# Analysis

Analysis is a broad branch of mathematics that encompasses many fields, generally sharing a basis in calculus. Historically, it has played a crucial role in solving problems in physics and engineering; recent years have seen surprising applications to solve problems in other mathematical fields like the Poincaré Conjecture in topology.

In our Department, research in analysis is conducted in various subfields: partial differential equations, dynamical systems, mathematical physics, geometric analysis, and ergodic theory.

**Partial Differential Equations & Dynamical Systems**

The research of Maxim Braverman includes work on the index theory and determinants of elliptic operators; he has also done interdisciplinary work in hydrodynamics. Robert McOwen studies solvability and regularity properties of elliptic operators in domains and on singular and noncompact manifolds. Martin Schwarz works in nonlinear analysis: Liouville Theorem for partial differential equations that are completely integrable. Mikhail Shubin works on various linear and non-linear partial differential equations, as well as the spectral theory of elliptic operators, especially Schrödinger operators. Peter Topalov studies Hamiltonian partial and ordinary differential equations, and dynamical systems. And Solomon Jekel uses topological methods to find closed orbits for dynamical systems on spheres in order to obtain periodic solutions to delay equations, which are ordinary differential equations in which the rate of change of the process described by the equation at any time is allowed to depend on the behavior of the process at earlier times. Ting Zhou works on partial differential equations and inverse problems, with applications to imaging, tomography, transformation optics and cloaking.

**Mathematical Physics **

Maxim Braverman has studied Dirac and Schrödinger operators as well as superconductivity. Chris King works on problems in mathematical physics using a variety of methods, including matrix analysis and convex analysis. Martin Schwarz works on Maxwell Higgs and other nonlinear problems from the physical sciences. Mikhail Shubin uses various areas of analysis, especially the spectral theory of Schrödinger operators and non-commutative geometry, to study problems in mathematical physics.

**Geometric Analysis **

Maxim Braverman has studied various problems in differential geometry including analytic torsion. Robert McOwen studies the application of nonlinear PDEs to conformal metrics in differential geometry. Mikhail Shubin works on various applications of analysis to problems in topology and differential geometry and non-commutative geometry to problems in mathematical physics. And Peter Topalov applies various analytic techniques to problems in Riemannian and symplectic geometry.

**Ergodic Theory **

Ergodic theory is the study of measure-preserving transformations, such as the different ways of mixing two fluids (e.g. gin and vermouth). Stanley Eigen currently works in ergodic theory and some of its connection with tilings and group theory. Arshag Hajian studies infinite-measure ergodic theory with applications to algebra.