Mohamed Omar

Department of Mathematics	Email: mlastname@math.ucdavis.edu
University of California, Davis

Welcome to my webpage. I am a PhD candidate at the University of California, Davis. My advisor is Jesus De Loera. I am supported by an NSERC PGS D doctoral grant. Before coming to Davis, I completed the Double Honours program in Pure Mathematics and Combinatorics & Optimization (2006), and the Master's program in Combinatorics & Optimization (2007) under the supervision of Ian Goulden, both at the University of Waterloo.

Research

I am interested in the interplay between algebra and combinatorics. In particular, I am interested in non-linear algebraic techniques applied to problems in graph theory and combinatorial optimization. These algebraic methods include real/complex algebraic geometry, combinatorial commutative algebra, and some algebraic topology. I also appreciate asymptotics and applications.

Articles in Preparation

De Loera, J., Hillar, C., Malkin, P., Omar, M., Iterative Algebraic Algorithms for the Recognition of Combinatorial Properties and Integer Hull Descriptions.

Boucher, C., Omar M., On Sampling and Counting Consensus Strings.

Articles

Eagle, C., Gao, Z., Omar, M., Panario, D., Richmond, B. Distribution of the Number of Encryptions in Revocation Schemes for Stateless Receivers. Discrete Mathematics and Theoretical Computer Science, pp. 195-206, (2008).

Omar, M., Combinatorial Approaches To The Jacobian Conjecture. Master's Thesis, University of Waterloo (2007).

Omar, M., Panario, D., Richmond, B., Whitely, J. Asymptotics of Largest Components in Combinatorial Structures. Algorithmica. Vol. 46, Issue: 3-4. pp.493-503 (2006).

Research Notes / Current Reading

I am currently studying graph automorphisms using Algebraic Curves (both real and complex). I am particularly interested in how to use the theory of real and complex algebraic curves to detect when a graph has trivial automorphism group.

I am curious about theta bodies (certain semidefinite relaxations of convex bodies), in the context of the paper Theta Bodies for Polynomial Ideals (Gouveia, Parrilo, Thomas) and their ability to provide insight on combinatorial problems.
A natural related topic to graph automorphisms is the study of permutation polytopes. I've been particularly curious about the combinatorial significance of their f-vectors. Not much is known about permutation polytopes. The paper Permutation polytopes and indecomposable elements in permutation groups (Guralnick, Perkinson) provides dimension bounds and characterizes the diameter of these polytopes. Furthermore, the paper On permutation polytopes (Baumeister, Haase, Nill, Paffenholz) classifies permutation polytopes in low dimensions (2,3,4).

Teaching

(Summer 2009) Combinatorics - MATH 145

Math Contests

I am interested in and heavily involved with mathematics competitions at the high school level. This year, my participation includes:

Explore Math: Coaching and Lecturing for the Northern California ARML Team

Explore Math is an NSF VIGRE funded high school enrichment initiative organized and run solely by graduate students at UC Davis. ARML (American Regions Mathematics League) is a national mathematics competition for high school students held on the last weekend of May at four sites around the United States. I was the primary instructor for the Northern California ARML team in the Winter 2009 Quarter. I am a coach this quarter.

Contest Writing for mathleague.org:

The website mathleague.org provides regional and state mathematics competitions for math circles and schools in over 10 states. I am in charge of writing the Sprint contests this year (a 1 hour, 30 question multiple choice competition). I am a contest writer for mathleague.org, a national program that provides various mathematics competitions to high school math circles. In the past, I have been involved with Canadian Mathematics Competitions run by the Centre for Education in Mathematics & Computing.