Ben Green: Quadratic forms in 8 prime variables

17th November, 2021

I will discuss a recent paper of mine, the aim of which is to count the number of prime solutions to Q(p1,..,p8) = N, for a fixed quadratic form Q and varying N. The traditional approach to problems of this type, the Hardy-Littlewood circle method, does not quite suffice. The main new idea is to involve the Weil representation of the symplectic groups Sp8(ℤ/qℤ). I will explain what this is, and what it has to do with the original problem. I hope to make the talk accessible to a fairly general audience.

Kaisa Matomäki: Singmaster’s conjecture in the interior of Pascal’s triangle

3rd November, 2021

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.

Natasha Morrison: Uncommon systems of equations

20th October, 2021

A system of linear equations L over 𝔽q is common if the number of monochromatic solutions to L in any two-colouring of (𝔽q)n is asymptotically at least the expected number of monochromatic solutions in a random two-colouring of (𝔽q)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.

Jared Duker Lichtman: Twin primes & a modified linear sieve

6th October, 2021

The linear sieve is a powerful tool to tackle problems related to the primes, when combined with equidistribution estimates for the remainder. In 1977 Iwaniec introduced a well-factorable modification of the linear sieve to prove there are infinitely many integers n such that n2+1 has at most two prime factors. Furthermore, the (well-factorable) linear sieve leads to the best known upper bounds for twin primes. These bounds use work of Bombieri, Friedlander, and Iwaniec from 1986, showing these sieve weights equidistribute primes of size x in arithmetic progressions to moduli up to x4/7. This level was recently increased to x7/12 by Maynard.We introduce a new modification of the linear sieve whose weights equidistribute primes of size x to level x10/17. As an application we refine a 2004 upper bound for twin primes of Wu, which gives the largest percent improvement since the work of Bombieri, Friedlander, and Iwaniec.

Will Sawin: Sums in progressions over 𝔽q[T], the symmetric group, and geometry

30th September, 2021

I will discuss some recent progress in analytic number theory for polynomials over finite fields, giving strong new estimates for the number of primes in arithmetic progressions, as well as for sums of some arithmetic functions in arithmetic progressions. The strategy of proof is fundamentally geometric, and I will explain some of the geometric ideas in the proof, including how we can use the representation of the symmetric group to handle many different arithmetic functions in a uniform way.

Sam Chow: Galois groups of random polynomials

19th July, 2021

How often is a quintic polynomial solvable by radicals? We establish that the number of such polynomials, monic and irreducible with integer coefficients in [-H, H], is O(H3.91). More generally, we show that if n ≥ 3 and n ≠ 7, 8, 10 then there are O(Hn-1.017) monic, irreducible polynomials of degree n with integer coefficients in [-H, H] and Galois group not containing An. Save for the alternating group and degrees 7, 8, 10, this establishes a 1936 conjecture of van der Waerden, that irreducible non-Sn polynomials are substantially rarer than reducible polynomials.

Huy Pham: Subset sums, completeness and colorings

7th June, 2021

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.

Guy Moshkovitz: An Optimal Inverse Theorem

31st May, 2021

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.

Lisa Sauermann: Finding solutions with distinct variables to systems of equations over 𝔽p

24th May, 2021

Let us fix a prime p and a homogeneous system of m linear equations aj,1x1+ . . . +aj,kxk=0 for j=1, . . ., m with coefficients aj,i ∈ 𝔽p. Suppose that k ≥ 3m, that a_{j,1}+ . . . +a_{j,k}=0 for j=1,. . . ,m and that every m × m minor of the m × k matrix (a_{j,i})_{j,i} is non-singular. Then we prove that for any (large) n, any subset A ⊆ (𝔽p)n of size |A| greater than C Γn contains a solution (x1, . . .,xk) ∈ Ak to the given system of equations such that the vectors x1, . . .,xk ∈ A are all distinct. Here, C and Γ are constants only depending on p, m and k such that Γ < p. The crucial point here is the condition for the vectors x1, . . .,xk in the solution (x1, . . .,xk) ∈ Ak to be distinct. If we relax this condition and only demand that x1, . . .,xk are not all equal, then the statement would follow easily from Tao's slicerank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slicerank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments.

Allysa Lumley: Primes in short intervals – Heuristics and calculations

17th May, 2021

We formulate, using heuristic reasoning, precise conjectures for the range of the number of primes in intervals of length y around x, where y ≪ (log x)2. In particular, we conjecture that the maximum grows surprisingly slowly as y ranges from log x to (log x)2. We will show that our conjectures are somewhat supported by available data, though not so well that there may not be room for some modification. This is joint work with Andrew Granville.

Orit Raz: Expanding polynomials and the discretized Elekes-Rónyai theorem

10th May, 2021

Elekes and Rónyai have characterized real bivariate polynomials which have a small image over large Cartesian products. Besides being an interesting problem in itself, the Elekes-Rónyai setup, and certain generalizations thereof (such as those considered by Elekes and Szabó), arise in many Erdős-type problems in combinatorial geometry and additive combinatorics. In the talk I will give some overview of this topic, and then tell about a recent result (joint with J. Zahl) in which cardinality of a finite set is replaced by either the δ-covering number of a set, or its Hausdorff dimension.

Benny Sudakov: Small doubling, atomic structure and l-divisible set families

4th May, 2021

A central theme in additive combinatorics is the study of the structure of sets with small doubling, i.e. sets A such that A + A has size not much larger than A. In this talk we discuss a problem of a similar flavour for set systems. Given a set-family F, let F*F={A ∩ B | A,B ∈ F}. We measure the size of F by the dimension of the subspace spanned by the characteristic vectors of the sets in F over some field. What can we say about F if F*F is not much larger than F? Observe that if S is an atomic set-family, i.e., the ground set is split into disjoint subsets and S contains their arbitrary unions, then S* S = S. Our structure theorem shows that this is essentially the only possible example: any set-family F with small 'doubling' must be close to being atomic. We will use this theorem to solve a 40-year old problem of Frankl and Odlyzko on set families with restricted intersections.

Dor Minzer: New bounds on the size of product free sets in the alternating group

26th April, 2021

A set of permutations A is said to contain a product if there are two permutations in it whose product also lies in A; otherwise A is called product free. We show that the density of a product free set A ⊆ An is at most O(1/√n). This result is asymptotically tight, and improves on earlier results of Gowers (O(1/n1/3)) and Eberhard (O(log7/2 n /√n)). We also show stability results. Our proof uses recent refinements of the hypercontractive inequality over Sn for the class of "global functions", and in particular a version of the level 1 inequality in this setting. Based on a joint work with Peter Keevash and Noam Lifshitz.

Lola Thompson: Summing μ(n): an even faster elementary algorithm

12th April, 2021

We present a new elementary algorithm for computing the Mertens function, which improves on existing combinatorial algorithms. While there is an analytic algorithm due to Lagarias-Odlyzko with computations based on the integrals of ζ(s), our algorithm has the advantage of being easier to implement. The new approach roughly amounts to analysing the difference between a model that we obtain via Diophantine approximation and reality, and showing that it has a simple description in terms of congruence classes and segments. This simple description allows us to compute the difference quickly by means of a table lookup.

Corrine Yap: A Topological Turán Problem

15th March, 2021

The classical Turán problem asks: given a graph H, how many edges can an n-vertex graph have while containing no isomorphic copy of H? By viewing (k+1)-uniform hypergraphs as k-dimensional simplicial complexes, we can ask a topological version of this, first posed by Nati Linial: given a k-dimensional simplicial complex S, how many facets can an n-vertex k-dimensional simplicial complex have while containing no homeomorphic copy of S? Until recently, little was known for k greater than 2. In this talk, we give an answer for general k using dependent random choice - a method that has produced powerful results in extremal graph theory, additive combinatorics, and Ramsey theory.

Aled Walker: Extremal problems for GCDs

8th March, 2021

In this talk we will discuss how ideas from Koukoulopoulos and Maynard's proof of an old conjecture in diophantine approximation led to some surprisingly subtle questions in combinatorial number theory. These questions ask for a bound on the maximal size of two finite sets of natural numbers A and B with the property that 1% of the pairs (a,b) have a large greatest common divisor. We will also describe our recent joint work with Ben Green, in which we proved close-to-optimal bounds on these problems.

Harald Helfgott: Expansion, divisibility and parity

5th March, 2021

We will discuss a graph that encodes the divisibility properties of integers by primes. We show that this graph is shown to have a strong local expander property almost everywhere. We then obtain several consequences in number theory, beyond the traditional parity barrier. For instance: for λ the Liouville function (that is, the completely multiplicative function with λ(p) = -1 for every prime), (1/log x) ∑n ≤ x λ(n) λ(n+1)/n = O(1/√(log log x)), which is stronger than a well-known result by Tao. We also manage to prove, for example, that λ(n+1) averages to 0 at almost all scales when n restricted to have a specific number of prime divisors Ω(n)=k, for any "popular" value of k (that is, k = log log N + O(√(log log N)) for n ≤ N. We will give a quick overview of the combinatorial ideas behind the proof.

Jimmy He: Random walks on finite fields with deterministic jumps

1st March, 2021

Recently, Chatterjee and Diaconis showed that most bijections, if applied between steps of a Markov chain, cause the resulting chain to mix much faster. However, explicit examples of this speedup phenomenon are rare. I will discuss recent work studyingsuch walks on finite fields where the bijection is algebraically defined. This work gives a large collection of examples where this speedup phenomenon occurs. These walks can be seen as a non-linear analogue of the Chung-Diaconis-Graham process, where the bijectionis multiplication by a non-zero element of the finite field. This work is partially joint with Huy Pham and Max Xu.

Swastik Kopparty: Geometric rank of tensors and the subrank of matrix multiplication

22nd February, 2021

I will talk about a new notion of rank for tensors called geometric rank, and discuss some of its basic properties, as well as its relationship with other well-studied notions of rank like subrank, slice rank and analytic rank. As an application, we will see a proof of tightness of an old bound of Strassen on the subrank of the matrix multiplication tensor.