We discuss efficient numerical algorithms for solving parameterized partial differential equations. These include reduced-basis methods, in which parameterized approximate solutions are constructed from a space of dimension significantly smaller than the dimension of the spatial discretization; stochastic Galerkin methods, in which a large deterministic solution is specified to produce approximate solutions that are easily evaluated; and stochastic collocation methods, in which approximation based on interpolation using so-called sparse grid methods.  We outline the properties and costs of these methods and compare their performance on benchmark problems.

Numerous manifestations of wave localization permeate acoustics, quantum physics, mechanical and energy engineering. It was used in construction of noise abatement walls, LEDs, optical devices, to mention just a few applications. Yet, no systematic methods could predict the exact spatial location and frequencies of the localized waves.

In this talk I will present recent results revealing a new criterion of localization, tuned to the aforementioned questions, and will illustrate our findings in the context of the boundary problems for the Laplacian and bilaplacian, $div A\nabla$,  and (continuous) Anderson and Anderson-Bernoulli models on a bounded domain. Via a new notion of ``landscape" we connect localization to a certain multi-phase free boundary problem and identify location, shapes, and energies of localized eigenmodes. The landscape further provides estimates on the rate of decay of eigenfunctions and delivers accurate bounds for the corresponding eigenvalues, in the range where both classical Agmon estimates and Weyl law may fail.   

In an award winning 2014 paper, Damanik, Fillman, and Gorodetski rigorously established a framework for investigating Schrodinger operators on the real line whose potentials are generated by ergodic subshifts. In the case of the Fibonacci subshift, they also described the asymptotic behavior in the large energy and small coupling settings when the potential pieces are characteristic functions of intervals of equal length. These estimates relied on explicit formulae and calculations, and thus could not be immediately generalized. In joint work with Fillman, we show that when the potential pieces are square integrable, the Hausdorff dimension of the spectrum tends to one in the large energy and small coupling settings.

In collaboration with Dominik Wodarz, we would like to announce a new NSF funded project, and are hoping to interest students to join the team. This will provide the opportunity to perform novel mathematical work in the field of virus dynamics, and at the same time to apply the mathematical work to experimental and clinical data in the context of human immunodeficiency virus (HIV). The evolution of the virus within patients is an important determinant of the disease process, and is also an important reason why treatments and vaccines can fail. Recent experimental data indicate that “social interactions” among different HIV mutants within the same patient can determine evolutionary outcomes, and this has so far not been investigated mathematically, in the context of evolutionary theory. The aim of the funded project is to fill this gap. This will provide new information that will be crucial to advance our understanding of the disease, and to design more effective vaccination approaches.

The talk will start with some remarks on the role that zeta functions and Tuberian theorems have played in number theory in the last 180 years starting essentially with Dirichlet's proof of his Arithmetic Progression Theorem. The remainder of the talk will be devoted to giving a survey of recent applications of Tauberian theorems to counting arithmetic objects.