Michael Tait: Research Statement
نویسنده
چکیده
My research uses algebraic and geometric methods to prove theorems in extremal combinatorics. Going the other way, I also use combinatorial methods to prove algebraic results. Algebraic methods are deeply embedded in my work and nearly all of my success in graph theoretic research has come from attacking purely combinatorial problems through the lens of algebra, combinatorial number theory, or finite geometry. Conversely, my work in graph theory has allowed me to understand algebraic structure through combinatorial methods, yielding theorems in the same areas of algebra, combinatorial number theory, and finite geometry. As an overview, being able to work in the intersection of several different areas of mathematics has been fruitful and exciting. Interesting and open problems abound, and in this document I will describe a selection of problems that I have worked on and plan to work on in the future. Section 5 contains several specific problems that would make good projects for undergraduate or graduate students.
منابع مشابه
My favorite application using eigenvalues: Eigenvalues and the Graham-Pollak Theorem
The famous Graham-Pollak Theorem states that one needs at least n− 1 complete bipartite subgraphs to partition the edge set of the complete graph on n vertices. Originally proved in conjunction with addressing for networking problems, this theorem is also related to perfect hashing and various questions about communication complexity. Since it’s original proof using Sylvester’s Law of Intertia,...
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Crossing Number of Alternating Knots in S × I
A hundred years ago, Tait conjectured that the number of crossings in a reduced alternating projection of an alternating knot is minimal. This statement was proven in 1986 by Kauffman, Murasugi and Thistlethwaite, [6], [10], [11], working independently. Their proofs relied on the new polynomials generated in the wake of the discovery of the Jones polynomial. We usually think of this result as a...
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We establish a conjecture of Graham and Lovász that the (normalized) coefficients of the distance characteristic polynomial of a tree are unimodal; we also prove they are log-concave. Email addresses: [email protected] (Ghodratollah Aalipour), [email protected] (Aida Abiad), [email protected] (Zhanar Berikkyzy), [email protected] (Leslie Hogben), [email protected] (Fra...
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In this personal statement, I give a non-technical description of my research. I only cite my own papers, numbering them as they appear on my CV. This non-technical description mostly discusses my work in symplectic geometry, but also some recent work in string topology. In the separate self-contained research statement, I provide a technical description of the research. In the research stateme...
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