Forthcoming events in this series


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.

Mon, 19 Feb 2024
16:00
L2

On entropy of arithmetic functions

Fei Wei
(University of Oxford)
Abstract

In this seminar, I will talk about a notion of entropy of arithmetic functions and some properties of this entropy.  This notion was introduced to study Sarnak's Moebius Disjointness Conjecture.

Mon, 12 Feb 2024
16:00
L2

Higher descent on elliptic curves

Sven Cats
(University of Cambridge)
Abstract

Let $E$ be an elliptic curve over a number field $K$ and $n \geq 2$ an integer. We recall that elements of the $n$-Selmer group of $E/K$ can be explicitly written in terms of certain equations for $n$-coverings of $E/K$. Writing the elements in this way is called conducting an explicit $n$-descent. One of the applications of explicit $n$-descent is in finding generators of large height for $E(K)$ and from this point of view one would like to be able to take $n$ as large as possible. General algorithms for explicit $n$-descent exist but become computationally challenging already for $n \geq 5$. In this talk we discuss combining $n$- and $(n+1)$-descents to $n(n+1)$-descent and the role that invariant theory plays in this procedure.

Mon, 05 Feb 2024
16:00
L2

TBC

TBC
(TBC)
Abstract

TBC

Mon, 29 Jan 2024
16:00
L2

Quantitative bounds for a weighted version of Chowla's conjecture

Cédric Pilatte
(University of Oxford)
Abstract

The Liouville function $\lambda(n)$ is defined to be $+1$ if $n$ is a product of an even number of primes, and $-1$ otherwise. The statistical behaviour of $\lambda$ is intimately connected to the distribution of prime numbers. In many aspects, the Liouville function is expected to behave like a random sequence of $+1$'s and $-1$'s. For example, the two-point Chowla conjecture predicts that the average of $\lambda(n)\lambda(n+1)$ over $n < x$ tends to zero as $x$ goes to infinity. In this talk, I will discuss quantitative bounds for a logarithmic version of this problem.

Mon, 22 Jan 2024
16:00
L2

Computing Tangent Spaces to Eigenvarieties

James Rawson
(University of Warwick)
Abstract

Many congruences between modular forms (or at least their q-expansions) can be explained by the theory of $p$-adic families of modular forms. In this talk, I will discuss properties of eigenvarieties, a geometric interpretation of the idea of $p$-adic families. In particular, focusing initially on the well-understood case of (elliptic) modular forms, before delving into the considerably murkier world of Bianchi modular forms. In this second case, this work gives numerical verification of a couple of conjectures, including BSD by work of Loeffler and Zerbes.

Mon, 15 Jan 2024
16:00
L2

A friendly introduction to Shimura curves

Håvard Damm-Johnsen
(University of Oxford)
Abstract

Modular curves play a key role in the Langlands programme, being the simplest example of so-called Shimura varieties.  Their less famous cousins, Shimura curves, are also very interesting, and very concrete. 
In this talk I will give a gentle introduction to the arithmetic of Shimura curves, with lots of explicit examples. Time permitting, I will say something about recent work about intersection numbers of geodesics on Shimura curves.

Mon, 27 Nov 2023
16:00
C1

On two variations of Mazur's deformation functor

Simon Alonso
(ENS de Lyon)
Abstract

In 1989, Mazur defined the deformation functor associated to a residual Galois representation, which played an important role in the proof by Wiles of the modularity theorem. This was used as a basis over which many mathematicians constructed variations both to further specify it or to expand the contexts where it can be applied. These variations proved to be powerful tools to obtain many strong theorems, in particular of modular nature. In this talk I will give an overview of the deformation theory of Galois representations and describe two variants of Mazur's functor that allow one to properly deform reducible residual representations (which is one of the shortcomings of Mazur's original functor). Namely, I will present the theory of determinant-laws initiated by Bellaïche-Chenevier on the one hand, and an idea developed by Calegari-Emerton on the other.
If time permits, I will also describe results that seem to indicate a possible comparison between the two seemingly unrelated constructions.

Mon, 20 Nov 2023
16:00
L1

Post-Quantum Cryptography (and why I’m in the NT corridor)

Patrick Hough
(University of Oxford)
Abstract

In this talk I will give a brief introduction to the field of post-quantum (PQ) cryptography, introducing a few of the most popular computational hardness assumptions. Second, I will give an overview of a recent work of mine on PQ electronic voting. I’ll finish by presenting a short selection of ‘exotic’ cryptographic constructions that I think are particularly hot at the moment (no, not blockchain). The talk will be definitionally light since I expect the area will be quite new to many and I hope this will make for a more engaging introduction.

Mon, 13 Nov 2023
16:00
C3

Modular generating series

Mads Christensen
(University College London)
Abstract

For many spaces of interest to number theorists one can construct cycles which in some ways behave like the coefficients of modular forms. The aim of this talk is to give an introduction to this idea by focusing on examples coming from modular curves and Heegner points and the relevant work of Zagier, Gross-Kohnen-Zagier and Borcherds. If time permits I will discuss generalizations to other spaces.

Mon, 06 Nov 2023
16:00
L1

A Basic Problem in Analytic Number Theory

George Robinson
(University of Oxford)
Abstract

I will discuss a basic problem in analytic number theory which has appeared recently in my work. This will be a gentle introduction to the Gauss circle problem, hopefully with a discussion of some extensions and applications to understanding L-functions.

Mon, 30 Oct 2023
16:00
C2

Hodge theory in positive characteristic

Inés Borchers Arias
(University of Oxford)
Abstract

I will introduce the Hodge-de-Rham spectral sequence and formulate an algebraic Hodge decomposition theorem. Time permitting, I will sketch Deligne and Illusie’s proof of the Hodge decomposition using positive characteristic methods.

Mon, 16 Oct 2023
16:00
C3

Avoiding Problems

Francesco Ballini
(University of Oxford )
Abstract

In 2019 Masser and Zannier proved that "most" abelian varieties over the algebraic numbers are not isogenous to the jacobian of any curve (where "most" refers to an ordering by some suitable height function). We will see how this result fits in the general Zilber-Pink Conjecture picture and we discuss some (rather concrete) analogous problems in a power of the modular curve Y(1).

Mon, 09 Oct 2023
16:00
C3

Primes in arithmetic progressions to smooth moduli

Julia Stadlmann
(University of Oxford)
Abstract

The twin prime conjecture asserts that there are infinitely many primes p for which p+2 is also prime. This conjecture appears far out of reach of current mathematical techniques. However, in 2013 Zhang achieved a breakthrough, showing that there exists some positive integer h for which p and p+h are both prime infinitely often. Equidistribution estimates for primes in arithmetic progressions to smooth moduli were a key ingredient of his work. In this talk, I will sketch what role these estimates play in proofs of bounded gaps between primes. I will also show how a refinement of the q-van der Corput method can be used to improve on equidistribution estimates of the Polymath project for primes in APs to smooth moduli.

Mon, 12 Jun 2023
16:00
C3

Probabilistic aspects of the Riemann zeta function

Khalid Younis
(University of Warwick)
Abstract

A central topic of study in analytic number theory is the behaviour of the Riemann zeta function. Many theorems and conjectures in this area are closely connected to concepts from probability theory. In this talk, we will discuss several results on the typical size of the zeta function on the critical line, over different scales. Along the way, we will see the role that is played by some probabilistic phenomena, such as the central limit theorem and multiplicative chaos.

Mon, 05 Jun 2023
16:00
C3

On Sarnak's Moebius Disjointness Conjecture

Fei Wei
(University of Oxford)
Abstract

It is known that there exists certain randomness in the values of the Moebius function. It is widely believed that this randomness predicts significant cancellations in the summation of the Moebius function times any 'reasonable' sequence. This rather vague principle is known as an instance of the 'Moebius randomness principle'. Sarnak made this principle precise by identifying the notion 'reasonable' as deterministic. More precisely, Sarnak's Moebius Disjointness Conjecture predicts the disjointness of the Moebius function from any arithmetic functions realized in any topological dynamical systems of zero topological entropy. In this talk, I will firstly introduce some background and progress on this conjecture. Secondly, I will talk about some of my work on this. Thirdly, I will talk some related problems to this conjecture.

Mon, 29 May 2023
16:00
C3

TBD

Mon, 22 May 2023
16:00
C3

The modular approach for solving $x^r+y^r=z^p$ over totally real number fields

Diana Mocanu
(University of Warwick)
Abstract

We will first introduce the modular method for solving Diophantine Equations, famously used to
prove the Fermat Last Theorem. Then, we will see how to generalize it for a totally real number field $K$ and
a Fermat-type equation $Aa^p+Bb^q=Cc^r$ over $K$. We call the triple of exponents $(p,q,r)$ the 
signature of the equation. We will see various results concerning the solutions to the Fermat equation with
signatures $(r,r,p)$ (fixed $r$). This will involve image of inertia comparison and the study of certain
$S$-unit equations over $K$. If time permits, we will discuss briefly how to attack the very similar family
of signatures $(p,p,2)$ and $(p,p,3)$. 

Mon, 15 May 2023
16:00
C3

Ranges of polynomials control degree ranks of Green and Tao over finite prime fields

Thomas Karam
(University of Oxford)
Abstract

Let $p$ be a prime, let $1 \le t < d < p$ be integers, and let $S$ be a non-empty subset of $\mathbb{F}_p$ (which may be thought of as being $\{0,1\}$). We will establish that if a polynomial $P:\mathbb{F}_p^n \to \mathbb{F}_p$ with degree $d$ is such that the image $P(S^n)$ does not contain the full image $A(\mathbb{F}_p)$ of any non-constant polynomial $A: \mathbb{F}_p \to \mathbb{F}_p$ with degree at most $t$, then $P$ coincides on $S^n$ with a polynomial $Q$ that in particular has bounded degree-$\lfloor d/(t+1) \rfloor$-rank in the sense of Green and Tao, and has degree at most $d$. Likewise, we will prove that if the assumption holds even for $t=d$ then $P$ coincides on $S^n$ with a polynomial determined by a bounded number of coordinates and with degree at most $d$.

Mon, 08 May 2023
16:00

TBD

TBD
Mon, 01 May 2023
16:00
C3

Combinatorics goes perverse: An Erdős problem on additive Sidon bases

Cédric Pilatte
Abstract

In 1993, Erdős, Sárközy and Sós posed the question of whether there exists a set $S$ of positive integers that is both a Sidon set and an asymptotic basis of order $3$. This means that the sums of two elements of $S$ are all distinct, while the sums of three elements of $S$ cover all sufficiently large integers. In this talk, I will present a construction of such a set, building on ideas of Ruzsa and Cilleruelo. The proof uses a powerful number-theoretic result of Sawin, which is established using cutting-edge algebraic geometry techniques.

Mon, 24 Apr 2023
16:00
C3

The weight part of Serre's conjecture

Martin Ortiz
(UCL (LSGNT))
Abstract

Serre's conjecture (now a theorem) predicts that an irreducible 2-dimensional odd
Galois representation of $\mathbb Q$ with coefficients in $\bar{\mathbb F}_p$ comes from the mod p reduction of
a modular form. A key feature is that two modular forms of different weights can have the same
mod p reduction. Fixing a modular form $f$, the weight part of Serre's conjecture seeks to find all
the possible weights where one can find a modular form congruent to $f$ mod $p$. The recipe for these
weights was conjectured by Serre, and it depends only on the local Galois representation at $p$. I
will explain the ideas involved in Edixhoven's proof of the weight part, and if time allows, I
will briefly say something about what the generalizations beyond $\operatorname{GL}_2/\mathbb Q$ might look like. 

Mon, 06 Mar 2023
16:00

TBD