Forthcoming events in this series


Mon, 10 Mar 2025
16:00
C4

Sums of integers divisible by the sum of their digits

Kate Thomas
(University of Oxford)
Abstract

A base-g Niven number is an integer divisible by the sum of its digits in base-g. We show that any sufficiently large integer can be written as the sum of three base-3 Niven numbers, and comment on the extension to other bases. This is an application of the circle method, which we use to count the number of ways an integer can be written as the sum of three integers with fixed, near-average, digit sum. 

Mon, 03 Mar 2025
16:00
C6

From the classical to the $\mathrm{GL}_m$ large sieve

Alexandru Pascadi
(University of Oxford)
Abstract

The large sieve inequality for Dirichlet characters is a central result in analytic number theory, which encodes a strong orthogonality property between primitive characters of varying conductors. This can be viewed as a statement about $\mathrm{GL}_1$ automorphic representations, and it is a key open problem to prove similar results in the higher $\mathrm{GL}_m$ setting; for $m \ge 2$, our best bounds are far from optimal. We'll outline two approaches to such results (sketching them first in the elementary case of Dirichlet characters), and discuss work-in-progress of Thorner and the author on an improved $\mathrm{GL}_m$ large sieve. No prior knowledge of automorphic representations will be assumed.

Mon, 24 Feb 2025
16:00
C4

Modularity of certain trianguline Galois representations

James Kiln
(Queen Mary University of London)
Abstract

A generalisation of Wiles’ famous modularity theorem, the Fontaine-Mazur conjecture, predicts that two dimensional representations of the absolute Galois group of the rationals, with a few specific properties, exactly correspond to those representations coming from classical modular forms. Under some mild hypotheses, this is now a theorem of Kisin. In this talk, I will explain how one can p-adically interpolate the objects on both sides of this correspondence to construct an eigensurface and “trianguline” Galois deformation space, as well as outline a new approach to proving a theorem of Emerton, that these spaces are often isomorphic.

Mon, 17 Feb 2025
16:00
C6

TBC

Jori Merikowski
(University of Oxford)
Abstract

TBC

Mon, 17 Feb 2025
16:00
C6

Hoheisel's theorem on primes in short intervals via combinatorics

Jori Merikoski
(Oxford)
Abstract

Hoheisel's theorem states that there is some $\delta> 0$ and some $x_0>0$ such that for all $x > x_0$ the interval $[x,x+x^{1-\delta}]$ contains prime numbers. Classically this is proved using the Riemann zeta function and results about its zeros such as the zero-free region and zero density estimates. In this talk I will describe a new elementary proof of Hoheisel's theorem. This is joint work with Kaisa Matomäki (Turku) and Joni Teräväinen (Cambridge). Instead of the zeta function, our approach is based on sieve methods and ideas coming from additive combinatorics, in particular, the transference principle. The method also gives an L-function free proof of Linnik's theorem on the least prime in arithmetic progressions.

Mon, 10 Feb 2025
16:00
C4

A new axiom for $\mathbb{Q}_p^{ab}$ and non-standard methods for perfectoid fields

Leo Gitin
(University of Oxford)
Abstract

The class of henselian valued fields with non-discrete value group is not well-understood. In 2018, Koenigsmann conjectured that a list of seven natural axioms describes a complete axiomatisation of $\mathbb{Q}_p^{ab}$, the maximal extension of the $p$-adic numbers $\mathbb{Q}_p$ with abelian Galois group, which is an example of such a valued field. Informed by the recent work of Jahnke-Kartas on the model theory of perfectoid fields, we formulate an eighth axiom (the discriminant property) that is not a consequence of the other seven. Revisiting work by Koenigsmann (the Galois characterisation of $\mathbb{Q}_p$) and Jahnke-Kartas, we give a uniform treatment of their underlying method. In particular, we highlight how this method yields short, non-standard model-theoretic proofs of known results (e.g. finite extensions of perfectoid fields are perfectoid).

Mon, 03 Feb 2025
16:00
C6

Progress towards the Keating-Snaith conjecture for quadratic twists of elliptic curves

Nathan Creighton
(University of Oxford)
Abstract

The Keating-Snaith conjecture for quadratic twists of elliptic curves predicts the central values should have a log-normal distribution. I present recent progress towards establishing this in the range of large deviations of order of the variance. This extends Selberg’s Central Limit Theorem from ranges of order of the standard deviation to ranges of order of the variance in a variety of contexts, inspired by random walk theory. It is inspired by recent work on large deviations of the zeta function and central values of L-functions.
 

Mon, 27 Jan 2025
16:00
C4

Applied analytic number theory

Cédric Pilatte
(University of Oxford)
Abstract

The security of many widely used communication systems hinges on the presumed difficulty of factoring integers or computing discrete logarithms. However, Shor's celebrated algorithm from 1994 demonstrated that quantum computers can perform these tasks in polynomial time. In 2023, Regev proposed an even faster quantum algorithm for factoring integers. Unfortunately, the correctness of his new method is conditional on an ad hoc number-theoretic conjecture. Using tools from analytic number theory, we establish a result in the direction of Regev's conjecture. This enables us to design a provably correct quantum algorithm for factoring and solving the discrete logarithm problem, whose efficiency is comparable to Regev's approach. In this talk, we will give an accessible account of these developments.

Mon, 02 Dec 2024
16:00
C3

TBC

Leo Gitin
(University of Oxford)
Abstract

TBC

Mon, 25 Nov 2024
16:00
C3

Gap distributions and the Metric Poissonian Property 

Sophie Maclean
(King's College London)
Abstract
When studying dilated arithmetic sequences, it is natural to wonder about their distribution. Whilst it is relatively achievable to ascertain whether the resulting sequence is equidistributed, is it much more difficult to say much about gap size between consecutive elements of the new set? In this talk I will explore the gap distributions in dilated arithmetic sequences modulo 1, including what it means for a sequence to have the metric poissonian property. I will also give an overview of the current progress and what I am aiming to discover in my own work.
 
 
Mon, 18 Nov 2024
16:00
C3

Heegner points and Euler systems

Andrew Graham
(University of Oxford)
Abstract

Heegner points are a powerful tool for understanding the structure of the group of rational points on elliptic curves. In this talk, I will describe these points and the ideas surrounding their generalisation to other situations.

Mon, 04 Nov 2024
16:00
C3

Approximating Primes

Lasse Grimmelt
(University of Oxford)
Abstract

A successful strategy to handle problems involving primes is to approximate them by a more 'simple' function. Two aspects need to be balanced. On the one hand, the approximant should be simple enough so that the considered problem can be solved for it. On the other hand, it needs to be close enough to the primes in order to make it an admissible to replacement. In this talk I will present how one can construct general approximants in the context of the Circle Method and will use this to give a different perspective on Goldbach type applications.

Mon, 28 Oct 2024
16:00
C3

An introduction to modularity lifting

Dmitri Whitmore
(University of Cambridge)
Abstract
The (global) Langlands programme is a vast generalization of classical reciprocity laws. Roughly, it predicts a correspondence between:
1) modular forms (and their generalizations, automorphic forms)
2) representations of the Galois group of a number field.
While many constructions of Galois representations from automorphic forms exist, the converse direction is often harder to establish. The main tools to do so are modularity lifting theorems and are proved via the Taylor-Wiles method, originating from Wiles' proof of Fermat's Last Theorem.
 
I will introduce these ideas and their applications, focusing particularly on the problem of modularity of elliptic curves. I will then briefly discuss a generalization of the Taylor-Wiles method developed in my thesis which led to new modularity theorems in the setting of quadratic extensions of totally real fields by building of work of Boxer-Calegari-Gee-Pilloni.
Mon, 21 Oct 2024
16:00
C3

Monochromatic non-commuting products

Matt Bowen
(University of Oxford)
Abstract

We show that any finite coloring of an amenable group contains 'many' monochromatic sets of the form $\{x,y,xy,yx\},$ and natural extensions with more variables.  This gives the first combinatorial proof and extensions of Bergelson and McCutcheon's non-commutative Schur theorem.  Our main new tool is the introduction of what we call `quasirandom colorings,' a condition that is automatically satisfied by colorings of quasirandom groups, and a reduction to this case.

Mon, 14 Oct 2024
16:00
C3

Self-Similar Sets and Self-Similar Measures

Constantin Kogler
(University of Oxford)
Abstract

We give a gentle introduction to the theory of self-similar sets and self-similar measures. Connections of this topic to Diophantine approximation on Lie groups as well as to additive combinatorics will be exposed. In particular, we will discuss recent progress on Bernoulli convolutions. If time permits, we mention recent joint work with Samuel Kittle on absolutely continuous self-similar measures. 
 

Mon, 10 Jun 2024
16:00
L2

Duffin-Schaeffer meets Littlewood - a talk on metric Diophantine approximation

Manuel Hauke
(University of York)
Abstract

Khintchine's Theorem is one of the cornerstones in metric Diophantine approximation. The question of removing the monotonicity condition on the approximation function in Khintchine's Theorem led to the recently proved Duffin-Schaeffer conjecture. Gallagher showed an analogue of Khintchine's Theorem for multiplicative Diophantine approximation, again assuming monotonicity. In this talk, I will discuss my joint work with L. Frühwirth about a Duffin-Schaeffer version for Gallagher's Theorem. Furthermore, I will give a broader overview on various questions in metric Diophantine approximation and demonstrate the deep connection to both analytic and combinatorial number theory that is hidden inside the proof of these statements.

Mon, 03 Jun 2024
16:00
L2

Upper bounds on large deviations of Dirichlet L-functions in the Q-aspect

Nathan Creighton
(University of Oxford)
Abstract

Congruent numbers are natural numbers which are the area of right angled triangles with all rational sides. This talk will investigate conjectures for the density of congruent numbers up to some value $X$. One can phrase the question of whether a natural number is congruent in terms of whether an elliptic curve has non−zero rank. A theorem of Coates and Wiles connects this to whether the $L$-function associated to this elliptic curve vanishes at $1$. We will mention the conjecture of Keating on the asymptotic density based on random matrix considerations, and prove Tunnell’s Theorem, which connects the question of whether a natural number is a congruent number to counting integral points on varieties. Finally, I will hint at some future work I hope to do on non-vanishing of the $L$-functions.

Mon, 27 May 2024
16:00
L2

Special values of L-functions

Aleksander Horawa
(University of Oxford)
Abstract

In 1735, Euler observed that $ζ(2) = 1 + \frac{1}{2²} + \frac{1}{3²} + ⋯ = \frac{π²}{6}$. This is related to the famous identity $ζ(−1) "=" 1 + 2 + 3 + ⋯ "=" \frac{−1}{12}$. In general, values of the Riemann zeta function at positive even integers are equal to rational numbers multiplied by a power of $π$. The values at positive odd integers are much more mysterious; for example, Apéry proved that $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ⋯$ is irrational, but we still don't know if $ζ(5) = 1 + \frac{1}{2⁵} + \frac{1}{3⁵} + ⋯$ is rational or not! In this talk, we will explain the arithmetic significance of these values, their generalizations to Dirichlet/Dedekind L−functions, and to L−functions of elliptic curves. We will also present a new formula for $ζ(3) = 1 + \frac{1}{2³} + \frac{1}{3³} + ...$ in terms of higher algebraic cycles which came out of an ongoing project with Lambert A'Campo.

Mon, 20 May 2024
16:00
L2

Inhomogeneous multiplicative diophantine approximation

Kate Thomas
(University of Oxford)
Abstract

Introducing an inhomogeneous shift allows for generalisations of many multiplicative results in diophantine approximation. In this talk, we discuss an inhomogeneous version of Gallagher's theorem, established by Chow and Technau, which describes the rates for which we can approximate a typical product of fractional parts. We will sketch the methods used to prove an earlier version of this result due to Chow, using continued fraction expansions and geometry of numbers to analyse the structure of Bohr sets and bound sums of reciprocals of fractional parts.

Mon, 13 May 2024
16:00
L2

Eigenvarieties and p-adic propagation of automorphy

Zachary Feng
(University of Oxford)
Abstract

Functoriality is a key feature in Langlands’ conjectured relationship between automorphic representations and Galois representations; it predicts that certain Galois representations are automorphic, i.e. should come from automorphic representations. We discuss the idea of $p$-adic propagation of automorphy, which seeks to establish the automorphy of everything in a “neighborhood” given the automorphy of something in that neighborhood. The “neighborhoods” that we consider will be the irreducible components of a $p$-adic analytic space called the eigenvariety, which parameterizes $p$-adic automorphic representations. This technique was introduced by Newton and Thorne in their proof of symmetric power functoriality, and can be adapted to investigate similar problems.

Mon, 06 May 2024
16:00
L2

On twisted modular curves

Franciszek Knyszewski
(University of Oxford)
Abstract

Modular curves are moduli spaces of elliptic curves equipped with certain level structures. This talk will be concerned with how the attendant theory has been used to answer questions about the modularity of elliptic curves over $\mathbb{Q}$ and over quadratic fields. In particular, we will outline two instances of the modularity switching technique over totally real fields: the 3-5 trick of Wiles and the 3-7 trick of Freitas, Le Hung and Siksek. The recent work of Caraiani and Newton over imaginary quadratic fields naturally leads one to consider the descent theory of 'twisted' modular curves, and this will be the focus of the final part of the talk.

Mon, 29 Apr 2024
16:00
L2

New Lower Bounds For Cap Sets

Fred Tyrrell
(University of Bristol)
Abstract

A cap set is a subset of $\mathbb{F}_3^n$ with no solutions to $x + y + z = 0$ other than when $x = y = z$, or equivalently no non-trivial $3$-term arithmetic progressions. The cap set problem asks how large a cap set can be, and is an important problem in additive combinatorics and combinatorial number theory. In this talk, I will introduce the problem, give some background and motivation, and describe how I was able to provide the first progress in 20 years on the lower bound for the size of a maximal cap set. Building on a construction of Edel, we use improved computational methods and new theoretical ideas to show that, for large enough $n$, there is always a cap set in $\mathbb{F}_3^n$ of size at least $2.218^n$. I will then also discuss recent developments, including an extension of this result by Google DeepMind.

Mon, 22 Apr 2024
16:00
L2

On Unique Sums in Abelian Groups

Benjamin Bedert
(University of Oxford)
Abstract

In this talk, we will study the problem in additive combinatorics of determining for a finite Abelian group $G$ the size of its smallest subset $A\subset G$ that has no unique sum, meaning that for every two $a_1,a_2\in A$ we can write $a_1+a_2=a’_1+a’_2$ for different $a’_1,a’_2\in A$. We begin by using classical rectification methods to obtain the previous best lower bounds of the form $|A|\gg \log p(G)$, which stood for 50 years. Our main aim is to outline the proof of a recent improvement and discuss some of its key notions such as additive dimension and the density increment method. This talk is based on Bedert, B. On Unique Sums in Abelian Groups. Combinatorica (2023).

Mon, 04 Mar 2024
16:00
L2

The dispersion method and beyond: from primes to exceptional Maass forms

Alexandru Pascadi
(University of Oxford)
Abstract
The dispersion method has found an impressive number of applications in analytic number theory, from bounded gaps between primes to the greatest prime factors of quadratic polynomials. The method requires bounding certain exponential sums, using deep inputs from algebraic geometry, the spectral theory of GL2 automorphic forms, and GLn automorphic L-functions. We'll give a broad outline of this process, which combines various types of number theory; time permitting, we'll also discuss the key ideas behind some new results.
 
Mon, 26 Feb 2024
16:00
L2

The Metaplectic Representation is Faithful

Christopher Chang, Simeon Hellsten, Mario Marcos Losada, and Sergiu Novac.
(University of Oxford)
Abstract

Iwasawa algebras are completed group rings that arise in number theory, so there is interest in understanding their prime ideals. For some special Iwasawa algebras, it is conjectured that every non-zero such ideal has finite codimension and in order to show this it is enough to establish the faithfulness of the modules arising from the completion of highest weight modules. In this talk we will look at methods for doing this and apply them to the specific case of the metaplectic representation for the symplectic group.