Title: The essential coexistence phenomenon in Hamiltonian dynamics

Date: 03/26/2019

Time: 3:00 PM - 4:00 PM

Place: C117 Wells Hall

We construct an example of a Hamiltonian flow $f^t$ on a $4$-dimensional smooth manifold $\mathcal{M}$ which after being restricted to an energy surface $\mathcal{M}_e$ demonstrates essential coexistence of regular and chaotic dynamics, that is, there is an open and dense $f^t$-invariant subset $U\subset\mathcal{M}_e$ such that the restriction $f^t|U$ has non-zero Lyapunov exponents in all directions (except the direction of the flow) and is a Bernoulli flow while on the boundary $\partial U$, which has positive volume, all Lyapunov exponents of the system are zero.
This is a continuation of the talk given in previous weeks.

Speaker: Yash Jhaveri, Institute for Advanced Study

Title: Higher Regularity of the Singular Set in the Thin Obstacle Problem

Date: 03/27/2019

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

In this talk, I will give an overview of some of what is known about solutions to the thin obstacle problem, and then move on to a discussion of a higher regularity result on the singular part of the free boundary. This is joint work with Xavier Fernández-Real.

Zemke shows that the map on knot Floer homology induced by a ribbon concordance is injective in his paper.
I will be talking about its applications and proof.

Title: Cluster Structures on Double Bott-Samelson Cells

Date: 03/28/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

Let $G$ be a Kac-Peterson group associated to a symmetrizable generalized Cartan matrix. Let $(b, d)$ be a pair of positive
braids associated to the root system. We define the double Bott-Samelson cell associated to $G$ and $(b,d)$ to be the moduli space of configurations of flags satisfying certain relative position conditions. We prove that they are affine varieties and their coordinate rings are upper cluster algebras. We construct the Donaldson-Thomas transformation on double Bott-Samelson cells and show that it is a cluster transformation. In the cases where $G$ is semisimple and the positive braid $(b,d)$ satisfies a certain condition, we prove a periodicity result of the Donaldson-Thomas transformation, and as an application of our periodicity result, we obtain a new geometric proof of Zamolodchikov's periodicity conjecture in the cases of $D\otimes A_n$. This is joint work with Linhui Shen.

Title: Variance Swaps on Time-Changed Markov Processes

Date: 03/28/2019

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

We prove that a variance swap has the same price as a co-terminal European-style contract, when the underlying is a Markov process, time-changed by a general continuous stochastic clock, which is allowed to have general correlation with the driving Markov process, which is allowed to have state-dependent jump distributions. The European contract’s payoff function satisfies an ordinary integro-differential equation, which depends only on the dynamics of the Markov process, not on the clock. In some examples, the payoff function that prices the variance swap can be computed explicitly. Joint work with Peter Carr and Matt Lorig.

Title: Jensen–Polya Program for the Riemann Hypothesis and Related Problems

Date: 03/28/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In 1927 Polya proved that the Riemann Hypothesis is equivalent to the hyperbolicity of Jensen polynomials for Riemann’s Xi-function. This hyperbolicity had only been proved for degrees $d=1,2,3$. We prove the hyperbolicity of all (but possibly finitely many) the Jensen polynomials of every degree $d$. Moreover, we establish the outright hyperbolicity for all degrees $d< 10^{26}$. These results follow from an unconditional proof of the "derivative aspect" GUE distribution for zeros. This is joint work with Michael Griffin, Larry Rolen, and Don Zagier.

Title: Distinguished Undergraduate Lecture: Why Does Ramanujan, “The Man Who Knew Infinity,” Matter?

Date: 03/29/2019

Time: 4:10 PM - 5:00 PM

Place: B119 Wells Hall

Srinivasa Ramanujan, one of the most inspirational figures in the history of mathematics, was a poor gifted mathematician from lush south India who left behind three notebooks that engineers, mathematicians, and physicists continue to mine today. Born in 1887, Ramanujan was a two-time college dropout. He could have easily been lost to the world, a thought that scientists cannot begin to absorb. He died in 1920. Prof. Ono will explain why Ramanujan matters today, and will share several clips from the film, “The Man Who Knew Infinity,” starring Dev Patel and Jeremy Irons. Professor Ono served as an associate producer and mathematical consultant for the film.
Bio: Ken Ono is the Asa Griggs Candler Professor of Mathematics at Emory University and the Vice President of the American Mathematical Society. He is considered to be an expert in the theory of integer partitions and modular forms. He has been invited to speak to audiences all over North America, Asia and Europe. His contributions include several monographs and over 170 research and popular articles in number theory, combinatorics and algebra. He received his Ph.D. from UCLA and has received many awards for his research in number theory, including a Guggenheim Fellowship, a Packard Fellowship and a Sloan Fellowship. He was awarded a Presidential Early Career Award for Science and Engineering (PECASE) by Bill Clinton in 2000 and he was named the National Science Foundation’s Distinguished Teaching Scholar in 2005. In addition to being a thesis advisor and postdoctoral mentor, he has also mentored dozens of undergraduates and high school students. He serves as Editor-in-Chief for several journals and is an editor of The Ramanujan Journal. He was also an associate producer of the 2016 Hollywood film “The Man Who Knew Infinity” which starred Jeremy Irons and Dev Patel.