## Analysis

Department of
Mathematics and Statistics

Our research group in Analysis works in a variety of fields, including operator theory, harmonic analysis, partial differential equations, complex analysis, functional analysis, mathematical physics, and approximation theory. Their work includes connections to image processing and geometry/topology. Here are their personal research statements.

### Ivana Alexandrova

"My research lies in the areas of partial differential equations and mathematical physics, more specifically, in scattering theory, semi-classical analysis, magnetic Hamiltonians, and the Aharonov--Bohm effect. Scattering theory studies interacting physical systems on scales of time and/or distance much larger than the interaction region itself. Semi-classical analysis studies the dependence of differential and pseudodifferential operators, their solutions and other related objects on a small parameter. Originally motivated by Bohr correspondence principle, which asserts that classical mechanics is the limit of quantum mechanics, as Planck's constant approaches 0, which corresponds to a change of scales, the tools and techniques of semi-classical analysis have found numerous applications to many areas of science in which a small parameter plays a crucial role, for example, the inverse of the square root of the nuclear mass in the Born--Oppenheimer approximation, the adiabatic parameter in adiabatic theory, the magnetic field strength in solid-state physics, the inverse of the square root of the energy in high-energy spectral problems, the inverse of the norm of the position in scattering theory, and others.

"Magnetic Hamiltonians describe physical systems in the presence of magnetic fields. The Aharonov--Bohm effect concerns such physical systems in which the magnetic field is non-zero only at a point. The Aharonov--Bohm effect is considered one of the most significant quantum mechanical phenomenon. It refers to the situation in which an electrically charged particle interacts with the magnetic field even in regions of space where the field vanishes. Unexplainable from the point of view of classical mechanics, the Aharonov--Bohm effect suggests that magnetic potentials have a physical meaning themselves and that potential energy and not forces and force fields should be taken as the starting point of formulating physical theory, contrary to what has been done since Newton's equations of motion."

### Marius Beceanu

"My research interests lie in the area of partial differential equations. I study evolution equations such as Schroedinger's equation and the wave equation, both at the linear and at the nonlinear level. I work on problems related to soliton stability and prove linear dispersive estimates such as Strichartz estimates, pointwise decay estimates, and wave operator boundedness.

"My most recent project is about the semilinear Schroedinger's equation with a random time-dependent potential. We prove that the solution completely scatters, regardless of possible bound states of the Hamiltonian."

### Yunlong Feng

"My research interests lie in the areas of machine learning and statistical learning theory. Driven by the increasing need of dealing with data from real-world applications, my research is to sift information through data from theoretical and practical ways. In real-world applications, data may exhibit various patterns or trends. For example, they may be contaminated, inherently dependent, or possess certain tensorial structures. On the practical side, I am enthusiastic in developing applicable learning algorithms to deal with various types of data while on the theoretical side, I am interested in assessing their learning performance by using statistical and mathematical tools. Several topics that I have been working on include robust learning, kernel methods, tensor-based learning, and learning with non-i.i.d observations. The long-term goal of my research is to develop generic tools to advance big and complex data analytics."

### Joshua Isralowitz

"My research is in harmonic analysis and operator theory. Specifically, I am interested in “size estimates” of various operators on Banach spaces of holomorphic functions, and more recently, in weighted norm inequalities for Calderon--Zygmund and related operators (and in particular, matrix weighted norm inequalities.)

"My most recent project involves proving matrix weighted Poincaré and Sobolev inequalities, with an eye towards proving new Hölder regularity results for degenerate elliptic partial differential equations. I also have possible plans in the future to look at problems in nonhomogeneous harmonic analysis and time-frequency analysis (as it relates to the bilinear Hilbert transform)."

### Michael Stessin

"My research interests lie in the areas of functional analysis, complex analysis and approximation theory. Among main topics of my concentration are spectral theory of tuples of noncommuting operators, operator theory in spaces of analytic functions including composition and Toeplitz operators, operator theoretical methods in recovery problems in functional spaces and applications to image processing, and n-widths estimates with applications to numerical analysis."

### Rongwei Yang

"My research is in multi-variable operator theory. The theory has two major components: on commuting operator tuples and on non-commuting operator tuples. In the commuting front, my research is focused on invariant subspaces in the Hardy space over the bidisk. Much progresses have been made in the past 20 years. Now the theory is relatively mature and it now can serve as a model for studies in other but similar settings. In the non-commuting front, my research is laid out around a new notion of joint spectrum (projective spectrum) for operator tuples. The project is relatively new. But results so far have indicated fascinating connections with Geometry and Topology."

### Yiming Ying

"My research is in statistical machine learning and learning theory with emphasis on developing theoretically sound and practically efficient algorithms suitable for big data analysis. I am enthusiastic about machine learning applications to real data problems in computer vision and biomedical research. The goal of my research is to address the computational and statistical challenges arising in modern data analysis, helping the discovery of hidden information from increasingly big and complex data.

"For this purpose, I am particularly interested in reproducing kernels in Hilbert spaces, regularization techniques, learning theory, convex optimization and computational statistics."

### Kehe Zhu

"My research interests are in function theoretic operator theory and the theory of Banach spaces of holomorphic functions on various domains. Among other related topics, I study “size estimate” properties (boundedness, compactness, and Schatten class membership) and spectral properties of Toeplitz, Hankel, and composition operators on a number of spaces of holomorphic functions, and study various properties of Banach spaces of holomorphic functions on various domains, such as invariant subspaces, zero sets, sampling, and interpolation properties."