���ϳԹ���Դ

Algebra & Geometry Seminar
Tuesday, November 26th, 2024

Time: 3:30 p.m.  Place: Jeffery Hall, Room 422

Speaker:  Rikuto Ito (Nagoya University)

Title: Potential modularity of K3 surfaces with complex multiplication

Abstract: Let X be a K3 surface defined over a number field k. Let T(X) be the transcendental lattice of H^2(X, Q(1)) of rank 22-ρ(X) where ρ(X) is the Picard number of X. We prove that T_ℓ(X, k) is modular if X has complex multiplication over k.

PDEs & Applications Seminar

Friday, November 22nd, 2024

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

Speaker: Catherine Sulem (University of Toronto)

Title: Bloch-Floquet band gaps for linearized water waves over a periodic bottom

Abstract: A central object in the analysis of the water wave problem is the Dirichlet-Neumann operator. This work concerns the study of its spectrum in the context of the water wave system linearized near equilibrium in a domain with a variable bottom, assumed to be a smooth periodic function. We use the analyticity of the Dirichlet-Neumann operator with respect to the bottom variation and combine it with general properties of elliptic systems and spectral theory for self-adjoint operators to develop a Bloch-Floquet theory and describe the structure of its spectrum. We find that under some conditions on the bottom variations, the spectrum is composed of bands separated by gaps, with explicit formulas for their sizes and locations. This is a joint work with Christophe Lacave and Matthieu Ménard.

Math & Stats Department Colloquium
Friday, November 22nd, 2024

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

Speaker: Catherine Sulem (University of Toronto)

Title: A Hamiltonian approach to nonlinear modulation of surface water waves in the presence of linear shear currents

Abstract: This is a study of the water wave problem in a two-dimensional domain in the presence of constant vorticity. The goal is to describe the effects of uniform shear flow on the modulation of weakly nonlinear surface waves. Starting from the Hamiltonian formulation of the water wave problem and using techniques of Hamiltonian transformation theory,  we derive a Hamiltonian, high-order Nonlinear Schrödinger equation (often referred to as Dysthe equation) for the time evolution of the wave envelope. Consistent with previous studies, we observe that the uniform shear flow tends to enhance or weaken the modulational instability of Stokes waves depending on its direction and strength. This model is tested against direct numerical simulations of the full Euler equations and against a related Dysthe equation recently derived by Curtis, Carter and Kalisch (2018).  This is a joint work with Philippe Guyenne and Adilbek Kairzhan.

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.