Thomas Karam

Maths bio:

As an undergraduate I was a student at the ENS in Paris (2014-2018), 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 a total absence of structure. Timothy Gowers brought me with him from Paris to Cambridge for my doctoral studies under him (2018-2022), during which I started by working on strategies aimed at making progress on some of the remaining central open combinatorial problems. One of these was the polynomial density Hales-Jewett conjecture, on which I eventually managed to make a dent, but the most fruitful effect that it has had on my research so far has been to give (chronologically speaking) the first motivation to work on basic properties which I believe will ultimately turn out to be more important by themselves. The objects that these properties address are (i) the ranks of tensors, and (ii) polynomials in finite fields with restricted variables (for instance Boolean variables).

In 2022 I moved to Oxford. There, I have continued to release work related to the above topics, but the majority of my research has revolved around quite different additional themes: geometric inequalities on volumes, extremal and stability questions for these inequalities (as well as for discrete sets), and number theory. Peter Keevash often played a key role in giving me advice on these areas, where again there is no shortage of deep questions, but also no shortage of more approachable cases for which there is a clear reason why they constitute meaningful progress towards the former. Once the resulting manuscripts have reached a sufficiently high standard I will release them publicly as well and summarise them by adding more to the next section.

Main contributions:

A recurring theme in several of my results in the first two areas below 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.

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.

• 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 with 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.

• For notions of rank defined in terms of how the entries of a rank-1 tensor may split in terms of its coordinates, the implications between the boundedness of these different notions may be completely characterised.

• These results have analogues for covering numbers of lattices. In some special cases presenting a correspondence between them and ranks of tensors, the above results can thereby be recovered through simpler proofs.
  
  

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 polynomial F_p^n -> F_p that is not approximately equidistributed on {0,1}^n coincides on {0,1}^n with a polynomial that may be expressed in terms of a bounded number of polynomials with strictly smaller degree. This extends a result of Green and Tao from 2007.

• If a polynomial in several variables does not even have full range on {0,1}^n, or has small range in a sense that can be characterised in terms of one-variable polynomials, then the degrees of the polynomials in the decomposition can be required to be smaller still.

• 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. This extends to group homomorphisms and polynomial conditions.

The third contribution is to combinatorial problems motivated largely by the polynomial density Hales-Jewett conjecture, which aims at unifying the different generalisations of van der Waerden’s theorem on the existence of arbitrarily long arithmetic progressions in colourings of the integers with finitely many colours, and also motivated by a special case of this common generalisation asking for the existence of a power difference in dense families of subsets of grids [n]^d.

• It is not possible to obtain a counterexample to the full conjecture that may be defined purely from modular considerations.

• Probabilistic techniques involving coverings provide natural reductions of this special case, and also reveal a difficulty gap distinguishing it from similar-looking statements.

• Already in the special case, we cannot specify too strongly the structure of the power difference: it cannot be required to be a union of at most d/2 intervals.