Geometry & Topology
The Mathematical Study of Shapes and Spaces
Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.
My primary research interests are in topology and differential geometry. One current area of interest is bifurcation theory, the study of how the set of solutions to an equation varies as a parameter in the equation is varied. The relationship between symmetries of an equation and its bifurcations is very interesting. Bifurcation theory uses tools from analysis, linear algebra, and topology. The theory sheds light on questions in pure mathematics, such as the study of 3- and 4- dimensional manifolds (generalizations of surfaces) as well as applied problems.
My research interests are in low-dimensional topology and geometry and range from classical knot theory to the topology of 3-manifolds and smooth 4-manifolds. The tools I use in my research are various gauge theories including Donaldson and Seiberg-Witten theory but most prominently discovered Heegaard Floer theory. I am interested in question pertaining to knot concordance, specifically torsion in the smooth knot concordance group. With regards to smooth 4-manifolds, I am interested in better understanding the Heegaard Floer invariants of Lefschetz fibrations.
I study the fixed point theory of continuous maps on compact spaces, such as a torus and generalizations called Nilmanifolds and Solvmanifolds. Examples of these include the famous Klein Bottle or the collection of n by n upper triangular matrices with 1s on the diagonal. Certain properties of the fixed points of a map on one of these spaces are homotopy invariant, i.e., they don't change when the map is deformed. These properties are studied using techniques from group theory, combinatorics, and lots and lots of Linear Algebra.
My research is in low dimensional topology and knot theory. My work has focused on symmetries of knots, relationships between knot invariants and invariants of three-manifolds, and an equivalence relation known as knot-concordance which brings in four-dimensional space. The techniques used in my work come from algebraic and geometric topology.