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PDEs & Applications Seminar

Friday, November 15th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Christopher Kennedy (Queen's University)

Title: Interaction between long internal waves and free surface waves in deep water (Part II)

Abstract: We present a study of the two-dimensional water wave problem consisting of a density-stratified fluid composed of two immiscible layers separated by a sharp interface. A goal is to describe the interaction between long, larger amplitude, nonlinear waves on the interface and modulated, smaller amplitude, free wave packets on the surface when the lower fluid is infinitely deep. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we describe the resonant interaction of the waves by a system of equations where the internal wave solves a high-order Benjamin-Ono equation coupled to a linear Schroedinger equation for the time evolution of the wave envelope of the free surface. Next, we establish a local well-posedness result for the BO-Schroedinger system in the physical regime where the densities of the two fluid layers are close. Neglecting the higher-order coupling terms, we perform a gauge transformation to eliminate the higher-order non-linear terms and reformulate our BO equation, from which our proof follows by a fixed-point argument. This is a joint work with A. Kairzhan and C. Sulem.

Math & Stats Number Theory Seminar
Thursday, November 28th, 2024

Time: 4:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Anurag Sahay (Purdue University)

Title: The moments of the Hurwitz zeta function with irrational shifts

Abstract: The Hurwitz zeta function is a shifted integer analogue of the Riemann zeta function, with a shift parameter $0 < \alpha \leqslant 1$. We will consider moments of the Hurwitz zeta function on the critical line with a focus on the case where the shift $\alpha$ is irrational. We will briefly review rational $\alpha$, which leads naturally into moments of products of Dirichlet $L$-functions. Heuristics involving random matrix theory can then be used to predict an asymptotic formula for all integer moments. For irrational $\alpha$, we will discuss recent work joint with Winston Heap investigating these moments, where we proved a sharp upper bound for the fourth moment of the order $T(\log T)^2$ assuming that $\alpha$ is not too well-approximable by rationals (concretely, when its irrationality exponent $\mu(\alpha)$ is less than $3$). We also put forth a conjecture for higher moments that suggests that the distribution of the Hurwitz zeta function with irrational shifts on the critical line is approximately Gaussian. This contrasts with the Riemann zeta function (and other $L$-functions from arithmetic), where the analogous fourth moment is of order $T(\log T)^4$ and where the distribution is approximately log-Gaussian instead of Gaussian.

Math & Stats Number Theory Seminar
Thursday, November 7th, 2024

Time: 4:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Ram Murty (Queen's University)

Title: ADELIC ARITHMETIC

Abstract: The adelic perspective introduced by Emil Artin and made famous by his student John Tate in his celebrated thesis, has been central in numerous advances of number theory, especially in the context of automorphic L-functions.  It has had limited impact though on classical analytic number theory. But that will soon change.  I will discuss a paper by Kubota and Sugita that uses adelic probability theory to deduce some classical limit theorems of probabilistic number theory.

 

PDEs & Applications Seminar

Friday, November 8th, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: Christopher Kennedy (Queen's University)

Title: Interaction between long internal waves and free surface waves in deep water

Abstract: We present a study of the two-dimensional water wave problem consisting of a density-stratified fluid composed of two immiscible layers separated by a sharp interface. A goal is to describe the interaction between long, larger amplitude, nonlinear waves on the interface and modulated, smaller amplitude, free wave packets on the surface when the lower fluid is infinitely deep. Starting from the Hamiltonian formulation of this problem and using techniques from Hamiltonian transformation theory, we describe the resonant interaction of the waves by a system of equations where the internal wave solves a high-order Benjamin-Ono equation coupled to a linear Schroedinger equation for the time evolution of the wave envelope of the free surface. Next, we establish a local well-posedness result for the BO-Schroedinger system in the physical regime where the densities of the two fluid layers are close. Neglecting the higher-order coupling terms, we perform a gauge transformation to eliminate the higher-order non-linear terms and reformulate our BO equation, from which our proof follows by a fixed-point argument. This is a joint work with A. Kairzhan and C. Sulem.

Math & Stats Department Colloquium
Friday, November 8th, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Sanjeena Dang (Subedi) (Carleton University)

Title: Clustering microbiome data using a mixture of logistic normal multinomial distributions

Abstract: The human microbiome plays an important role in human health and disease status. Next-generating sequencing technologies allow for quantifying the composition of the human microbiome. Clustering these microbiome data can provide valuable information by identifying underlying patterns across samples. However, clustering these datasets is challenging. Taxa count data in microbiome studies are typically high-dimensional, over-dispersed, and can only reveal relative abundance, and therefore often are treated as compositional. Analysing such compositional data presents many challenges because they are restricted to a simplex. I will present recent advances in clustering microbiome data using a mixture of logistic normal multinomial models. In a logistic normal multinomial model, the relative abundance of the microbiome is mapped from a simplex to a latent variable in the real Euclidean space using the additive log-ratio transformation. While a logistic normal multinomial approach brings flexibility for modelling the data, it comes with a heavy computational cost as the parameter estimation typically relies on Bayesian techniques. In our work, we utilize an efficient framework for parameter estimation using variational Gaussian approximations (VGA). Adopting a variational Gaussian approximation for the posterior of the latent variable reduces the computational overhead substantially. Some other recent and ongoing developments using extensions of LNM distribution to cluster microbiome data will be discussed.

 

Math & Stats Number Theory Seminar
Thursday, October 10th, 2024

Time: 4:30 p.m.  Place: Jeffery Hall, Room 202

Speaker: Becca Carter (Queen's University)

Title: Diameters of directed graphs

Abstract: This talk explores spectral bounds on the diameter of graphs, with a focus on both undirected and directed graphs. I will present two key results: an improved spectral bound for undirected graphs and a novel bound for directed graphs. The improved bound for undirected graphs refines the classic result of Fan Chung from her 1989 paper "Diameters and Eigenvalues". For directed graphs, we introduce a new spectral bound using similar techniques. Central to both results is the representation of graphs as Markov chains, leveraging the transition matrix to derive bounds on the diameter. This spectral approach offers new insights into the structural properties of graphs, with potential applications in network theory.

Dynamics, Geometry and Groups Seminar
Friday, November 1st, 2024

Time: 2:30 p.m.  Place: Jeffery Hall, Room 234

Speaker: Joel Kamnitzer (McGill)

Title: Commutative subalgebras of universal enveloping algebras and moduli spaces of cactus flower curves

Abstract: A classic problem in representation theory is to decompose tensor product representations. I will explain how to do this using Gaudin algebras -- certain commutative subalgebras in tensor products of universal enveloping algebras. These commutative subalgebras are parametrized by the Deligne-Mumford space of genus 0 curves. I will explain how a modification of this construction leads to a new moduli space, called the moduli space of cactus flower curves, which is closely connected with matroid Schubert varieties.

PDEs & Applications Seminar

Friday, November 1st, 2024

Time: 9:30 a.m.  Place: Jeffery Hall, Room 422

Speaker: José Palacios Armesto (University of Toronto)

Title: Linearized dynamic stability for vortices of Ginzburg-Landau evolutions

Abstract: We consider the problem of dynamical stability for the vortex of the Ginzburg-Landau model. Vortices are one of the main examples of topological solitons, and their dynamic stability is the basic assumption of the asymptotic "particle plus field" description of interacting vortices. In this talk we focus on co-rotational perturbations of vortices and establish a variety of pointwise dispersive and decay estimates for their linearized evolution in the relativistic (or Klein-Gordon) case. One of the main ingredients is the construction of the distorted Fourier transform associated with the (two) linearized operators at the vortex. The general approach follows that of Krieger-Schlag-Tataru and Krieger-Miao-Schlag in the context of stability of blow-up for wave maps and relies on the spectral analysis of Schrodinger operators with strongly singular potentials (see also Gezstesy-Zinchenko). However, since the vortex is not given by an explicit formula, and one of the operators appearing in the linearization has zero energy solutions that oscillate at infinity, the linear analysis requires some additional work. In particular, to construct the distorted Fourier basis and to control the spectral measure some additional arguments are needed, compared to previous works. This is joint work with Fabio Pusateri.