Analysis, Geometry and Dynamics

Geometric Analysis

Jie Qing works on conformal geometry, the AdS-CFT correspondence, and most recently in general relativity. In conformal geometry, only the angle between two vectors can be measured but not the vector's lengths. The AdS-CFT correspondence relates the Riemannian or pseudo-Riemannian (general relativistic) geometry on one manifold to the conformal geometry of a manifold that bounds it. AdS-CFT has roots going back about 30 years but became very popular in the last decade due to its relevance to string theory. Qing has some of the strongest uniqueness results available in AdS/CFT for conformal spheres.

Tony Tromba's work is on minimal surfaces. These are surfaces that minimize area among all surfaces bounding a given curve. He has developed an index theory for minimal surfaces paralleling Morse's index theory for geodesics. He has written several books on the subject.  

Longzhi Lin works on geometric PDEs— more precisely, geometric flows including mean curvature flow and harmonic map heat flow, harmonic maps, minimal surfaces, surfaces of constant mean curvature, and min-max theory. His most recent work is concerned with the min-max construction of minimal surfaces with free boundary in any Riemannian manifold, and the energy convexity of intrinsic bi-harmonic maps.

Random Matrices and Operator Theory

Torsten Ehrhardt uses operator theory in order to study the spectral properties and the asymptotics of Toeplitz, Hankel, Wiener-Hopf, and related operators. Part of his work is motivated by applications to random matrix theory and statistical physics. His research interests also include Wiener-Hopf factorization theory, Riemann-Hilbert problems, as well as the theory of Banach algebras.

Roozbeh Gharakhloo studies the asymptotic behavior of principal objects in Mathematical Physics and Random Matrix Theory which have a characterization in terms of structured determinants and/or a system of orthogonal polynomials. A primary objective in his work has been to expand the application of the Riemann-Hilbert approach to studying the asymptotic properties of various structured determinants, including Toeplitz and Hankel determinants, as well as their altered forms like Toeplitz+Hankel, bordered Toeplitz, framed Toeplitz, and more general forms such as pj-qk or slant Toeplitz determinants.

Mechanics, Symplectic Geometry, and Dynamical Systems

Symplectic geometry is the geometry underlying classical mechanics. It is also important to quantum mechanics and low-dimensional topology and is an active area of research. In the department, three of our faculty represent symplectic geometry and its links to mechanics and dynamics.

Viktor Ginzburg's recent work is at the interface of symplectic geometry and dynamical systems. Its main theme is the existence and non-existence of periodic orbits of Hamiltonian systems. The methods are essentially symplectic topological such as Floer theory. He has worked on the Hamiltonian Seifert conjecture, constructing examples of such systems without periodic orbits, and on the existence results such as the Conley Conjecture establishing unconditional existence of infinitely many periodic orbits in many cases. His work has applications to various systems originating in physics, e.g., the motion of a charge in a magnetic field. 

Debra Lewis is an expert in relative equilibria for Hamiltonian systems and their stability, as well as bifurcations of relative equilibria. A relative equilibrium for a dynamical system is a solution that is an equilibrium modulo a symmetry group action. Her work is an interplay between group theory and symplectic geometry, and uses a good deal of symbolic manipulation.

Richard Montgomery is known for his work on N-body problems and sub-Riemannian geometry. He and Alain Chenciner (Paris 7 and the Bureau des Longitudes, Paris) rediscovered, and proved the existence of, a figure-eight-shaped orbit for the Newtonian 3-body problem. This result, and the introduction of variational and group-theoretic methods that they pioneered, has driven much recent work in celestial mechanics. He has one basic result in the field ("geodesics which do not satisfy the geodesic equations") and has written a well-received book on it. His work on sub-Riemannian geometry grew out of applications to nonlinear control theory, and physical chemistry (e.g., NMR, Berry's phase).

Inverse problems in PDEs and integral geometry

François Monard studies parameter-reconstruction ("inverse") problems in PDEs and integral geometry, with applications to imaging sciences. His most recent work is concerned with the inversion of X-ray/Radon transforms (mappings which integrate a geometric object along a given family of curves) and their generalizations to non-Euclidean geometries and to the non-abelian integration of matrix fields.

Spectral Geometry

Pedro Morales-Almazán is interested in regularization problems arising in quantum field theory and asymptotic methods in number theory. His focus has been the use of spectral zeta functions arising from pseudo-differential operators in Riemannian manifolds to compute Casimir energy and vacuum energy of quantum fields. He is also interested in number theory problems involving regularization and resummation methods, specifically using zeta function regularizations.

Geometric Group Theory

Kasia Jankiewicz works in geometric group theory and its connections to geometric topology, especially in low-dimensions. Among the main themes of her research are metric notions of negative and non-positive curvature, group theoretic analogues of topological properties of 3-manifolds, and Coxeter and Artin groups.

Riemannian Geometry

Jiayin Pan is interested in Riemannian geometry, more precisely, manifolds with Ricci curvature bounded below and their limit spaces under Gromov-Hausdorff convergence. His research concerns the fundamental groups of open manifolds with nonnegative Ricci curvature, as well as the structure of Ricci limit spaces.

Fluids and Partial Differential Equations

Emeritus Professor Maria Schonbek's research centers on non-linear diffusive equations, such as the Navier-Stokes ones for viscous fluids, or the quasi-geostrophic ones describing "wind" between areas of different pressure. These partial differential equations can be used to describe ocean currents, water flow in a pipe, and other fluid behavior. The main direction of her research is towards understanding the qualitative behavior of solutions, in particular their energy asymptotics and decay rates. Questions of existence and regularity of solutions are also considered in her work.