Georgia Institute of Technology College of Sciences

Travelling pulses and waves are a rich subset of feasible patterns in reaction-diffusion systems. Many have investigated their existence, stability, and other properties, but what happens if the deterministic dynamics is affected by random occurrences? How does the interplay between diffusion and noise influence the velocity, curvature, and stability of multidimensional patterns?

We consider reaction-diffusion systems with multiplicative noise on a spatial domain of dimension two or higher. The noise process is white in time, coloured in space, and invariant under translations; based on applications. Inspired by previous works on the real line, we establish the multidimensional stability of planar waves on a cylindrical domain on time scales that are exponentially long with respect to the noise strength. In the deterministic setting, multidimensional stability of planar waves on the whole space has been obtained, and we show to what extend we can do this in the stochastic case.

The metastability result above is achieved by means of a stochastic phase tracking mechanism that can be maintained over such long-time scales. The corresponding mild formulation of our problem features stochastic integrals with respect to anticipating integrands, which hence cannot be understood within the well-established setting of Itô-integrals. To circumvent this problem, we exploit and extend recently developed theory concerning forward integrals.

The Cwikel-Lieb-Rozenblum (CLR) inequality is a semi-classical estimate on the number of bound states for Schrödinger operators. In this talk I will give a brief overview of the CLR inequality and present a substantial refinement of Cwikel’s original approach which leads to an astonishingly good bound for the constant in the CLR inequality. Our new proof highlights a natural but overlooked connection of the CLR inequality with bounds for maximal Fourier multipliers from harmonic analysis and leads to a variational problem that can be reformulated in terms of a variant of Hadamard’s three-lines lemma. The solution of this variational problem relies on some interesting complex analysis techniques. (Based on joint work with T. Carvalho-Corso, D. Hundertmark, P. Kunstmann, S. Vugalter)

I will discuss the numerical approximation of differential operators on unknown manifolds where the manifolds are identified by a finite sample of point cloud data. While our formulation is general, we will focus on Laplacian operators whose spectral properties are relevant to manifold learning. I will report the spectral convergence results of these formulations with Radial Basis Functions approximation and their strengths/weaknesses in practice. Supporting numerical examples, involving the spectral estimation of various vector Laplacians will be demonstrated. Applications to solve elliptic PDEs will be discussed. To address the practical issue with the RBF approximation, I will discuss a weak approximation with a higher-order local mesh method that not only promotes sparsity but also allows for an estimation of differential operators with nontrivial Cristoffel symbols such as Bochner and Hodge Laplacians.

We show that for a fixed $q$, the number of $q$-ary $t$-error correcting codes of length $n$ is at most $2^{(1 + o(1)) H_q(n,t)}$ for all $t \leq (1 - q^{-1})n - 2\sqrt{n \log n}$, where $H_q(n, t) = q^n / V_q(n,t)$ is the Hamming bound and $V_q(n,t)$ is the cardinality of the radius $t$ Hamming ball. This proves a conjecture of Balogh, Treglown, and Wagner and makes progress towards a 2005 question of Sapozhenko.