Department of

# Mathematics

Seminar Calendar
for Graduate Geometry Learning Seminar events the year of Thursday, October 11, 2018.

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events for the
events containing

Questions regarding events or the calendar should be directed to Tori Corkery.
    September 2018          October 2018          November 2018
Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
1       1  2  3  4  5  6                1  2  3
2  3  4  5  6  7  8    7  8  9 10 11 12 13    4  5  6  7  8  9 10
9 10 11 12 13 14 15   14 15 16 17 18 19 20   11 12 13 14 15 16 17
16 17 18 19 20 21 22   21 22 23 24 25 26 27   18 19 20 21 22 23 24
23 24 25 26 27 28 29   28 29 30 31            25 26 27 28 29 30
30


Friday, September 7, 2018

1:00 pm in 2 Illini Hall,Friday, September 7, 2018

#### Dynamics and Rigidity 1: Introduction to Ergodic Theory

###### Venkata Sai Narayana Bavisetty

Abstract: This talk will be an introduction to Ergodic Theory with the goal being to develop the necessary background to prove and understand the ergodicity of the geodesic flow on hyperbolic manifolds.

Wednesday, September 12, 2018

1:00 pm in 2 Illini Hall,Wednesday, September 12, 2018

#### Dynamics and Rigidity 2: Ergodicity of Geodesic Flow

###### Venkata Sai Narayana Bavisetty

Abstract: I will first present a proof of the ergodicity of the geodesic flow for compact hyperbolic surfaces and then will generalize these ideas to prove ergodicity of the geodesic flow for compact hyperbolic n-manifolds.

Friday, September 14, 2018

1:00 pm in 2 Illini Hall,Friday, September 14, 2018

#### Dynamics and Rigidity 3: Mostow Rigidity Theorem

###### Cameron Rudd

Friday, September 28, 2018

1:00 pm in 2 Illini Hall,Friday, September 28, 2018

#### Dynamics and Spectral Theory 1: Weyl’s Law and Geodesic Flow

Abstract: In this talk I’ll introduce some results relating the dynamics of geodesic flow and the eigenvalues of the Laplace operator. We’ll begin by investigating how the asymptotics of eigenvalue growth can change in the presence of dynamical hypotheses, and use these results to motivate this interplay of analysis and geometry. This talk will be focused on examples and should have minimal analytic prerequisites.

Wednesday, October 3, 2018

1:00 pm in 2 Illini Hall,Wednesday, October 3, 2018

#### Dynamics and Spectral Theory 2: Hearing the length spectrum

Abstract: In this talk, we’ll introduce the wave kernel, and demonstrate how an analysis of its singularities can (sometimes!) determine the lengths of closed geodesics on compact manifolds. We’ll focus on surfaces, and study the relation between geodesic length rigidity and Laplace spectral rigidity. This will necessarily involve some results for pseudodifferential operators, however this will be presented alongside a “user’s guide” to pseudodifferential operators.

Friday, October 5, 2018

1:00 pm in 2 Illini Hall,Friday, October 5, 2018

#### Dynamics and Spectral Theory 3: Microlocal Lifts and Quantum Unique Ergodicity

Abstract: Concluding our series, we’ll generalize the asymptotics of the first talk to prove a “local” Weyl’s law. Then, we'll use this to prove an Ergodic theorem in the spirit of the Classical-Quantum Correspondence: the "quantum average" of an operator is equal to the "phase space average" of its principal symbol, "almost always". I’ll explain how this can be turned into mathematical statement, how to improve on it, and explain why both number theorists and geometers might care.

Friday, October 12, 2018

1:00 pm in 2 Illini Hall,Friday, October 12, 2018

#### Clifford algebras and $K$-theory I: The structures and representations of Clifford algebras

###### Ningchuan Zhang

Abstract: In this talk, I’ll introduce Clifford algebras and study their structures. The goal is to demonstrate a periodicity phenomenon arising from the representations of Clifford algebras that resembles the Bott periodicity for $K$-theory.

Wednesday, October 17, 2018

1:00 pm in 2 Illini Hall,Wednesday, October 17, 2018

#### Clifford algebras and K-theory II: The Atiyah-Bott-Shapiro construction

###### Ningchuan Zhang

Abstract: Abstract: In this talk, I’ll construct the Atiyah-Bott-Shapiro map that relates the periodicity of Clifford algebras to the Bott periodicity of $K$-theory. From there, I will explain the Thom isomorphism theorem for $K$ ($KO$)-theory and its relations to $\mathrm{Spin}^c$ ($\mathrm{Spin}$)-structures on a vector bundle. I’ll also talk about how the ABS construction is related to the $\hat{A}$-genus if time allows.

Friday, October 26, 2018

1:00 pm in 2 Illini Hall,Friday, October 26, 2018

#### Hodge theory and singularities I

###### Sungwoo Nam

Abstract: In this talk, I will describe classical Hodge theory on compact Kahler manifolds and discuss its application to geometry of complex manifolds, such as hard Lefschetz theorem and Lefschetz (1,1) theorem. Then I will try to introduce(or at least motivate, if time does not permit) extension of pure Hodge theory needed to study singularities, mixed Hodge structures and variation of Hodge structures.

Wednesday, October 31, 2018

1:00 pm in 2 Illini Hall,Wednesday, October 31, 2018

#### Hodge theory and singularities II

###### Sungwoo Nam

Abstract: After motivating singularities, I’ll introduce Deligne's mixed Hodge structure, which, unlike pure Hodge structure, sees singularities. Then I’ll discuss(with examples of Riemann surfaces) how one can use it to study singularities. I’ll end by connecting it to an isolated singularity coming from a family, focusing on Lefschetz degeneration.