Contact for seminar credit: Matthew Blair, blair ["at"] math.unm.edu

Talks

Monday, August 24

Speaker: Matthew Blair

Title: Introduction to Fourier restriction and L^p bounds for eigenfunctions of the Laplacian

Abstract: In this talk, we will introduce the restriction problem for the Fourier transform, which asks when the Fourier transform of a function can be meaningfully restricted to a hypersurface. The problem is often characterized in terms of L^p bounds on the restriction, which can lead to interesting estimates on solutions to PDE, such as the wave, Schrodinger and Helmholtz equations. We then consider a generalization of the latter to compact Riemannian manifolds, which considers L^p bounds on eigenfunctions of the Laplacian in the high frequency limit. Here it is interesting to consider how the geometry of the manifold and the underlying geodesic flow influences these estimates. This talk will provide some foundations for future talks and possibly for the New Mexico Analysis Seminar at UNM in spring of 2016.

Monday, August 31

Speaker: Matthew Blair

Title: Introduction to Fourier restriction and L^p bounds for eigenfunctions of the Laplacian, Part II

Abstract: This is a continuation of the previous lecture, focusing on eigenfunctions of the Laplacian as a generalization of Fourier restriction on the sphere.

Monday, September 14

Speaker: Bishnu Sedai

Title: Trace formulas for a Laplacian perturbed by multiplication by a constant.

Abstract: In this talk, We will see a trace formula for the remainder of the Taylor approximation of an operator function for n=1 and n=2, n is a natural number, and for a Hilbert space H equal to L^2([0,pi]), unbounded self-adjoint operator H_0 equal to the negative Laplacian, bounded self-adjoint operator V equal to multiplication by a constant, and a sufficiently smooth compactly supported function f.

Monday, September 21

Speaker: Bishnu Sedai

Title: Trace formulas for a Laplacian perturbed by multiplication by a constant, Part II.

Abstract: In this talk, We will see a trace formula for the remainder of the Taylor approximation of an operator function for n=1 and n=2, n is a natural number, and for a Hilbert space H equal to L^2([0,pi]), unbounded self-adjoint operator H_0 equal to the negative Laplacian, bounded self-adjoint operator V equal to multiplication by a constant, and a sufficiently smooth compactly supported function f.

Monday, September 28

No Seminar

Monday, October 5

Speaker: Bishnu Sedai

Title: Trace formulas for a Laplacian perturbed by multiplication by a constant, Part III.

Abstract: In this talk, we will see a trace formula for the remainder of the Taylor approximation of an operator function of low order and for a Hilbert space H equal to L^2([0,pi]), unbounded self-adjoint operator H_0 equal to the negative Laplacian, bounded self-adjoint operator V equal to multiplication by a constant, and a sufficiently smooth compactly supported function f.

Monday, November 9

Speaker: Bishnu Sedai

Title: Perturbation of self-adjoint operators with compact resolvents.

Abstract: We will discuss integral representations for the trace of Taylor remainders of operator functions corresponding to self-adjoint perturbations of self-adjoint operators, particularly of self-adjoint operators with resolvent belonging to Schatten-von Neumann ideal of compact operators S^n.

Monday, November 16

Speaker: Maxim Zinchenko

Title: Nonlinear Analogs of Paley–Wiener Theorem.

Abstract: In this talk I will review the classical Paley–Wiener Theorem and its nonlinear analogs appearing in the spectral theory of Jacobi and CMV operators.

Monday, November 23

Speaker: Stephen Lau

Title: On exact transparent boundary conditions for the wave equation.

Abstract: Consider the Cauchy problem for the ordinary (3+1) wave equation. Reduction of the spatial domain to a half-space involves an exact transparent boundary condition enforced on a planar boundary. This boundary condition is most easily formulated in terms of the tangential-Fourier and time-Laplace transform of the solution. Using the Schwartz theory of distributions, we examine two other formulations: (i) the nonlocal spacetime form and (ii) its 3-dimensional (tangential/time) Fourier transform. The spacetime form features a convolution between two tempered distributions. This is joint work with Jim Ellison and Klaus Heinemann.