Kevin Ford: Prime gaps, probabilistic models, the interval sieve, Hardy-Littlewood conjectures and Siegel zeros

3rd September, 2020

Motivated by a new probabilistic interpretation of the Hardy-Littlewood k-tuples conjectures, we introduce a new probabilistic model of the primes and make a new conjecture about the largest gaps between the primes below x. Our bound depends on a property of the interval sieve which is not well understood. We also show that any sequence of integers which satisfies a sufficiently uniform version of the Hardy-Littlewood conjectures must have large gaps of a specific size. Finally, assuming that Siegel zeros exist we show the existence of gaps between primes which are substantially larger than the gaps which are known unconditionally. Much of this work is joint with Bill Banks and Terry Tao.

Will Sawin: The quadratic Bateman-Horn conjecture over 𝔽q[u]

17th August, 2020

The Bateman-Horn conjecture predicts the fraction of integers n such that n2+1 is prime, and makes similar predictions for polynomials of higher degree. In joint work in progress with Mark Shusterman, we prove an analogue of the n2+1 case, replacing natural numbers n with polynomials in 𝔽q[u], which for instance counts the fraction of polynomials f such that f2+u is an irreducible polynomial. The proof combines geometric methods, unusual algebraic properties of polynomials, and some (very) classical number theory.

Ofir Gorodetsky: The distribution of squarefree integers in short intervals

11th August, 2020

The squarefree integers are divisible by no square of a prime. It is well known that they have a positive density within the integers. We consider the number of squarefree integers in a random interval of size H : |{n ∈ [x,x+H] : n is squarefree}|, where x is a random number between 1 and X. The variance of this quantity has been studied by R. R. Hall in 1982, obtaining asymptotics in the range H less than X2/9, with a proof method that stays in 'physical space'. Keating and Rudnick recently conjectured that his result persists for the entire range H less than X1-ε. We make progress on this conjecture, with properties of the Riemann zeta function playing a role in our results. We will show how, on RH, one can verify the conjecture for H up to X2/3.

Akshat Mudgal: Diameter free estimates for Vinogradov systems

10th August, 2020

A classical object of study in additive number theory has been the Vinogradov system, that is, the system defined by the equations x1j+ . . . + xsj = y1j+ . . . + ysj (j = 1, . . ., k). In particular, given a finite set A of integers, finding sharp upper bounds for the number of solutions Js,k(A) to this system, when all the variables lie in the set A, has been an important topic of work. Recently, two major approaches have been developed to tackle this problem - the efficient congruencing method of Wooley, and the decoupling techniques of Bourgain-Demeter-Guth. Both these methods give upper bounds for Js,k(A) in terms of s,k, and the cardinality |A| of A, and the diameter X of A. In particular, when X is large in terms of |A|, say when X is much larger than exp(exp(|A|)), these bounds perform worse than the trivial estimates. In this talk, we present new upper bounds for Js,2(A) which depend only on |A| and s. These improve upon, and generalize, a previous result of Bourgain and Demeter.

Julian Sahasrabudhe: Zeros of cosine polynomials, a problem of Littlewood

4th August, 2020

Let f be a {0,1}-cosine polynomial with n terms. In his 1986 monograph, J.E. Littlewood considered the minimum number of zeros that such polynomials have in [0,2π] and conjectured that the number of such roots is "n-1 or not much less". While it is now known that there exist cosine polynomials with considerably fewer roots, much less is known about the lower bound and, in fact, it was a long standing problem just to show that the number of such zeros tends to infinity with n. We will discuss the resolution of this conjecture and also mention some more recent progress on the upper bound.

Xuancheng Shao: Gowers uniformity of primes in arithmetic progressions

27th July, 2020

A celebrated theorem of Green-Tao asserts that the set of primes is Gowers uniform, allowing them to count asymptotically the number of k-term arithmetic progressions in primes up to a threshold. In this talk I will discuss results of this type for primes restricted to arithmetic progressions. These can be viewed as generalizations of the classical Bombieri-Vinogradov theorem. I will also discuss a number of applications; for example, the set of primes p obeying explicit bounded gaps.

Nikos Frantzikinakis: Ergodic properties of the Liouville function and applications

20th July, 2020

The Liouville function is a multiplicative function that encodes important information related to distributional properties of the prime numbers. A conjecture of Chowla states that the values of the Liouville function fluctuate between plus and minus in such a random way, that all sign patterns of a given length appear with the same frequency. The Chowla conjecture remains largely open and in this talk we will see how ergodic theory combined with some feedback from number theory allows us to establish two variants of this conjecture. Key to our approach is an in-depth study of measure preserving systems that are naturally associated with the Liouville function. The talk is based on joint work with Bernard Host.

Sarah Peluse: An asymptotic version of the prime power conjecture for perfect difference sets

13th July, 2020

A subset D of a finite cyclic group ℤ/mℤ is called a "perfect difference set" if every nonzero element of ℤ/mℤ can be written uniquely as the difference of two elements of D. If such a set exists, then a simple counting argument shows that m=n2+n+1 for some nonnegative integer n. Singer constructed examples of perfect difference sets in ℤ/(n2+n+1)ℤ whenever n is a prime power, and it is an old conjecture that these are the only such n for which a perfect difference set exists. In this talk, I will discuss a proof of an asymptotic version of this conjecture: the number of n less than N for which ℤ/(n2+n+1)ℤ contains a perfect difference set is asymptotically the number of prime powers less than N.

Thomas Bloom: Breaking the logarithmic barrier in Roth’s theorem on progressions

10th July, 2020

(joint work with Olof Sisask) We present an improvement to Roth's theorem on arithmetic progressions, by showing that if A ⊂ [N] has no non-trivial three-term arithmetic progressions then |A| ≪ N/(log N)1+c for some positive absolute constant c. In particular, this establishes the first non-trivial case of a conjecture of Erdős on arithmetic progressions.

Sophie Stevens: An update on the sum-product problem

6th July, 2020

In new work with Misha Rudnev, we prove a stronger bound on the sum-product problem, showing that

max(|A + A|, |AA|) ≥ |A|4/3 + 2/1167 − o(1)

for any finite set A of real numbers. This builds upon the work of Solymosi, Konyagin and Shkredov, although our paper is self-contained. I will give an overview of the arguments, both old and new, and describe some consequences of the new arguments.

James Maynard: Primes in arithmetic progressions to large moduli

2nd July, 2020

How many primes are there which are less than x and congruent to a modulo q? This is one of the most important questions in analytic number theory, but also one of the hardest - our current knowledge is limited, and any direct improvements require solving exceptionally difficult questions to do with exceptional zeros and the Generalized Riemann Hypothesis! If we ask for 'averaged' results then we can do better, and powerful work of Bombieri and Vinogradov gives good answers for q less than the square-root of x. For many applications this is as good as the Generalized Riemann Hypothesis itself! Going beyond this 'square-root' barrier is a notorious problem which has been achieved only in special situations, perhaps most notably this was the key component in the work of Zhang on bounded gaps between primes. I'll talk about recent work going beyond this barrier in some new situations. This relies on fun connections between algebraic geometry, spectral theory of automorphic forms, Fourier analysis and classical prime number theory. The talk is intended for a general audience.

Zachary Chase: A random analogue of Gilbreath’s conjecture

29th June, 2020

Given a sequence a1, a2, . . . of integers, one can form the sequence |a1 - a2|, |a2, a3|, . . .. Gilbreath's conjecture is that if you start with the sequence of the primes and iterate this consecutive differencing procedure, then the first term of every sequence (besides the initial one) is a 1. We prove the conclusion of Gilbreath's conjecture for a suitably random initial sequence instead of the primes.

Marina Iliopoulou: A discrete Kakeya-type inequality

22nd June, 2020

The Kakeya conjectures of harmonic analysis claim that congruent tubes that point in different directions rarely meet. In this talk we discuss the resolution of an analogous problem in a discrete setting (where the tubes are replaced by lines), and provide some structural information on quasi-extremal configurations.

Ben Green: On a conjecture of Gowers and Long

15th June, 2020

In a very interesting paper, Gowers and Long discussed binary operations * on finite sets which are somewhat associative in the sense that x * (y * z) = (x * y) * z for 1 percent (say) of all triples (x,y,z). They presented an example of such an operation which, they conjectured, is not closely related to any genuine group operation. I will discuss a proof of their conjecture, which uses a number of tools from (nonabelian) additive combinatorics.

Jozsef Solymosi: Sums and products along edges of sparse graphs

2nd June, 2020

In their seminal paper Erdős and Szemerédi formulated conjectures on the size of sum set and product set of integers. The strongest form of their conjecture is about sums and products along the edges of a graph, when we consider sums and products of some pairs only. With Noga Alon and Imre Ruzsa we showed that this strong form of the Erdős-Szemerédi conjecture does not hold. In this talk I will list some related problems and recent results.

Ashwin Sah: Diagonal Ramsey via effective quasirandomness

28th May, 2020

We improve the upper bound for diagonal Ramsey numbers to R(k+1,k+1) ≤ exp(-c(log k)2)(2k)!/(k!)2 for k ≥ 3. To do so, we build on a quasirandomness and induction framework for Ramsey numbers introduced by Thomason and extended by Conlon, demonstrating optimal 'effective quasirandomness' results about convergence of graphs. This optimality represents a natural barrier to improvement.

Dmitrii Zhelezov: Sets inducing large additive doubling

25th May, 2020

Rephrasing the celebrated Freiman lemma in additive combinatorics, one can show that a finite set in ℤd containing a K-dimensional simplex has additive doubling at least ~K. We will discuss a novel framework for studying how such induced doubling can be inherited from a more general class of multi-dimensional subsets. It turns out that subsets of so-called quasi-cubes induce large doubling no matter the dimension of the ambient set. Time permitting, we will discuss how it allows to deduce a structural theorem for sets with polynomially large additive doubling and an application to the ”few products, many sums” problem of Bourgain and Chang.

Tomasz Schoen: Improved bound in Roth’s theorem

22nd May, 2020

I sketch a proof of a new bound in Roth's theorem on arithmetic progressions: if A ⊆ {1,...,N} does not contain any non-trivial three-term arithmetic progression then |A| ≪ (log log N)^3+o(1)N/log N.

Yufei Zhao: Popular common difference

19th May, 2020

Green proved the following strengthening of Roth's theorem: for every positive ϵ, there is some n(ϵ) such that for every N ≥ n(ϵ) and A ⊂ [N] with |A| = αN, there is some nonzero d such that A contains at least (α3 − ϵ)N three-term arithmetic progressions with common difference d (i.e., a popular common difference with frequently at least roughly the random bound). I'll discuss some extensions and generalizations of this result:

   •  How large does n(ϵ) have to be for the result to hold? (It turns out that a tower-type bound is necessary)
   •  Besides 3-term arithmetic progressions, is there a similar result for other patterns?
   •  What about patterns in higher-dimensional patterns?

Giorgis Petridis: Energy in the affine group and a question of Yufei Zhao

12th May, 2020

We will survey new and old results on the "energy'' of a set of affine transformations and see applications in the geometric side of additive combinatorics, in combinatorial geometry and in questions on growth in the affine group. In particular we will answer to the affirmative a question of Yufei Zhao.