Research Area: Topology and Geometry
Topology is the study of continuity, from defining this basic notion to its application in the study of spaces. Geometry builds on topology, analysis and algebra to study the property of shapes and space.
Noncommutative geometry is the geometric approach to the study of noncommutative algebras, which finds its roots in mathematical physics, representation theory of groups, and the study of singular spaces from the world of differential geometry. Our focus is primarily on noncommutative metric geometry, where we study quantum metric spaces, i.e. noncommutative generalizations of the algebras of Lipschitz functions over metric spaces. Our purpose is to develop a geometric framework for the study of quantum metric spaces which arise from various fields, such as mathematical physics, dynamical systems, differential geometry, and more. A key tool in this framework is a generalization of the Gromov-Hausdorff distance to the noncommutative realm, which enables the exploration of the topology and geometry of classes of quantum metric spaces. We thus become able to construct finite dimensional approximations for C*-algebras, establish the continuity of various families of quantum metric spaces and associated structures, and investigate questions from mathematical physics and C*-algebra theory from a new perspective inspired by metric geometry.
The noncommutative algebras studied by noncommutative geometers typically fit within the realm of functional analysis, i.e. the analysis of infinite dimensional topological vector spaces and related concepts. The techniques used in their study borrows from differential geometry, algebraic and differential topology, topological group theory, abstract harmonic analysis, and metric geometry.
Topological Ramsey Theory
Knot Theory and the Yang-Baxter Equation
Knot theory is concerned with topological properties of knots and similar objects. Knots are often distinguished by means of algebraic invariants, such as the Jones polynomial or the knot quandle. We study algebraic structures associated with knots, such as quandles and racks (which encode the classical Reidemeister moves in algebraic form).
The Yang-Baxter equation was first introduced in statistical mechanics but now appears in many areas of mathematics. We are interested in set-theoretical solutions of the Yang-Baxter equation, particularly those arising naturally in knot theory and in the theory of braid groups.