In 2001 Croot resolved an old conjecture of Erdős and Graham, proving that in any finite colouring of the positive integers there is a (non-trivial) monochromatic solution to 1/n₁+...+1/n_k = 1 with all n_i distinct. A natural generalisation, also conjectured by Erdos and Graham, is that in fact any set of positive density contains such a solution. We will discuss the proof of this conjecture, which extends Croot's method, and uses Fourier analysis coupled with elementary number theoretic and combinatorial arguments.

A key tool in modern additive combinatorics, going back to Gowers' proof of Szemeredi's theorem, is that counts of linear configurations are controlled by Gowers norms. For example, if S and T are two dense sets and |S-T|_U^k-1 is small then S and T have roughly the same number of k-term arithmetic progressions. Like many other core arguments in the field, this is proven by (k-1) applications of the Cauchy--Schwarz inequality. Generalizations of this statement quickly become subtle. For example, linear configurations (x, x+z, x+y, x+y+z, x+2y+3z, 2x+3y+6z) are controlled by the U²-norm (i.e., by normal Fourier analysis) but it is not at all straightforward to prove this just with Cauchy--Schwarz; whereas controlling (x, x+z, x+y, x+y+z, x+2y+3z, 13x+12y+9z) requires the U³-norm (i.e., quadratic Fourier analysis). A conjecture of Gowers and Wolf (resolved combining work of Gowers--Wolf, Green--Tao, Hatami--Hatami--Lovett and Altman) gives a condition to determine the smallest U^k-norm required for a given configuration, but the proofs require deep structure theorems and (unlike Cauchy--Schwarz arguments) give very weak bounds. In this talk, I will describe how it is (sometimes) possible to find the missing Cauchy--Schwarz arguments by "mining proofs". The equality cases of these inequalities correspond (it turns out) to facts about functional equations. For example, the k-term progression case states the following: if f₁,...,f_k are functions such that f₁(x)+f₂(x+h)+...+f_k(x+(k-1)h) = 0 for all x,h, then each f_i must be a polynomial of degree at most k-2. This statement is not completely obvious but has a short elementary proof. Given such an elementary proof (at least, one of a special type), we can recover an iterated Cauchy--Schwarz proof of the corresponding inequality -- albeit a very long and complicated one that would be hard to discover by hand. This answers the Gowers--Wolf question with polynomial bounds, and hopefully other questions where the availability of complicated Cauchy--Schwarz arguments is a limiting factor.

In 1971, David Singmaster conjectured that any natural number greater than one only appears in Pascal's triangle a bounded number of times. In the talk I will discuss what is known about this conjecture, concentrating on a recent result in joint work with Maksym Radziwill, Xuancheng Shao, Terence Tao, and Joni Teräväinen that establishes the conjecture in the interior region of Pascal's triangle.

While the problem is combinatorial, we use number theoretic and analytic tools. In particular an important analytic input in our proof is Vinogradov's estimate for exponential sums over primes.

A system of linear equations L over 𝔽_q is common if the number of monochromatic solutions to L in any two-colouring of (𝔽_q)ⁿ is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of (𝔽_q)ⁿ. Motivated by existing results for specific systems (such as Schur triples and arithmetic progressions), as well as extensive research on common and Sidorenko graphs, the systematic study of common systems of linear equations was recently initiated by Saad and Wolf. Building on earlier work of of Cameron, Cilleruelo and Serra, as well as Saad and Wolf, common linear equations have been fully characterised by Fox, Pham and Zhao. In this talk I will discuss some recent progress towards a characterisation of common systems of two or more equations. In particular we prove that any system containing an arithmetic progression of length four is uncommon, confirming a conjecture of Saad and Wolf. This follows from a more general result which allows us to deduce the uncommonness of a general system from certain properties of one- or two-equation subsystems.

We develop novel techniques which allow us to prove a diverse range of results around subset sums, including solutions to several longstanding open problems. These include: solutions to three problems of Burr and Erdős on Ramsey complete sequences, for which Erdős later offered a combined total of $350; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and Erdős on the minimum number of colours needed to colour the positive integers less than n so that n cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erdős and Graham on sets of integers avoiding a given subset sum; and, answering a question of Tran, Vu and Wood, a homogeneous strengthening of a seminal result of Szemerédi and Vu on long arithmetic progressions in subset sums.

The partition rank and the analytic rank of a tensor measure algebraic structure and bias, respectively. We prove that they are equivalent up to a constant, over any large enough finite field (independently of the number of variables) of any characteristic. The proof constructs rational maps computing a rank decomposition for successive derivatives, on a carefully chosen subset of the kernel variety associated with the tensor. Proving the equivalence between these two quantities is the main question in the "bias implies low rank" line of work in higher-order Fourier analysis, and was reiterated by multiple authors. Joint work with Alex Cohen.