"Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. "
- David Hilbert
Click for my Research Statement
My research area falls under the umbrella of algebraic combinatorics and categorical combinatorics. I currently have two research programs. One is to apply the notion of partial symmetries and inverse semigroups to graphs and other combinatorial objects. The standard notions of symmetry, via the automorphism group, is a coarse method and many extremely disparate objects will have isomorphic automorphism groups. The inverse semigroup is capable of distinguishing many of these objects. For example graphs have isomorphic inverse semigroups if and only if they are isomorphic as graphs. Another research program is using an generalization of the Noether Isomorphism Theorem to graph categories, and applying this theorem to reformulate classic graph theory conjectures and finding new attacks on these problems.
Additionally, I'm generally curious about all sorts of mathematics, and the interconnections between areas of mathematics with each other, as well as connections between mathematics and other disciplines. For example I'm a fan of boxing, and in light of Muhammed Ali's recent passing, I've submitted an expository paper applying ranking algorithms to the weighted directed graph of professional boxing.
The Two Cultures of Mathematics by Tim Gowers
Mathematical development throughout history mimics the development of student's understanding of mathematics. As we progress, the mathematics becomes more abstract, and in this abstraction lies the power of mathematics. Seemingly disparate problems or subjects are revealed to be two shadows cast by a more general mathematical object or idea.
In the last century some of the greatest strides in mathematics were made in the fields of algebra, geometry and topology. In the groundbreaking work led by Alexander Grothendieck, category theory was used to find connections between these areas and by looking at the relationship between mathematical objects and analogous relationships between these areas, mathematicians were able to create a much more abstract theory which revolutionized these areas forever.
Another area that exploded in the last century was my own, Combinatorics. From graph theory to matroids to design theory and more, much of what we think of as combinatorics was developed in the 20th century. However, combinatorics has a reputation as been filled with disparate and unrelated topics. One should remember, however, that topics within geometry and algebra were once thought to be disparate, and these areas likewise disparate from each other. Applying an increasingly abstract viewpoint allowed the connections between these topics to be revealed.
The categorification of Algebra, Geometry and Topology led to the groundbreaking work by Grothendieck, which in turn leads to modern attempts at unifying mathematics such as the Langlands Program and the L-functions and Modular Forms Data Base. It is high time that the same tools be brought to bear on the area of Combinatorics. Such an approach would not only provide a novel perspective of combinatorial objects which may lead to solutions to long standing conjectures, but will also uncover the hidden connections between subjects within combinatorics, and between combinatorics and the other branches of mathematics.