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Current research areas include analysis, statistics, differential equations, dynamical systems, engineering, architecture and urbanism, logic, model theory, mathematical biology, population dynamics, mathematics of finance, non-associative algebras, geometry, functional analysis, numerical analysis, number theory, random matrix theory, mathematics education.

Regularity for Fully Nonlinear P-Laplacian Parabolic Systems: the Degenerate Case 2014 Differential Equations

Abstract: This paper introduces new nonlinear heat approximation and L∞ preserving homotopy techniques to investigate regularity properties of bounded weak solutions of strongly coupled p-Laplacian parabolic systems which consist of more than one equation defined on a domain of any dimension. The main results imply everywhere Holder continuity of bounded weak solutions and the global existence of strong solutions to nonlinear p-Laplacian systems.

Unitary invariants on the unit ball of B(H)^n 2013 Operator Theory

Abstract: In this paper, we introduce a unitary invariant

defined in terms of the characteristic function , the noncommutative Poisson kernel , and the defect operator associated with . We show that the map detects the pure row isometries in the closed unit ball of and completely classify them up to a unitary equivalence. We also show that detects the pure row contractions with polynomial characteristic functions and completely noncoisometric row contractions, while the pair is a complete unitary invariant for these classes of row contractions.

The unitary invariant is extracted from the theory of characteristic functions and noncommutative Poisson transforms, and from the geometric structure of row contractions with polynomial characteristic functions which are studied in this paper. As an application, we characterize the row contractions with constant characteristic function. In particular, we show that any completely noncoisometric row contraction with constant characteristic function is homogeneous, i.e., is unitarily equivalent to for any free holomorphic automorphism of the unit ball of .

Under a natural topology, we prove that the free holomorphic automorphism group is a metrizable, -compact, locally compact group, and provide a concrete unitary projective representation of it in terms of noncommutative Poisson kernels.

Ultraproducts and metastability 2013 Model Theory

Abstract. Given a convergence theorem in analysis, under very general conditions a model-theoretic compactness argument implies that there is a uniform bound on the rate of metastability. We illustrate with three examples from ergodic theory.