The Swinnerton-Dyer Seminar 2026-2027
Welcome to Warwick's Swinnerton-Dyer Seminar (formerly the Junior Number Theory Seminar)!
The seminar takes place during term time on Mondays from 11am - 12pm in
- B3.02 (terms 1 and 3),
- B3.01 (term 2).
The seminar is organised by Joseph Cohen and Rubin Pal.
If you want to give a talk in this seminar, contact one of the organisers!
Term 1
Date |
Speaker |
Title |
|---|---|---|
| 5 October 2026 | Edison Au-Yeung (Warwick) | The elliptic sieve and ideal norms |
| 12 October 2026 | Jiewei Xiong (Reading) | A history of the Zilber–Pink conjecture |
| 19 October 2026 | Joseph Harrison (Warwick) | Effectivity in the semi-abelian Mordell conjecture |
| 26 October 2026 | Leonardo Franchi (Cambridge) | Product-free subsets of (0,1) and a strengthened Brunn-Minkowski inequality |
| 2 November 2026 | Owen Jones (Warwick) | Zero-one laws in Diophantine approximation and a mixed Gallagher's theorem |
| 9 November 2026 | Paweł Nosal (Warwick) | The size of the squarefree sum of a typical Legendre symbol is $o(\sqrt{x})$ |
| 16 November 2026 | Nathan Creighton (Oxford) | Shifted convolutions of divisor functions |
Abstracts
Week 1 - 5 October 2026
Edison Au-YeungLink opens in a new window (University of Warwick) - The elliptic sieve and ideal norms
A theorem of Landau and Ramanujan states that almost all integers are not sums of two squares (when ordered by their absolute value). Equivalently, almost all integers are not norms from the Gaussian rationals $\mathbb{Q}(i)$. A version of this norm result for elliptic curves $E$ was proved by Bhakta et al. in 2023 using an argument known as the elliptic sieve. In this talk, I will first give an overview of the elliptic sieve, followed by a discussion of a version of the proof in which the norm result applies to abelian extensions of imaginary number fields.
Week 2 - 12 October 2026
Jiewei XiongLink opens in a new window (University of Reading) - A history of the Zilber–Pink conjecture
The talk doesn't contain any new mathematical content, rather it's a genealogical survey of the Zilber–Pink conjecture (both its original forms and subsequent variants) and preceding related problems, with some partial results (due to other people). I will use the motif of finiteness in diophantine geometry as a guide but if time allows I will also tell the story from the Hodge theoretic point of view.
Week 3 - 19 October 2026
Joseph HarrisonLink opens in a new window (University of Warwick) - Effectivity in the semi-abelian Mordell conjecture
Let $G$ be a commutative algebraic group over $\mathbb{C}$, not containing a subgroup isomorphic to the additive group of complex numbers. Let $X$ be a closed subvariety of $G$ and $\Gamma$ a finitely generated subgroup of $G$. From the perspective of the group structure the set $X$ is unstructured, the notable exception being when $X$ is a translate of an algebraic subgroup. Similarly, $\Gamma$ is unstructured for the Zariski topology on $G$. A generalisation of the classical Mordell conjecture asks for a description of the Zariski closure of $X\cap\Gamma$. This was achieved for abelian varieties by Faltings and semi-abelian varieties by Vojta, but the proofs are ineffective. I will discuss an effective approach for certain $X$ and $\Gamma$ that does not make use of Diophantine approximation, linear forms in logarithms or the Chabauty—Skolem methods.
Week 4 - 26 October 2026
Leonardo FranchiLink opens in a new window (University of Cambridge) - Product-free subsets of (0,1) and a strengthened Brunn-Minkowski inequality
Week 5 - 2 November 2026
Owen JonesLink opens in a new window (University of Warwick) - Zero-one laws in Diophantine approximation and a mixed Gallagher's theorem
Motivated by the Borel–Cantelli lemmas in probability theory, zero-one laws have been ubiquitous in Diophantine approximation for over a century. Examples range from Khintchine's theorem in 1924 to the Duffin–Schaeffer conjecture, whose celebrated proof was published by Koukoulopoulos and Maynard in 2020. In this talk, we discuss a general method for proving results of this type which extends to inhomogeneous and higher dimensional settings. We then outline how it could provide an alternative proof of a recent result of Chow-Jones.
Week 6 - 9 November 2026
Paweł NosalLink opens in a new window (University of Warwick) - The size of the squarefree sum of a typical Legendre symbol is $o(\sqrt{x})$
In a recent work of Harper, he showed that for a large prime $r$ , the partial sum of length $x$ of a typical Dirichlet character modulo $r$ exhibits ''better than squareroot cancellation". His proof of this surprising statement follows a randomisation argument that relates averages of characters to those of random multiplicative functions. In this talk, I will outline some of the ideas from that proof and show how it can be adapted to study cancellation in squarefree partial sum of the Legendre symbol of a typical prime. This is based on upcoming work.
Week 7 - 16 November 2026
Nathan CreightonLink opens in a new window (University of Oxford) - Shifted convolutions of divisor functions
A classical problem in analytic number theory is to understand the shifted correlations $$\sum_{n\le x}d_k(n)d_\ell(n+h),$$ where $d_k$ denotes the $k$-fold divisor function and $h\neq 0$ is fixed. While asymptotic formulas have long been known when one of $k,\ell$ is at most $2$, the case $k,\ell\geq 3$ remained open.
In this talk, I will discuss a recent AI-assisted proof of Miguel Walsh, which establishes the expected leading asymptotic for all fixed $k,\ell$ and nonzero shifts $h$. I will outline the main ideas of the proof, which combines a density model for divisor functions with a novel application of weighted dilation graphs. In particular, I will explain how the graph machinery transforms the original fixed-shift correlation into averages over more tractable shifts involving independent products of primes, allowing the problem to be completed using Fourier-analytic methods.