Department of

Mathematics


Seminar Calendar
for Graduate Geometry Learning Seminar events the year of Friday, September 14, 2018.

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More information on this calendar program is available.
Questions regarding events or the calendar should be directed to Tori Corkery.
     August 2018           September 2018          October 2018    
 Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa   Su Mo Tu We Th Fr Sa
           1  2  3  4                      1       1  2  3  4  5  6
  5  6  7  8  9 10 11    2  3  4  5  6  7  8    7  8  9 10 11 12 13
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                        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

Hadrian Quan

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

Hadrian Quan

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

Hadrian Quan

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.