Presently my research interests include the following:
The Nobel-winning theory of Anderson localization is a central topic in dynamical systems and spectral theory. Originally formulated for a single particle in disordered medium, it has since been extended to nonlinear and few-body systems. However, a rigorous mathematical theory of localization in the genuine many-body setting (i.e., where the number of particles tends to infinity) remains elusive. Resolving this problem is crucial for understanding new physical phenomena observed in recent laboratory experiments.
Heat kernels and resolvents are ubiquitous in analysis. Their estimates play an important role in harmonic, microlocal, and geometric analysis, and are the most studied cases of a broader concept known as operator kernel estimates. Given a self-adjoint operator and a sufficiently regular real function, the operator kernel can be defined as a natural generalization of the classical kernel function of integral operators. We would like to develop further operator-theoretic techniques for studying the decay properties of generic operator kernels on graphs and manifolds.
Consider nonlinear evolution equations that admit large families of soliton solutions, such as the Allen-Cahn and Ginzburg-Landau equations, as well as many geometric flows. The solitons form lower-dimensional structures in the configuration space. Our goal is to derive corresponding lower-dimensional effective dynamics that approximate the full evolution near the solitons. These effective dynamics typically involve far fewer degrees of freedom, and can be viewed as a rigorous form of renormalization.
Typical problems in engineering that involve control systems and signal processing can be formulated mathematically as systems of differential equations. Practical objectives then amount to deriving suitable a priori estimates for those equations. We aim to extract the mathematical problems arising from real-life applications, including optics, metrology, robotics, etc., and develop the necessary analytical tools to address them according to realistic demands.