Department of

# Mathematics

Seminar Calendar
for Graduate Geometry and Topology Seminar events the year of Tuesday, October 26, 2021.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2021          October 2021          November 2021
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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!

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.

Friday, September 10, 2021

4:00 pm in Altgeld Hall 347,Friday, September 10, 2021

#### What is an Alternating Knot?

###### Brannon Basilio

Abstract: One class of knots which is straight forward to understand through its definition is alternating knots. However, although these classes of knots are abundant in knots with a low number of crossings, it becomes increasingly rare to find and even harder to detect as we encounter knots with higher crossing numbers. Alternating knots have certain properties that no other classes of knots have. For example, the degree of its Alexander polynomial is equal to twice its genus. Thus, we may use this to find when a knot is not alternating. However, proving a knot is alternating is a difficult task. We thus state a recent result giving an equivalent definition of knots using only the topology of its complement in S^3. This talk does not assume any knowledge about knots, and relevant definitions will be covered throughout the talk.

Friday, September 17, 2021

4:00 pm in Altgeld Hall 347,Friday, September 17, 2021

#### Lie Groups of Diffeomorphisms

###### Wilmer Smilde (UIUC)

Abstract: The diffeomorphism group of a smooth manifold is really large, yet it is still possible to make sense of it as an infinite-dimensional Lie group. Things get more complicated when we look at subgroups preserving geometric structures. In this talk we will look at a few things: the smooth structure on the group of diffeomorphisms that makes it into a Lie group; subgroups preserving several geometric structures, such as metrics, complex structures, symplectic structures; how Lie groupoids can be used to describe the symmetries of several "intransitive" geometric structures, such as manifolds with a distinguished hypersurface (e.g. a boundary) and foliations. The main focus is on the geometry, so no technical knowlegde of infinite-dimensional manifolds is required. Also, Lie groupoids will only be used at the very end to provide context to some examples, so no worries if you have never seen them before. Hope to see many of you on Friday!

Friday, September 24, 2021

4:00 pm in Altgeld Hall 347,Friday, September 24, 2021

#### A Tour in Symplectic Geometry

###### Jason Liu (UIUC)

Abstract: Symplectic manifolds are smooth even dimensional manifolds equipped with a special two form. They play an important role in many real-world problems, especially some physics problems. In this talk, I’ll give a tour in symplectic geometry starting from the basic definitions and examples. With basic definitions, we can discuss well-known theorems and some open problems. Some famous results like Gromov’s Non-squeezing Theorem and Delzant Theorem will be included. Time permitting, I’ll describe the complexity one problem that I’m currently working on. No prior knowledge about symplectic geometry is required. I’ll try to demonstrate the geometric stories in a more intuitive way, leaving the technical detail aside.

Friday, October 1, 2021

4:00 pm in Altgeld Hall 347,Friday, October 1, 2021

#### Counting the number of essential surfaces in a 3-manifold

###### Chaeryn Lee (UIUC)

Abstract: Essential surfaces play an important role in understanding the nature of 3-manifolds. A natural question that may arise is to see how many distinct essential surfaces a 3-manifold has. In this talk we will look at a certain algorithm that enables us to count the number of isotopy classes of essential surfaces in 3-manifolds. In particular, we will count such surfaces in terms of their Euler characteristic to construct its generating function and then present results about this series.

Friday, October 8, 2021

4:00 pm in Altgeld Hall 347,Friday, October 8, 2021

#### An Introduction to Geometric Quantization

###### Levi Poon (UIUC)

Abstract: The notion of quantization has its origin in physics, where we seek to relate classical systems to their quantum counterparts. It turns out that the mathematical structures of classical and quantum mechanics are surprisingly similar, despite their apparent differences, and it is possible to formalize and (greatly) generalize the process of quantization to a large class of symplectic manifolds. Geometric quantization is one such approach, with interesting connections to representation theory. In this talk, I will give an introduction to geometric quantization. If time permits, I will discuss some interesting questions inspired by this construction. No backgrounds in physics or symplectic geometry will be assumed.

Friday, October 15, 2021

4:00 pm in Altgeld Hall 347,Friday, October 15, 2021

#### Introduction to Geometric Group Theory and Hyperbolicity

###### Marissa Miller (UIUC)

Abstract: Geometric group theory is an area of math that relates groups to metric spaces via their actions on these spaces. Studying the geometric and topological properties of the metric spaces on which groups act can give information about the algebraic properties of a group, as well as the geometric properties of a group (a notion which I will make precise in the talk). I will discuss some of the basic ideas in geometric group theory, and I will provide some examples of theorems in geometric group theory using Gromov-hyperbolic groups.

Friday, October 22, 2021

4:00 pm in Altgeld Hall 347,Friday, October 22, 2021

#### The Alexander Polynomial, the Alexander Polynomial, and the Twisted Alexander Polynomial

###### Joseph Malionek (UIUC)

Abstract: The Alexander polynomial is a topological invariant of a space coming from a somewhat natural module structure on one of its covers. It was originally developed for knot exteriors, but since has been defined for more general topological spaces. This talk goes through the construction for knot exteriors, and then two levels of generalizations (As well as several examples). Some highlights: Covering spaces! Modules! T W I S T I N G. Familiarity with algebraic topology and a very small amount of module theory will be helpful.

Friday, October 29, 2021

2:00 pm in 347 Altgeld Hall,Friday, October 29, 2021

#### Localization and Laplacians

###### Gayana Jayasinghe (UIUC)

Abstract: Localization can be broadly described as a phenomenon where certain integrands turn out to only be supported near special points. The classic result of this form is stationary phase. I will talk about different instances of localization and the analysis involved. I'll begin with stationary phase and move onto Atiyah Bott localization, the Duistermaat-Heckman theorem, and Witten deformation. For more on Witten deformation, check out the Graduate Geometry and Topology Seminar this week.

4:00 pm in Altgeld Hall 347,Friday, October 29, 2021

#### Witten Deformation, or Turning Critical Points into Cohomology

###### Gayana Jayasinghe (UIUC)

Abstract: Witten deformed the de Rham complex to relate critical points to harmonic representatives of cohomology. I will explain this relationship, and sketch out Witten's proof of the Morse inequalities. I will then try and explain how these ideas have been developed on infinite dimensional spaces.

Friday, November 5, 2021

4:00 pm in Altgeld Hall 347,Friday, November 5, 2021

#### An introduction to supermanifolds

###### Yigal Kamel (UIUC)

Abstract: Super mathematics concerns the extension of commutative mathematics to include the anti-commutative setting. For example, a superalgebra, is a Z/2-graded algebra which decomposes into commutative and anti-commutative summands. Accordingly, supermanifolds are the corresponding Z/2-graded generalization of manifolds. In this talk, I will introduce supermanifolds from two perspectives: the concrete approach, which constructs supermanifolds out of charts to a superspace (as with ordinary manifolds), and the algebro-geometric approach which extends the sheaf of continuous functions on a manifold to the super setting.

Friday, November 19, 2021

4:00 pm in Altgeld Hall 347,Friday, November 19, 2021