do my best to gesture at which of these questions were inspired by conjectures of physicists (such as the AdS-CFT correspondence), while remaining firmly in the context of well-defined mathematical objects. At the end I'll discuss how the geometry of the hyperbolic space inside of Minkowski space can be used to prove theorems on such asymptotically hyperbolic spaces. Email basilio3 (at) illinois (dot) edu for Zoom details.

Friday, March 19, 2021

4:00 pm in Zoom,Friday, March 19, 2021

Free products of Abelian groups in Mod(S)

Chris Loa (UIUC)

Abstract: In 2002, Farb and Mosher introduced a notion of convex cocompactness for mapping class groups. The original notion of convex cocompactness comes from Kleinian groups, where it is a special case of geometric finiteness. In recent work Dowdall, Durham, Leininger, and Sisto have introduced a notion of “parabolic” geometric finiteness for mapping class groups. Examples include convex cocompact groups (as one would hope) and finitely generated Veech groups by work of Tang. In this talk we’ll construct a new family of examples of parabolically geometrically finite groups and show why they are undistorted in Mod(S). Please email basilio3 (at) illinois (dot) edu for Zoom information.

Friday, March 26, 2021

4:00 pm in Zoom,Friday, March 26, 2021

An introduction to CAT(0) cube complexes

Marissa Miller (UIUC)

Abstract: In this talk, I will introduce the notion of a CAT(k) metric spaces, which are spaces that have geometry comparable to complete simply connected surfaces of constant curvature k. We will specifically focus on CAT(0) spaces and will explore CAT(0) cube complexes in some detail, looking at various examples of these complexes and their relationships to questions in geometric group theory. Please email basilio3 (at) illinois (dot) edu for Zoom details.

Friday, April 2, 2021

4:00 pm in Zoom,Friday, April 2, 2021

Oriented Cohomology Theories

Tsutomu Okano (UIUC)

Abstract: I will discuss oriented cohomology theories in both topological and algebro-geometric settings. They naturally come equipped with useful tools such as Thom isomorphisms, Chern classes and Gysin maps. I will give a sketch of the Thom-Pontrjagin construction, from which it follows that complex cobordism is the universal (complex) oriented cohomology theory. For Zoom details, please email basilio3 (at) illinois (dot) edu.

Friday, April 9, 2021

4:00 pm in Zoom,Friday, April 9, 2021

The Yamabe Problem

Xinran Yu (UIUC)

Abstract: In a two-dimensional case, the fact that every Riemann surface has a metric with constant Gaussian curvature leads to a successful classification of Riemann surfaces. Generalizing this property to higher dimensions could be an interesting problem to consider. Thus we seek a conformal metric on a compact Riemannian manifold with constant scalar curvature. The Yamabe problem was solved in the 1980s, due to Yamabe, Trudinger, Aubin, and Schoen. Their solution to the Yamabe problem uses the techniques of calculus of variation and elliptic regularity of the Laplacian. The proof introduces a conformal invariant, so-called the Yamabe invariant, which shifts the focus from an analysis point of view to understanding a geometric invariant. The solution is separated nicely into two cases, regarding the dimension and flatness of a given Riemannian manifold. For Zoom details, please email basilio3 (at) illinois (dot) edu.

Friday, April 16, 2021

4:00 pm in Zoom,Friday, April 16, 2021

Geometric Aspects of Syntactic Categories

Joseph Rennie (UIUC)

Abstract: In this conceptual talk, a natural progression of abstractions will take us from ordinary metric-geometry to a kind of geometry over syntactic categories of first-order theories. No model theory background is assumed. The focus of the talk will be solely on the geometric side, but the talk will end where geometry and logic become indistinguishable. For Zoom details, please email basilio3 (at) illinois (dot) edu.

Friday, April 23, 2021

4:00 pm in Zoom,Friday, April 23, 2021

The Calabi conjecture on non-compact manifolds

Karthik Vasu (UIUC)

Abstract: The problem of finding Einstein metrics involves determining the metric tensor (which is solving for $n^2$ many functions) is in general hard. In the complex case where we have a K\"{a}hler structure this problem reduces to solving for one smooth function through a PDE. Aubin and Yau in their works around 1980 were able to solve this PDE in the zero and negative curvature case on compact manifolds. In this talk I will discuss the geometric setup of the compact case and results in the non-compact case that followed. I will also give a bit of background information to have a high level understanding of the problem. Please email basilio3 (at) illinois (dot) edu for Zoom info.

Friday, May 7, 2021

4:00 pm in Zoom,Friday, May 7, 2021

3-manifolds and number theory

Robert Dicks (UIUC)

Abstract: Arithmetic hyperbolic 3-manifolds and algebraic number theory are intertwined in many ways. For this talk, the speaker will discuss the relation between these 3-manifolds and automorphic forms. Some topics will include the connection between the Langlands program and rational homology 3-spheres and work of Dunfield and Ramakrishnan relating the cohomology of an arithmetic hyperbolic 3-manifold M to the number of its finite covers which fiber over the circle. For Zoom details, please email basilio3 (at) illinois (dot) edu.