Thomas Beck

Associate Professor

Mathematics Department at Rose Hill

Fordham University

John Mulcahy Hall 401

Bronx, NY 10458

Email: tbeck7 at fordham dot edu

I am an Associate Professor in the Department of Mathematics at Fordham University.

My research interests include partial differential equations and spectral theory. In general, I study problems related to Laplace eigenfunctions and minimizers of other variational problems in convex, or near-convex settings, with applications to level set and eigenvalue estimates, monotonicity formulae, and domain decomposition methods in numerical analysis.

My research has previously been supported by the NSF Grant DMS 2042654, 'Estimates on eigenvalues and eigenfunctions in convex settings'.

Previously, I was a Postdoctoral Research Associate in the Department of Mathematics at the University of North Carolina at Chapel Hill (Mentor: Jeremy Marzuola) and a C.L.E. Moore Instructor at MIT (Mentor: David Jerison). I obtained my PhD in Mathematics at Princeton University (Advisor: Charles Fefferman).

Recent Work

Uniform upper bounds on Courant sharp Neumann eigenvalues of chain domainsWe obtain upper bounds on the number of nodal domains of Laplace eigenfunctions on chain domains with Neumann boundary conditions. The chain domains consist of a family of planar domains, with piecewise smooth boundary, that are joined by thin necks. Our work does not assume a lower bound on the width of the necks in the chain domain. As a consequence, we prove an upper bound on the number of Courant sharp eigenfunctions that is independent of the widths of the necks.

Nodal Set Openings on Perturbed Rectangular DomainsWe study the effects of perturbing the boundary of a rectangle on the nodal sets of eigenfunctions of the Laplacian. Namely, for a rectangle of a given aspect ratio $N$, we identify the first Dirichlet mode to feature a crossing in its nodal set and perturb one of the sides of the rectangle by a close to flat, smooth curve. Such perturbations will often "open" the crossing in the nodal set, splitting it into two curves, and we study the separation between these curves and their regularity. The main technique used is an approximate separation of variables that allows us to restrict study to the first two Fourier modes in an eigenfunction expansion. We show how the nature of the boundary perturbation provides conditions on the orientation of the opening and estimates on its size. In particular, several features of the perturbed nodal set are asymptotically independent of the aspect ratio, which contrasts with prior works. Numerical results supporting our findings are also presented.

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