Ross M. Richardson's Research

An induced subgraph of the BGP webgraph, consisting of major Internet routers.

I was awarded my Ph.D. in Mathematics on September 4^th, 2007, and am now a leaf in the following tree. Thanks to my committee. I am moving to PNYLAB LLC, a wholly owned subsidiary of Edgestream Partners LP. My research interests are in combinatorics and graph theory. Specifically, I am interested in:

• Probabilistic methods in combinatorics.
• Geometric and computational aspects of combinatorics.
• Complex network modeling and analysis.

My advisor is Dr. Fan Chung Graham. Recently, UCSD was ranked number two in combinatorics, according to US News and World Reports.

Papers

• Combinatorial and Geometric Problems on Point Processes. Ph.D. Thesis.
• The Shape of Uniform Random Trees. In progress, (with Fan Chung and Paul Horn).
• Monthly Problem 11262: Statement, Solution. This is a cute little graph theory problem.
• Geometric Random Graphs determined by a Random Connection Model. In preparation. (with Fan Chung and Bill Aiello).
• On Locality in Geometric Random Graphs (Journal Version of below). To appear, Internet Mathematics. PDF
• Local/Global Phenomena in Random Geometric Graphs (To be presented at WAW2006). To appear, Proceedings of WAW2006. PDF (abstract). PDF.
• Maxima in High Dimensional Convex Sets. In preparation.
• Random Circumscribing Polytopes. In preparation.
• Weighted Laplacians and the Sigma function of a graph, Quantum Graphs and Their Applications, (B. Berkolaiko et. al. eds), Contemporary Math, v.415, AMS, Providence, RI. (with Fan Chung). PDF.
• An inscribing model for random polytopes.To appear, Discrete and Computational Geometry. (with Van Vu and Lei Wu). PDF.
• Random Inscribing Polytopes.European Conference on Combinatorics, Graph Theory, and Applications, DMTCS Proceedings Volume AE (2005). 263-266. (with Van Vu and Lei Wu). PDF.
• Senior Thesis-Designing Rigid Motions:Computing Curves in Lie Groups. Advised by Weiqing Gu, Harvey Mudd College

Computation and Simulation Projects

Presentations

• Dalhousie Mathematics Colloquium Talk. Dalhousie University, Halifax, Nova Scotia. Fall '07.
• Final Defense.
Sept. 4, 2007.
• The Condition Number of a Randomly Perturbed Matrix. Combinatorics Summer Reading Seminar 2007.
• New Models and Questions in Geometric Random Graphs. AMS Sectional, Tucson, AZ. Spring '07.
• Randomness and Regularity à la Szemerédi. Food for Though Graduate Seminar, UCSD. Spring '07.
• Local/Global Phenomena in Geometric Random Graphs. WAW2006. Fall '06.
• Advancement Talk, Fall '06.
• Three Lectures in Nankai. Nankai University, Center for Combinatorics, Tianjin, China. Summer '06.
  如果你找我的讲课的题解，请在这儿点击。
• Practical Graph Drawing + A Random Geometric Tree Model (5MB). Combinatorics Seminar. Summmer '06.
• Graphs and Probability; A Perfect Matching. Food for Thought Graduate Seminar, UCSD. Spring '06
• Random Inscribing Polytopes. Workshop on Probabilistic Combinatorics. BIRS, Banff, Canada. Fall '05.
• Weighted Laplacians and Sigma-Functions. Graduate Student Research Seminar. Fall '05.
• Random Polytopes, 3 lectures, Reading Seminar in Combinatorics at UCSD, Fall '04, Spring '05. Directed by Van Vu.

Conferences Attended

• STOC 2007, Summer 2007.
• Models and Algorithms for the Web Graph (WAW2006) .
• Mount Baldy Conference on Enumerative Combinatorics, Fall 06
• BIRS workshop on Probabilistic Combinatorics, Fall 05
• AMS Sectional, UCSB
• Mt. Baldy Conference on Geometry, Algebra, and Phylogenetic Trees, Fall 04
• MSRI Conference on Discrete and Computational Geometry, Fall 03
• Mount Baldy Conference on Applied Algebra and Combinatorics, Fall 02
• Mount Baldy Conference on Differential Geometry, Fall 01

UCSD Combinatorics Links

• The Combinatorics of Large Sparse Graphs. This was a course given by Dr. Graham, and I maintain the lecture notes here for perpetuity. Note that there will be a forthcoming monograph on this subject, which should subsume these notes.
• 2004-2005 Reading Seminar List.
• 2005 Summer Seminar
• Fall 06 reading course in Harmonic Analysis for Combinatorics and Additive Number Theory.

Research Links

These links are mostly for my own use, but may be of more broad interest.

• UCSD Geometry Seminar.
• UCSD STAR (CS Theory) Seminar
• Seminars and Colloquia at UCSD.
• MSRI Program in Discrete and Computational Geometry. This is ongoing during the Fall of '03. A great place to find the new and open in the field.
• The Open Problem Page - Problems in Computational Geometry
• Geometry Junkyard - Open Problems
• Open Problems - Graph Theory and Combinatorics
• Open Problems in Combinatorics
• Open Problems of Paul Erdős
• Lectures on Discrete Geometry. A fantastic reference for all things discrete geometric. Also a model of exposition.
• The Probabilistic Method. The are no reasonable alternatives to learning this fantastic material. Luckily, this is a good book.
• Spectral Graph Theory. Spectral Goodness. Chances are if you see me I'm carrying around some subset of the last three books.
• An introduction to Convex Geometry - K. Ball. This is a great introduction to the tools and methods of modern convex geometry, clearly written, without the need of great technical background. Highly recommended (complements Matousek's book nicely). Also, it's free!
• Volume Estimates and Rapid Mixing - B. Bollobás Geometry, Combinatorics, and Probability. Gotta love it.
• Convex Geometric Analysis - MSRI Book The modern direction of convex geometry. Lots of big names and well written documents here.
• Kim's Probabilistic Method Notes. An alternate source of prob. method which isn't derived from Alon and Spencer.
• Random Graph Lecture by Alan Frieze. A good survey of the state of random graphs.
• Uniform Random Spanning Trees. This survey paper by R. Pemantle is an example not only of good pedagogy, but also is a great introduction to a very successful theory which, along the way, gives the best motivation for the Laplacian I've seen.
• The Chen-Stein Method for Convergence of Distributions. The Chen-Stein method for proving convergence of distributions to normal, Poisson, etc. has become well known in the statistics community, but less so in the discrete world. This method is a little calculation heavy, but gives fantastic answers in a large quantity of cases.

I should mention the following people, who have supported me in research over my undergraduate tenure: Weiqing Gu, Sampath Kannan, Sanjeev Khanna, Elizabeth ‘Z’ Sweedyk.

This file last modified 06/11/08