2027 Graduate Summer School Course Descriptions

Week 1 (June 27- July 3): Oleksiy Klurman, Borys Kuca, and Sean Prendiville

Week 2 (July 4 - 10): Ben Green, Redmond McNamara, and Joel Moreira

Week 3 (July 11 - 17): Mariusz Mirek, Donald Robertson, and Joni Teräväinen

Graduate Program Lecturers: Ben Green, University of Oxford; Oleksiy Klurman, University of Bristol, UK; Borys Kuca, Jagiellonian University; Redmond McNamara, Northwestern University; Mariusz Mirek, Rutgers University; Joel Moreira, University of Warwick; Sean Prendiville, Lancaster University; Donald Robertson, University of Manchester; and Joni Teräväinen, University of Cambridge.

Ben Green

Title: Quantitative inverse theorems

Abstract: The Gowers norms are a fundamental family of norms in additive combinatorics. They are crucial in understanding questions about arithmetic progressions (such as Szemeredi's theorem) but have many other applications. Here, we will discuss inverse theorems for these norms, primarily for functions on $\{1,\dots, N\}$, but perhaps also in other contexts. The primary focus will be on the $U^3$-norm, which is the first nontrivial case and which already contains much interesting mathematics, but we will also discuss norms of higher order.

Oleksiy Klurman

Title: Multiplicative Functions and Applications

Abstract: The study of multiplicative functions plays a fundamental role in analytic number theory and, more recently, in arithmetic combinatorics. We will explore the classical results of Wirsing and Halász describing the behaviour of mean values of multiplicative functions and discuss modern perspectives on their proofs. We will also touch upon more recent advances in the study of correlations of multiplicative functions and develop a framework for the later lectures, in particular those by McNamara and Moreira.

Borys Kuca

Title: Ergodic methods in combinatorics: an introduction to the Host–Kra theory

Abstract: Since Furstenberg’s celebrated multiple recurrence result from the 1970s, ergodic methods have proved to be enormously powerful tools for solving problems in combinatorics and number theory. This course will serve as a broad introduction to this exciting and ever-expanding field. After a presentation of major results, we will delve into the rudiments of the Host–Kra theory and conclude with a sample of more recent developments.

Redmond McNamara

Title: The Sarnak and Chowla Conjectures

Abstract: The Sarnak and Chowla conjectures are two fundamental conjectures in analytic number theory. The Sarnak conjecture states that the Möbius function does not correlate with low-complexity, so-called “zero-entropy” functions. This is the critical step in proving different versions of the prime number theorem for different dynamical systems, which can, in fact, be broadly useful to number theorists. The Chowla conjecture states that for any fixed k, the function

n→µ(n)µ(n+ 1)···µ(n+ k)

should have average zero. It is the analogue of the twin prime conjecture for the Möbius function and, in fact, a sufficiently strong quantitative version of the Chowla conjecture would imply both the twin prime conjecture and that every sufficiently large even number is the sum of two primes (asymptotic Goldbach). These conjectures turn out to be intimately related: Chowla implies Sarnak and Sarnak almost implies Chowla. We will survey some recent results on these conjectures.

Mariusz Mirek

Title: Pointwise ergodic theorems along polynomials and primes

Abstract: The purpose of this minicourse is to introduce some of the quantitative tools that have proved effective in studying pointwise convergence problems in ergodic theory and beyond. Our main example will be the circle method, a remarkably successful technique from analytic number theory that led to a quantitative solution of Waring’s problem and has become an important tool in the study of polynomial ergodic averages.

We will explain how quantitative estimates, such as maximal, oscillation, and variational inequalities, can be used to obtain pointwise almost everywhere convergence and why these problems are often fundamentally different from questions of norm convergence. The circle method provides a particularly striking illustration of this distinction, showing how arithmetic structure can enter pointwise ergodic theory in an essential way.

We will conclude by discussing how these ideas apply to polynomial and prime ergodic averages and how the circle method has contributed to recent progress on the Furstenberg–Bergelson–Leibman conjecture, a major open problem in pointwise ergodic theory.

Joel Moreira

Title: Partition regularity of polynomial patterns

Abstract: The main goal of Ramsey theory is to understand which patterns are unavoidable when one partitions (or colours) a large, structured object into a few pieces. When colouring the natural numbers, an influential problem is that of characterizing which arithmetic patterns must always emerge. For example, a theorem of vanderWaerden (which turns 100 years old this year!) states that no matter how the natural numbers are coloured with finitely many colours, there are always arbitrarily long arithmetic progressions—sets of the form {a,a+ d,a+ 2d,...,a+ kd}—that are monochromatic (all of the same colour).

More recently, attention has shifted to polynomial patterns. For instance, it is still unknown whether any finite colouring of the natural numbers must always produce a monochromatic Pythagorean triple (i.e., a solution to x2 + y2 = z2).

Recent developments often combine insights from ergodic theory, additive combinatorics and number theory. In this minicourse we will survey some of these tools and explore some of the more recent advances.

Sean Prendiville

Title: Using additive combinatorics to find configurations in dense sets and colourings

Abstract: We demonstrate how, given a colouring of the positive integers, one can find arithmetic configurations all of whose elements receive the same colour. This will lead us to discuss a number of topics, including density theorems, quantitative bounds, density increment arguments and inverse theorems. Assuming certain inverse theorems as a black box (these being the topic of a later course), we show how to prove special cases of the (polynomial) Szemerédi theorem, as well as Rado’s theorem on partition regularity. An overarching theme will be classifying obstructions to uniformity, also known as the true complexity problem.

Prerequisites: Some familiarity with Fourier series would be helpful, but not essential.

Donald Robertson

Title: Infinitary patterns in sets of positive density

Abstract: This course will cover the ergodic-theoretic approach to the existence of infinite additive patterns in sets of natural numbers having positive upper density. We will focus on additive patterns such as sums of finitely many infinite sets and their refinements, explaining how the structure theory of measure-preserving transformations plays a role in producing such patterns. The comparison with techniques for finding finite patterns such as arithmetic progressions will be emphasized.

Joni Teräväinen

Title: Gowers uniformity of primes

Abstract: The Gowers norms were introduced by Tim Gowers to give a new proof of Szemerédi’s theorem. They measure pseudorandomness of functions by counting additive patterns and have become an important tool in additive combinatorics and analytic number theory. Their use in analytic number theory was pioneered by Green, Tao and Ziegler in their work on patterns in primes, establishing that a suitably normalised version of the primes is Gowers uniform in the sense of having small Gowers norms. Since then, Gowers norms have been found to have applications to a range of problems in prime number theory, including somewhere the connection to additive patterns is far from obvious.

In this course we will introduce the Gowers norms over the integers, establish some of their properties, and discuss their applications to problems in multiplicative number theory. A particular focus will be on results about primes in short intervals and the extent to which the Gowers uniformity of the primes can be shown to persist in such intervals.