Title: The essential coexistence phenomenon in Hamiltonian dynamics

Date: 03/12/2019

Time: 3:00 PM - 4:00 PM

Place: C304 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 in previous week.

Speaker: Kristen Hendricks, Michigan State University

Title: Classical and Modern Invariants of Knots

Date: 03/12/2019

Time: 4:00 PM - 5:00 PM

Place: C304 Wells Hall

We'll give a brief introduction to what knot theory is and why you might want to study it, and talk about some classical invariants of knots and what they detect. We'll then introduce a modern invariant called Heegaard Floer knot homology from the early 2000s, and talk about its properties and its relationship to classical invariants. This talk should be accessible to undergraduate students.

Speaker: Yusuf Mustopa, UMass Amherst/Tufts University

Title: Effective Global Generation on Varieties with Numerically Trivial Canonical Bundle

Date: 03/13/2019

Time: 3:00 PM - 4:00 PM

Place: C304 Wells Hall

Fujita’s freeness conjecture predicts that if X is a smooth projective variety and A is an ample divisor on X, then K+mA is basepoint-free when m is at least dim(X)+1. Although this statement is optimal (as can be seen when X is projective space) there are much better statements for abelian varieties and surfaces with numerically trivial canonical bundle. In this talk, I will discuss a result of Fujita type for smooth projective varieties having numerically trivial canonical bundle, as well as its application to moduli spaces of sheaves on abelian surfaces. This is joint work with Alex Kuronya.

Classically, Sliding Window Embeddings were used in the study of dynamical systems to reconstruct topology of underlying attractors from generic observation functions. In 2015, Perea and Harer studied persistent homology of sliding window embeddings from L^2 periodic functions. We define a quasiperiodic function as a superposition of periodic functions with incommensurate frequencies. As it turns out, sliding window embeddings of quasiperiodic functions are dense in high dimensional tori. In this talk, I will present some results for the quasiperiodic case.

Title: Local smoothing estimates for Schrödinger equations on hyperbolic space and applications

Date: 03/13/2019

Time: 4:10 PM - 5:00 PM

Place: C517 Wells Hall

We establish frequency-localized local smoothing estimates for Schrödinger equations on hyperbolic space. The proof is based on the positive commutator method and a heat flow based Littlewood-Paley theory. Our results and techniques are motivated by applications to the problem of stability of solitary waves to nonlinear Schrödinger-type equations on hyperbolic space.
The talk is based on joint work with Andrew Lawrie, Sung-Jin Oh, and Sohrab Shahshahani.

Title: Cluster structure on moduli spaces of local systems for general groups

Date: 03/14/2019

Time: 3:00 PM - 4:00 PM

Place: C204A Wells Hall

There have been several references in the literature devoted to the study of the cluster structures on moduli spaces of G-local systems, all of which are based on case by case study. In this talk, we present a systematic construction that works for all groups at once. As an application, we will investigate the principal series representations of quantum groups from the perspective of cluster theory.

The classical necklace problem asks, given q possible colors of beads, how many ways to string n beads around a necklace, counting rotations as the same. This has a nice solution using Mobius inversion from number theory. Amazingly, necklaces also give a way to picture the elements of a finite field with q^n elements, as well as a basis of the free Lie algebra with q generators.

Title: Casson's invariant for oriented integer homology spheres

Date: 03/18/2019

Time: 1:40 PM - 3:00 PM

Place: C304 Wells Hall

This talk is an exposition of Casson's invariant: we will cover the definition of Casson's invariant in terms of Heegaard decomposition and representation spaces, and show how it can be computed (and defined) in terms of the Alexander polynomial of knots.

Title: Local Class Field Theory is Easy! Part I: Introduction

Date: 03/18/2019

Time: 3:00 PM - 4:00 PM

Place: C517 Wells Hall

Local class field theory is about classifying all abelian extensions over a base local
field (for instances Q_p , R, C). It turns out that this extrinsic data is completely determined by
the intrinsic properties of the base field. We will start by reviewing some basic number theory
facts. We will then discuss statements in local class field theory and see some interesting
examples. We assume a basic knowledge of commutative algebra, infinite Galois theory and
number theory.

Given a von Neumann algebra M equipped with a trace, any self-adjoint operator in M can be thought of as a non-commutative random variable. For an n-tuple X of such operators, the free Stein information of X is a free probabilistic quantity defined by the behavior of a non-commutative Jacobian on the polynomial algebra generated by entries of X. It is a number in the interval [0,n] and its value can provide information about the entries of X as well as the von Neumann algebra they generate. In this talk, I will discuss these and other properties of the free Stein information and consider a few examples where it can be explicitly computed. This is based on joint work with Ian Charlesworth.

Title: A Strategy for Addressing Elementary Pre–Service Teachers’ Mathematics Anxiety

Date: 03/20/2019

Time: 12:00 PM - 1:00 PM

Place: 133F Erick

Mathematics anxiety among elementary preservice teachers is a well–documented phenomenon that greatly affects their ability to engage in teacher preparation courses (e.g., Dutton, 1951; Gresham, 2007; Sloan, 2010). One way for instructors to engage with PSTs is to interact with them informally (Lamport, 1993). Informal conversations present an opportunity to increase students’ confidence and address their anxiety regarding mathematics content. A potential venue for informal conversations is office hours; however, college students often do not take advantage of office hours that are offered. This talk will describe preliminary results of a policy designed to increase instances of informal interactions between students and their instructors during office hours, by solely providing homework solutions to students during office hours. Initial evidence from surveys and course evaluations suggests that students who come into office hours engage with the instructor on topics they did not intend to discuss before coming to office hours, and suggests that these conversations have the potential to help reduce mathematics anxiety.

Title: Poincare type inequalities via 1-dimensional Malliavin calculus

Date: 03/21/2019

Time: 3:00 PM - 3:50 PM

Place: C405 Wells Hall

We will review briefly 3 types of operators which are mapping spaces of real-valued functions which are defined on the real line equipped with standard normal probability measure. Those are the derivative, divergence and Ornstein-Uklenbeck operators. There are simple formulas that describe the relationships between those operators. Using those formulas the proofs of the following will be presented:
1. Poincare inequality : The variance of a function of N(0,1) is dominated by the second moment of its derivative.
2. An upper bound to the Wasserstein distance between the distribution of a function of N(0,1) (the function has mean 0 and standard deviation 1) and N(0,1) itself. This upper bound is (up to a constant) the multiplication of the L4 norm of the function derivative and the L4 norm of the function 2nd derivative.
The material is based on Nourdin and Peccati book.

Title: Cluster Structures on Double Bott-Samelson Cells

Date: 03/21/2019

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.

Speaker: Wilfrid Gangbo, University of California, Los Angeles

Title: A weaker notion of convexity for Lagrangians not depending solely on velocities and positions

Date: 03/21/2019

Time: 4:10 PM - 5:00 PM

Place: C304 Wells Hall

In dynamical systems, one often encounters actions $\mathcal{A}\equiv \int_{\Omega}L(x, v(x))\rho dx$ which depend only on $v$, the velocity of the system and on $\rho$ the distribution of the particles. In this case, it is well–understood that convexity of $L(x, \cdot)$ is the right notion to study variational problems. In this talk, we consider a weaker notion of convexity which seems appropriate when the action depends on other quantities such as electro–magnetic fields. Thanks to the introduction of a gauge, we will argue why our problem reduces to understanding the relaxation of a functional defined on the set of differential forms (Joint work with B. Dacorogna).

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: 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.