Franklin Kenter's Webpage

My email address is fkenter (AT) math (DOT) ucsd (ANOTHER DOT) edu

I am a third year doctoral student in the Department of Mathematics at University of California, San Diego in the field of combinatorics. My specific research interests include spectral graph theory, graph coloring, and random graph models. My mentor is Fan Chung Graham.

Current Teaching:

Spring 2011: Professor Keilburg MATH 102, Advanced Linear Algebra

Past Teaching:

Fall 2008: Professor Lunasin, MATH 10B, Calculus

Fall 2008: Professor Horton, MATH 20C, Calculus for Science and Engineering

Winter 2009: Professo Volpato, MATH 10B, Calculus

Summer 2009: Professor Ferry: Professor Ferry, MATH 10B, Calculus

Fall 2009: Professor Ferry, MATH 4C, Precalculus for Science and Engineering

Fall 2009: Professor Weinkove, MATH 10A, Calculus

Winter 2009: Professor Yoo, MATH 20A, Calculus for Science and Engineering

Winter 2009: Professor Verstraete, MATH154, Introduction to Combinatorics

Spring 2010: Professor Balog: MATH 109, Mathematical Reasoning

Summer 2010: Professor Peng: CSE 20, Discrete Mathematics

Fall 2010: Professor Wulbert: MATH 142A, Advanced Calculus

Winter 2011: Professor Wulbert: MATH 142B, Advanced Calculus

Research:

Publications:

A Generalization of Hoffman's Theorem for Hypergraphs (submitted to Journal of Graph Theory):

Hoffman proved that for a simple graph $G$, the chromatic number of $\chi(G)$ obeys $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ where $\lambda_1$ and $\lambda_n$ are the maximal and minimal eigenvalues of the adjacency matrix of $G$ respectively. Lov\'asz later showed that $\chi(G) \le 1 - \frac{\lambda_1}{\lambda_{n}}$ for any (perhaps negatively) weighted adjacency matrix.

We give a probabilistic proof of Lov\'asz's theorem. We then extend the technique to derive generalizations of Hoffman's theorem for colorings allowing a given proportion of edge-conflicts and for hypergraphs using the spectra of the underlying graph.

Isoperimetric Inequalities for Directed Graphs (joint work with Fan Chung; in preperation)

We establish several isopermetric inequalities concerning various types of discrepancies in a directed graph $G$. For any two given subsets of vertices, say $S$ and $T$, we examine the `flow' from $S$ to $T$ in the random walk on $G$ at the state of the stationary distribution. In particular, we focus on the discrepancy which is the difference between the actual flow and the expected flow (which depends only the stationary distribution). We will show that the discrepancy of flows from $S$ to $T$ can be bounded above by using the eigenvalues of certain symmetric matrices associated with $G$. We also show that the maximum discrepancy can be bounded below by the same eigenvalues to within a logarithmic factor. In addition, we consider a variation of the discrepancy which concerns the difference between the flow from $S$ to $T$ and the flow from $T$ to $S$. We will show that this variation of discrepancy can be similarly bounded from above and below by using eigenvalues of a skew-symmetric matrix associated with $G$. Furthermore, we introduce a quantum Laplacian which has complex-valued entries but is self-adjoint. We will illustrate that the eigenvalues of the quantum Laplacian can be used to capture several types of discrepancies.

Presentations:

AMS Western Sectional Meeting. May 2011.

Isoperimetric Inequalities for Directed Graphs [slides]

UCSD Graduate Seminar. November 2010.

Advanced Mathematics in Children's Games

UCSD Graduate Seminar. September 2010.

A Spectrum of Colorful Diameters

UCSD Graduate Seminar. April 2009.

Games:

I am an avid player of various strategy games (and some party games). Current favorites include:

Settlers of Catan, Carcassone, Dominion (I like this site better now: Isotropic), Agricola (a colleague of mine maintains the online version), Blokus, Ticket To Ride, and Wits and Wagers. I also play Magic: The Gathering occasionally (look at this page closely) among other trading card games. In the past, I played trading card games competitively; my winning decklists can be found here.

Games Vita:

World Championships: Star Trek CCG 2002, 2009

World Championship Runner-Up: Anachronism 2006

Regional Championships: Anachronism 2005, 2006; Ticket To Ride USA 2010; Star Trek CCG 2007, 2008, 2009, 2010

Other High Finishes: Lord of the Rings TCG 2004 Las Vegas Open (3^rd Place), Stargate TCG 2007 World Championships (4^th Place).