Mathematics Home / Colloquium: Spring 2017

Time & Location: All talks are on Thursdays in Dinwiddie 102 at 3:30 pm unless otherwise noted. Refreshments in Gibson 426 after the talk.

Comments indicating vacations, special lectures, or change in location or time are in green.

Organizer: Gustavo Didier

Abstract: Exact enumeration in probability and combinatorics often leads to rational generating functions. These, in turn, lead to limit shapes, often more exotic than the generic Gaussian shape arising from a Central Limit Theorem. This motivates the study of the asymptotics of the coefficients of a rational power series

F(Z) = P(Z) / Q(Z) = sum_R a_R Z^R

where Z = (z_1, ... , z_d) and R = (r_1, ..., r_d) are d-tuples. Estimating a_R from P and Q is both a theoretical problem and a problem in effective computation. I will discuss what we know about how to read limit shape behavior from P and Q (mostly from Q). The examples in the pictures will all be explained.

Global Regularity v.s. Finite Time Blowup for Compressible Euler Equations

Abstract:

As one of the oldest nonlinear PDE systems, the compressible Euler equations has been studied by many outstanding mathematicians. However, some basic questions, such as the global existence of classical solution v.s. finite time blowup, are still open even in one space dimension. In this lecture, we will report our recent progress in this direction, including a complete understanding on isentropic flows, and a refreshed understanding on general adiabatic flows. This lecture is based on joint works with H. Cai, G. Chen, S. Zhu, and Y. Zhu.

Jonathan MattinglyDuke (Host: Nathan Glatt-Holtz and Scott McKinley)

Abstract:

Ergodicity is one of the fundamental questions for a stochastic dynamical system, ensuring the convergence of long time averages of observable quantities to a statistical steady state independent of the initial condition.

I will explore why the ergodic theory of stochastic PDEs is different and how it underlines the basic difference between ODEs and PDEs. I will start at the beginning giving a crash course on the basic elements needed to prove an ergodic result. We will come to understand why sometimes ergodicity can be easy for hard PDEs. Time permitting I will touch on hypoellipticity in infinite dimensions and singular PDEs.

Patrick FlandrinCNRS and ENS-Lyon (host: Didier Gustavo)

Abstract:

Disentangling multicomponent nonstationary signals into coherent AM-FM modes is usually achieved by identifying « loud » time-frequency trajectories where energy is locally maximum. We will here present an alternative perspective that relies on « silent » points, namely spectrogram zeros. Based on the theory of Gaussian analytic functions, a number of results will be presented regarding the distribution of such zeros considered as a point process in the plane, with repulsive properties. The rationale and the implementation of the zeros-based approach for recovering signals embedded in noise will then be discussed, with an application to the extraction and characterization of actual gravitational wave chirps.

Non-Archimedean analytic curves in algebraic varieties

Abstract:

I will explain Serge Lang's conjectures relating four aspects of algebraic varieties: geometric structure, rational maps from group varieties, rational points, and holomorphic maps from the complex plane. I will then explain what happens if one considers non-Archimedean analytic maps instead of complex holomorphic maps and discuss a parallel conjectural framework. I will give a broad overview and intend that the talk will be accessible even to those with limited experience with algebraic geometry and non-Archimedean analysis.

Nonuniqueness of weak solutions to the SQG equation

Abstract:

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering an open problem posed by De Lellis and Szekelyhidi Jr. Moreover, we show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian. This talk is based on a joint work with T. Buckmaster and S. Shkoller.

Special Colloquium

The power of crystal bases

Abstract:

Crystal bases originally arose from the representation theory of quantum groups. They can, however, be defined from a purely combinatorial point of view. One of their features is that the character of a highest weight crystal in type A is a Schur function. Hence imposing crystal structures on combinatorial sets can be used to prove positive expansions of symmetric functions in terms of Schur functions. We will demonstrate this for Stanley symmetric function, which capture properties of reduced words of an element in the symmetric group. We shed new light on their Schur expansion by giving a crystal theoretic interpretation in terms of decreasing factorizations of permutations. Whereas

crystal operators on tableaux are coplactic operators, the crystal operators on decreasing factorization intertwine with the Edelman-Greene insertion.

It is fun to make noise

Abstract:

Motion of a particle in a noisy environment is described by a Newtons's equation with an additional term, which models the noise. General theory of such stochastic differential equations (SDE) is very well established mathematically, but particular SDE models of physical systems offer many surprises. I will describe an experiment which detected an unexpected force acting on a Brownian particle, and the resulting general theory of noise-induced drifts in dynamical systems. If time permits, I will include a very recent result on non-Markovian systems, which occur naturally in applications. The results I will present were obtained jointly with several collaborators, including Giovanni Volpe, Scott Hottovy, Austin McDaniel and Soon Hoe Lim.

Uniqueness of critical points in the variational problems

Abstract:

Variational methods play an essential role in many areas of mathematics such as analysis, geometry, or mathematical physics. One of the important goals is to understand properties of minimizers, or more generally critical points, of functionals represented by energy, entropy etc. Since the critical points are in often equivalent to solutions of differential equations, their analysis provide a valuable insight into dynamics of very complicated systems. Although basic properties include their existence, uniqueness, and regularity of critical points, the literature provide only very basic criteria for the uniqueness . To close this gap, we prove a unified and general criterion for the uniqueness of critical points of a functional in or without the presence of constraints such as positivity, boundedness, or fixed mass. Our method relies on convexity properties along suitable paths and significantly generalizes well-known uniqueness theorems. Due to the flexibility in the construction of the paths, our approach does not depend on the convexity of the domain and can be used to prove uniqueness in subsets, even if it does not hold globally. The results apply to all critical points and not only to minimizers, thus they provide uniqueness of solutions to the corresponding Euler-Lagrange equations.To illustrate our method we present a unified proof of known results, as well as new theorems.

This is a joint work with Denis Bonheure, Ederson Moreira dos Santos, Alberto Saldana, and Hugo Tavares