Wash U Graduate Program in Mathematics : Faculty Research Projects

The following is a list of research projects recently conducted by members of the Mathematics faculty. These projects were supported by grants from or contracts with the National Science Foundation or similar agencies.

• Unsolved problems in complex function theory and other problems in analysis where symmetrization can be expected to play a role: developing a comprehensive and unified theory of symmetrization starting from a ``master inequality''; applying these ideas to problems involving Bloch's Theorem, quasiconformal mapping and singular integrals.
• Qualitative properties of linear elliptic and parabolic partial differential equations with minimal smoothness conditions, and of nonlinear evolution equations. Develop aspects of harmonic analysis associated with these problems.
• Geometric methods in complex function theory.
• Operator Theory: study of operators on Hilbert spaces of analytic functions and interplay between operator theory and function theory.
• Theory of functions of several complex variables: Hardy spaces, Nevanlinna class, boundary limits along curves, and the theory of Lipschitz and Bloch spaces. Applications to regularity theory for the "d-bar" operator on weakly pseudoconvex domains using the language of invariant metrics; automorphism groups of weakly pseudoconvex domains. Formulate the "d-bar" Neumann problem in the Sobolev topology and prove the existence and regularity for the new solution on a strongly pseudoconvex domain.
• Spaces of analytic functions: operator theoretic consequences of the decomposition of functions in Bergman space and related spaces, deformation theory for uniform algebras in general and algebras of analytic functions in particular.
• Spaces of bounded analytic functions on plane domains with complicated boundaries, looking for a constructive proof of the Denjoy conjecture, and extension of the class of domains for which the corona theorem holds.
• Aspects of harmonic analysis: atomic, molecular, and maximal characterizations of Hardy spaces; spaces generated by blocks, a generalization of atomic theory.
• Harmonic analysis and operator theory: in particular, real and complex interpolation of Banach spaces and quasi-Banach spaces. Calderon-Zygmund operators and their generalizations, characterization of Besov-Lipschitz and Triebel-Lizorkin spaces.
• Wavelet theory: construction of new kinds of wavelets, efficient algorithms for signal analysis and other applications, paths connecting wavelets, geometrical aspects of wavelets.
• Asymptotic behavior of leaves of foliations of hyperbolic 3-manifolds by surfaces. Analyze to what extent the toplogical structure of the foliation determines the geometry of the manifolds.
• Classification of isoparametric hypersurfaces; Investigating the stability, as Lagrangian submanifolds in the complex hyperquadric via the Gauss map, of the image of the inhomogeneous isoparametric hypersurfaces; Proving taut submanifolds in Euclidean space are real algebraic to settle affirmatively a question of Kuiper; Construction of minimal tori in the complex projective space.
• Foliations of codimension one; in particular, foliations of knot complements, leafwise hyperbolic structures, smoothability of foliations, ergodic properties in the presence of Morse singularities and the topology, topological dynamics and measure-theoretic dynamics of ``strange attractors.''
• Analysis of the foliations induced by an Anosov flow on a 3-manifold. Relate this to homotopic properties of closed orbits of the flow and to metric properties of flow lines.
• Theoretical models and data analysis to be used to study intragenic recombination in bacteria and other aspects of population dynamics that can be inferred from statistical analysis of DNA sequences.
• Hardy spaces on trees and associated questions related to martingales and probability theory, applications to random walks and computer estimation of harmonic measure of the planar Cantor set.
• Models and data analysis used in the study of various medical problems, including: psychiatric problems of homeless men and women, and of disaster victims; hip fractures in free-living elderly; and analysis of Pet image data of schizophrenics versus normals.
• Commutative algebra and algebraic geometry using techniques of combinatorics and combinatorial group theory: automorphisms of affine and projective n-space. Cremona groups of dimension greater than two, development of mathematical techniques to settle the critical cases of the Jacobian conjecture by means of computer calculations.
• Survival analysis, longitudinal data analysis, joint modeling of longitudinal and survival data.
• Geometry of singular spaces, i.e. orbifolds, orbit spaces, and leaf spaces of foliations; cyclic cohomology and index theory; Hopf algebras and generalized symmetry in noncommutative geometry; Symplectic (Poisson) geometry and quantization.

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