Thomas Karam
Maths bio:
As an undergraduate I was a student at the ENS in Paris, where I was fortunate to essentially complete specialised masters in different areas of mathematics (“Probability and random models”, “Number theory, analysis, geometry”, and “Pure mathematics” which was more algebraically focused). When the time to choose a narrower thesis topic came nearer I looked for the object or the few objects that I felt would nonetheless have the widest possible explanatory power. Shannon entropy was such a notion that I became particularly fond of, which led me back then to for a while view my research programme for the next few years as trying to show that universality theorems in probability theory could be proved using entropy methods. I ultimately did not continue in this direction back then, but continue to believe in this programme, and would be happy to return to it with anyone who is interested.
The area that I went to after that was Ramsey theory, a field of combinatorics showing the impossibility of total lack of structure. Timothy Gowers brought me with him to Cambridge for doctoral study under his supervision after returning from his sabbatical in Paris, and I then spent most of the first 2.5 years or so of my Ph.D. working on strategies aimed at making progress on some of the remaining central open problems in that area. In particular, the polynomial density Hales-Jewett conjecture chronologically gave me the first motivation to work on properties of the objects for which I by now have something more definitive to show (those of the next section), although retrospectively I expect that these properties will ultimately turn out to be more important by themselves. Thereafter, I moved to Oxford, where the other areas that I have been most active in so far have been geometric inequalities, extremal combinatorics, and number theory. I very much welcome proposals for collaborations in each of them.
Main contributions:
The first main contribution of my work so far has been to help develop the basic theory of the ranks of tensors (such as the tensor rank from computational complexity theory, the slice rank of Tao, and the partition rank of Naslund). The rank of matrices is rather well-understood, but this is not at all the case for any of the various competing notions of rank on tensors, even for some of the analogues of the simplest properties of matrix rank.
The second contribution is to the understanding of objects defined using vector spaces over finite fields - such as the distribution of polynomials, and the solution sets of systems of conditions defined using linear forms and polynomials - once we restrict the alphabets of their variables, in particular to the Boolean values 0,1. Likewise, to whatever extent the unrestricted objects are understood, their restrictions to the Boolean cube {0,1}^n are less so.
A recurring theme in several of my results in both areas is that a very simple property which holds in a good or even "perfect" way for the well-understood objects fails rather strongly for the poorly understood object, but it is proved that the latter object still satisfies a not too complicated reformulation of the original property that is very much in its original spirit. Here are some instances.
A tensor with rank k does not necessarily have a k x k x ... x k subtensor with rank k, but it does have a subtensor of not too large size with rank that tends to infinity k.
A tensor with bounded slice rank does not in general have only one minimal-length decomposition up to a change of basis, but, up to a natural (and necessary) class of transformations, it has only at most a bounded number of such decompositions.
A high-rank polynomial over a finite field might not be equidistributed on {0,1}^n, but if so, then it coincides with a low-rank polynomial on {0,1}^n (extends Green and Tao 2007).
If a system of mod-p linear conditions has a dense solution set inside {0,1}^n, then the system of conditions no longer necessarily has bounded dimension, but its solution set can always be internally approximated by a system of conditions that does.
Manuscripts (arxiv versions, from most to least recent):
Simple proofs for lattice coverings and sparse tensors (15 pp).
The interplay between bounded ranks of tensors arising from partitions (25 pp).
On the expressive power of mod-p linear forms on the Boolean cube (26 pp).
Small sunflowers and the structure of slice rank decompositions (45 pp).
(with Timothy Gowers) Low-complexity approximations for sets defined by generalizations of affine conditions (26 pp).
Ranges of polynomials control degree ranks of Green and Tao over finite prime fields (25 pp).
(with Jan Draisma) On subtensors of high partition rank (10 pp).
(with Timothy Gowers) Equidistribution of high-rank polynomials with variables restricted to subsets of F_p (42 pp).