David Ellis

I am a Junior Research Fellow in Pure Mathematics at St John's College, Cambridge, and a member of the Combinatorics Group in the Department of Pure Mathematics and Mathematical Statistics (DPMMS).
My college webpage, containing a description of my research for non-specialists, can be found here.
My CV can be found here; a list of my publications can be found here.

Research

I work on a variety of problems in combinatorics. I am particularly interested in connections between combinatorics and other areas of mathematics. Recently, I have been working mainly on the inferface between combinatorics, Fourier analysis, representation theory / group theory and probability theory. Here are some selected publications and papers:

Intersecting families of permutations and other problems in extremal combinatorics, a version of my PhD Thesis, also available from the University Library, University of Cambridge, UK.

Intersecting families of permutations (joint with Ehud Friedgut and Haran Pilpel), Journal of the American Mathematical Society 24 (2011), pp. 649-682.

A proof of the Cameron-Ku conjecture, to appear in Journal of the London Mathematical Society.

Stability for t-intersecting families of permutations, Journal of Combinatorial Theory, Series A 118 (2011), pp. 208-227.

Irredundant families of subcubes, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 150, Issue 2 (2011), pp. 257-272.

Almost isoperimetric subsets of the discrete cube, Combinatorics, Probability and Computing 20 (2011), pp. 363-380.

Generating all subsets of a finite set with disjoint unions (joint with Benny Sudakov), Journal of Combinatorial Theory, Series A 118 (2001) pp. 2319-2345.

Triangle-intersecting families of graphs (joint with Yuval Filmus and Ehud Friedgut), to appear in Journal of the European Mathematical Society.

Setwise intersecting families of permutations, submitted.

Here are the slides from some selected talks:

Intersecting families of permutations. Variants of this talk were given at the One-day Colloquium in Combinatorics at QMUL, May 2009, at the Fete of Combinatorics and Computer Science in Keszthely, Hungary, August 2009, at the IPAM Long Program in Combinatorics at UCLA, October 2009, and at the Combinatorics Seminars at LSE, the University of Bristol, and the University of Cambridge.

Irredundant families of subcubes, talk given at the Combinatorics Seminar, UCLA, January 2010.

Triangle-intersecting families of graphs, talk given at the Combinatorics Seminar, Mathematical Institute, University of Oxford, November 2010. Similar talks (without slides) were given at the Combinatorics Seminar at the University of Cambridge, February 2011, at the DIMAP Seminar at the University of Warwick, February 2011, and at the Isaac Newton Institute, April 2011 (see here for video).

Algebraic methods in combinatorics (Graduate course, Lent 2011)

In the last fifty years, algebraic methods have been used with striking success in combinatorics. This course explores these methods, and some of their most beautiful applications. There are connections with combinatorial geometry, probability theory, and theoretical computer science. Please find lecture notes below. To all: comments are most welcome! If you have any suggestions of alternative proofs, or other insights on the material you feel could be included, please email me, at D.Ellis at dpmms dot cam dot ac dot uk.

Algebraic methods in combinatorics: an overview (Lecture 1)

The Frankl-Wilson theorem and some consequences in Ramsey theory and combinatorial geometry (Lectures 2-5)

Exact intersections (Lecture 6)

Saturated and weakly saturated hypergraphs (Lectures 6-7)

The Polynomial Method (Lectures 8-10)
Eigenvalue methods in extremal combinatorics: an overview (Lectures 11-12)

The expansion of random regular graphs (Lectures 13-14)

Sparse superconcentrators (Lecture 14)

Eigenvalues, random walks, and Ramanujan graphs (Lectures 15-16)