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# Pure Mathematics

### Operator Algebras

C∗-algebras (pronounced "C-star") are an important area of research in functional analysis, a branch of mathematics.

#### Valentin Deaconu

My area of research is Operator Algebras, which is part of functional analysis. Functional analysis is the study of spaces of functions and other Banach spaces, and is related to differential equations, linear algebra, topology and abstract algebra. More precisely, I study groupoid C*-algebras and K-theory. Groupoids are similar to groups, except that they have many units, and one can not compose just any two elements. Additional structure is necessary, like a topology and a family of measures, in order to define a groupoid C*-algebra, which sometimes looks like a set of (infinite) matrices with complex entries. K-theory is a generalized cohomology theory, which is used in algebraic topology in order to distinguish surfaces and other topological spaces. The methods in my research are also inspired from dynamical systems, and the applications are in quantum statistical mechanics.

#### Alex Kumjian

My field of research is a branch of Analysis called Operator Algebras. It is an intriguing mixture of Analysis and infinite-dimensional linear algebra. It is a relatively new field that has its origins in the mathematical formalism of quantum mechanics. I am particularly interested in operator algebras which arise from dynamical systems.

### Topology

Topology developed as a field of study out of geometry and set theory, through analysis of such concepts as space, dimension, and transformation.

#### Chris Herald

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.

#### Stanislav Jabuka

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.** **

#### Tynan Kelly

My research interests are in low-dimensional topology and knot theory. My work uses both algebraic and geometric techniques to study knots and their invariants, ranging from Kirby calculus and handlebodies to classical knot invariants (such as the Alexander and Jones polynomials and Casson-Gordon invariants) to more modern techniques such as the Heegaard-Floer and Khovanov homology theories. I am particularly interested in knot concordance and the role that knots play in the topology of 3- and 4-manifolds.

#### Ed Keppelmann

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.

#### Swatee Naik

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.

### Number Theory

#### Jing-Jing Huang

My primary research interests lie in the interaction of analytic number theory, diophantine geometry and harmonic analysis. That is to say, I am interested in using analytic methods (complex analysis, fourier analysis, etc) to solve number theoretic problems (finding integral/rational solutions to diophantine equations, the distribution of prime numbers, etc). My current research project is to study the distribution of rational points near a curved manifold, which (if solved in a satisfactory manner) will have major applications to problems in metric diophantine approximation.

### Algebra

#### Chris Rogers

My research explores problems at the interface between algebra and topology/geometry that can be studied using ideas and techniques from topology, especially homotopy theory. This sort of research is often referred to as “homotopical algebra”, and it includes classical homological algebra as a special case, as well as category theory and its higher analogs.

My work involves studying what are called “higher structures” in differential (and algebraic) geometry. Examples include simplicial manifolds, groupoids/stacks, and gerbes. The word “higher” is used to emphasize the fact that the collection of maps, or morphisms, between these geometric objects crucially form something more akin to a topological space, rather than just a set. That is, there are morphisms between objects, 2-morphisms between morphisms, and higher morphisms between these, just as we have homotopies between maps between spaces, homotopies between homotopies, and so on. As the analogy suggests, there is a close relationship between homotopy theory and the study of these higher structures.

Curiously, mathematical physics – especially quantum field theory – provides a rich source of research problems in the above areas. I am particularly interested in the roles that homotopy theory and higher category theory play in quantization, a procedure physicists use to pass from a classical description of a physical system to a quantum one.