Forthcoming events in this series


Mon, 07 Feb 2022

16:00 - 17:00
C2

TBA

Mon, 31 Jan 2022

16:00 - 17:00
L5

The Probabilistic Zeta Function of a Finite Lattice

Besfort Shala
Abstract

In this talk, we present our study of Brown’s definition of the probabilistic zeta function of a finite lattice, and propose a natural alternative that may be better-suited for non-atomistic lattices. The probabilistic zeta function admits a general Dirichlet series expression, which need not be ordinary. We investigate properties of the function and compute it on several examples of finite lattices, establishing connections with well-known identities. Furthermore, we investigate when the series is an ordinary Dirichlet series. Since this is the case for coset lattices, we call such lattices coset-like. In this regard, we focus on partition lattices and d-divisible partition lattices and show that they typically fail to be coset-like. We do this by using the prime number theorem, establishing a connection with number theory.

Mon, 24 Jan 2022

16:00 - 17:00
C2

TBA

Yifan Jing
Mon, 17 Jan 2022

16:00 - 17:00
C4

Classical Mechanics and Diophantine Equations

Jay Swar
Abstract

We'll sketch how the $K$-rational solutions of a system $X$ of multivariate polynomials can be viewed as the solutions of a "classical mechanics" problem on an associated affine space.

When $X$ has a suitable topology, e.g. if its $\mathbb{C}$-solutions form a Riemann surface of genus $>1$, we'll observe some of the advantages of this new point of view such as a relatively computable algorithm for effective finiteness (with some stipulations). This is joint work with Minhyong Kim.
 

Mon, 29 Nov 2021

16:00 - 17:00
C1

Convex Functions and Additive Structure

Peter Bradshaw
Abstract

It is a widely accepted philosophy in additive number theory that convex sets ought not to exhibit much additive structure. We could measure this by estimating the sizes of their sumsets. In this talk, we will hopefully move from the philosophical to the concrete, by giving ways to see that convex sets and functions have poor additive structure. We will also discuss some recent developments in the area.

Mon, 22 Nov 2021

16:00 - 17:00
L6

A Smörgåsbord of Number Theory (pre-PhDs Encouraged!)

George Robinson, Nadav Gropper, Michael Curran, Ofir Gorodetsky
Abstract

The speakers will be giving short presentations introducing topics in algebraic number theory, arithmetic topology, random matrix theory, and analytic number theory.

Undergrads and Master's students are encouraged to come and sample a taste of research in these areas.

 

Mon, 15 Nov 2021

16:00 - 17:00
C1

Polynomial Pell equation

Nikoleta Kalaydzhieva
Abstract

In a world of polynomial Pell’s equations, where the integers are replaced by polynomials with complex coefficients, and its smallest solution is used to generate all other solutions $(u_{n},v_{n})$, $n\in\mathbb{Z}$. One junior number theory group will embark on a journey in search of the properties of the factors of $v_{n}(t)$. There will be Galois extensions, there will be estimations and of course there will be loglogs.

Mon, 08 Nov 2021

16:00 - 17:00
C1

TBA

George Robinson
(Oxford)
Abstract

The Jacquet-Langlands correspondence gives a relationship between automorphic representations on $GL_2$ and its twisted forms, which are the unit groups of quaternion algebras. Writing this out in more classical language gives a combinatorial way of producing the eigenvalues of Hecke operators acting on modular forms. In this talk, we will first go over notions of modular forms and quaternion algebras, and then dive into an explicit example by computing some eigenvalues of the lowest level quaternionic modular form of weight $2$ over $\mathbb{Q}$.

Mon, 01 Nov 2021

16:00 - 17:00
C1

Convexity and squares in additive combinatorics

Akshat Mudgal
Abstract

A nice collection of problems in additive combinatorics focus on analysing solutions to additive equations over sequences that exhibit some flavour of convexity. This, for instance, includes genuine convex sequences as well as images of arbitrary sets under convex functions. In this talk, I will survey some of the literature surrounding these type of questions, along with some motivation from analytic number theory as well as the current best known results towards these problems.

Mon, 25 Oct 2021

16:00 - 17:00
C2

Hyperelliptic continued fractions

Francesco Ballini
(Oxford)
Abstract

We can define a continued fraction for formal series $f(t)=\sum_{i=-\infty}^d c_it^i$ by repeatedly removing the polynomial part, $\sum_{i=0}^d c_it^i$, (the equivalent of the integer part) and inverting the remaining part, as in the real case. This way, the partial quotients are polynomials. Both the usual continued fractions and the polynomial continued fractions carry properties of best approximation. However, while for square roots of rationals the real continued fraction is eventually periodic, such periodicity does not always occur for $\sqrt{D(t)}$. The correct analogy was found by Abel in 1826: the continued fraction of $\sqrt{D(t)}$ is eventually periodic if and only if there exist nontrivial polynomials $x(t)$, $y(t)$ such that $x(t)^2-D(t)y(t)^2=1$ (the polynomial Pell equation). Notice that the same holds also for square root of integers in the real case. In 2014 Zannier found that some periodicity survives for all the $\sqrt{D(t)}$: the degrees of their partial quotients are eventually periodic. His proof is strongly geometric and it is based on the study of the Jacobian of the curve $u^2=D(t)$. We give a brief survey of the theory of polynomial continued fractions, Jacobians and an account of the proof of the result of Zannier.

Mon, 18 Oct 2021

16:00 - 17:00
C1
Mon, 11 Oct 2021

16:00 - 17:00
C1

Computing p-adic L-functions of Hecke characters

Håvard Damm-Johnsen
(Oxford)
Abstract

In 1973, Serre defined $p$-adic modular forms as limits of modular forms, and constructed the Leopoldt-Kubota $L$-function as the constant term of a limit of Eisenstein series. This was extended by Deligne-Ribet to totally real number fields, and Lauder and Vonk have developed an algorithm for interpolating $p$-adic $L$-functions of such fields using Serre's idea. We explain what an $L$-function is and why you should care, and then move on to giving an overview of the algorithm, extensions, and applications.

Mon, 21 Jun 2021

16:00 - 17:00
Virtual

Correlations of almost primes

Natalie Evans
(KCL)
Abstract

The Hardy-Littlewood generalised twin prime conjecture states an asymptotic formula for the number of primes $p\le X$ such that $p+h$ is prime for any non-zero even integer $h$. While this conjecture remains wide open, Matom\"{a}ki, Radziwi{\l}{\l} and Tao proved that it holds on average over $h$, improving on a previous result of Mikawa. In this talk we will discuss an almost prime analogue of the Hardy-Littlewood conjecture for which we can go beyond what is known for primes. We will describe some recent work in which we prove an asymptotic formula for the number of almost primes $n=p_1p_2 \le X$ such that $n+h$ has exactly two prime factors which holds for a very short average over $h$.

Mon, 14 Jun 2021

17:30 - 18:30
Virtual

TBA

Mon, 07 Jun 2021

16:00 - 17:00
Virtual

Inverse Galois Theory as Thor's Hammer

Catherine Ray
Abstract

The action of the automorphisms of a formal group on its deformation space is crucial to understanding periodic families in the homotopy groups of spheres and the unsolved Hecke orbit conjecture for unitary Shimura varieties. We can explicitly pin down this squirming action by geometrically modelling it as coming from an action on a moduli space, which we construct using inverse Galois theory and some representation theory (a Hurwitz space). I will show you pretty pictures.

Mon, 31 May 2021

16:00 - 17:00
Virtual

Critical exponents for primitive sets

Jared Duker Lichtman
(Oxford)
Abstract

A set of positive integers is primitive (or 1-primitive) if no member divides another. Erdős proved in 1935 that the weighted sum $\sum 1/(n\log n)$ for n ranging over a primitive set A is universally bounded over all choices for A. In 1988 he asked if this universal bound is attained by the set of prime numbers. One source of difficulty in this conjecture is that $\sum n^{-\lambda}$ over a primitive set is maximized by the primes if and only if $\lambda$ is at least the critical exponent $\tau_1\approx1.14$.
A set is $k$-primitive if no member divides any product of up to $k$ other distinct members. In joint work with C. Pomerance and T.H. Chan, we study the critical exponent $\tau_k$ for which the primes are maximal among $k$-primitive sets. In particular we prove that $\tau_2<0.8$, which directly implies the Erdős conjecture for 2-primitive sets.

Mon, 24 May 2021

16:00 - 17:00
Virtual

Block graded relations among motivic multiple zeta values

Adam Keilthy
(Max-Planck-Institut für Mathematik)
Abstract

Multiple zeta values, originally considered by Euler, generalise the Riemann zeta function to multiple variables. While values of the Riemann zeta function at odd positive integers are conjectured to be algebraically independent, multiple zeta values satisfy many algebraic and linear relations, even forming a Q-algebra. While families of well understood relations are known, such as the associator relations and double shuffle relations, they only conjecturally span all algebraic relations. As multiple zeta values arise as the periods of mixed Tate motives, we obtain further algebraic structures, which have been exploited to provide spanning sets by Brown. In this talk we will aim to define a new set of relations, known to be complete in low block degree.

To achieve this, we will first review the necessary algebraic set up, focusing particularly on the motivic Lie algebra associated to the thrice punctured projective line. We then introduce a new filtration on the algebra of (motivic) multiple zeta values, called the block filtration, based on the work of Charlton. By considering the associated graded algebra, we quickly obtain a new family of graded motivic relations, which can be shown to span all algebraic relations in low block degree. We will also touch on some conjectural ungraded `lifts' of these relations, and if we have time, compare to similar approaches using the depth filtration.

Mon, 17 May 2021

16:00 - 17:00
Virtual

Distributions of Character Sums

Ayesha Hussain
(Bristol)
Abstract

Over the past few decades, there has been a lot of interest in partial sums of Dirichlet characters. Montgomery and Vaughan showed that these character sums remain a constant size on average and, as a result, a lot of work has been done on the distribution of the maximum. In this talk, we will investigate the distribution of these character sums themselves, with the main goal being to describe the limiting distribution as the prime modulus approaches infinity. This is motivated by Kowalski and Sawin’s work on Kloosterman paths.
 

Mon, 10 May 2021

16:00 - 17:00
Virtual

An asymptotic expansion for the counting function of semiprimes

Dragos Crisan
(Oxford)
Abstract

A semiprime is a natural number which can be written as the product of two primes. Using elementary methods, we'll explore an asymptotic expansion for the counting function of semiprimes $\pi_2(x)$, which generalises previous findings of Landau, Delange and Tenenbaum.  We'll also obtain an efficient way of computing the constants involved. In the end, we'll look towards possible generalisations for products of $k$ primes.

Mon, 03 May 2021

16:00 - 17:00

On maximal product sets of random sets

Daniele Mastrostefano
Abstract

For every positive integer N and every α ∈ [0,1), let B(N, α) denote the probabilistic model in which a random set A of (1,...,N) is constructed by choosing independently every element of (1,...,N) with probability α. We prove that, as N → +∞, for every A in B(N, α) we have |AA| ~ |A|^2/2 with probability 1-o(1), if and only if (log(α^2(log N)^{log 4-1}))(√loglog N) → ∞. This improves on a theorem of Cilleruelo, Ramana and Ramar\'e, who proved the above asymptotic between |AA| and |A|^2/2 when α =o(1/√log N), and supplies a complete characterization of maximal product sets of random sets.

Mon, 26 Apr 2021

16:00 - 17:00
Virtual

Motivic representations and finite rational points

Jay Swar
(Oxford)
Abstract

I will briefly introduce the Chabauty-Kim argument for effective finiteness results on "topologically rich enough" curves. I will then introduce the Fontaine-Mazur conjecture and show how it provides an effective proof of Faltings' Theorem.

In the case of non-CM elliptic curves minus a point, following work of Federico Amadio Guidi, I'll show how the relevant input for effective finiteness is provided by the vanishing of adjoint Selmer groups proven by Newton and Thorne.

Mon, 29 Mar 2021

16:00 - 17:00
Virtual

Intro to Lawrence-Venkatesh's proof of Mordell-Faltings

Jay Swar
Abstract

This talk will be the first in a spin-off series on the Lawrence-Venkatesh approach to showing that every hyperbolic curve$/K$ has finitely many $K$-points. In this talk, we will give the overall outline of the approach and prove several of  the preliminary results, such as Faltings' finiteness theorem for semisimple Galois representations.

Mon, 08 Mar 2021

16:00 - 17:00
Virtual

Chen's theorem

Julia Stadlmann
(Oxford)
Abstract

In 1966 Chen Jingrun showed that every large even integer can be written as the sum of two primes or the sum of a prime and a semiprime. To date, this weakened version of Goldbach's conjecture is one of the most remarkable results of sieve theory. I will talk about the big ideas which paved the way to this proof and the ingenious trick which led to Chen's success. No prior knowledge of sieve theory required – all necessary techniques will be introduced in the talk.

Mon, 01 Mar 2021

16:00 - 17:00
Virtual

Diophantine problems over local fields (and their extensions)

Konstantinos Kartas
Abstract

We will discuss the problem of deciding (algorithmically) whether a variety over a local field K has a K-rational point, surveying some known results. I will then allow K to be an infinite extension (of some arithmetic interest) of a local field and present some recent work.
 

Mon, 22 Feb 2021

16:00 - 17:00
Virtual

Wild Galois Representations

Nirvana Coppola
(Bristol)
Abstract

Let C be an elliptic or hyperelliptic curve over a p-adic field K. Then C is equipped with a Galois representation, given by the action of the absolute Galois group of K on the Tate module of C. The behaviour of this representation depends on the reduction type of C. We will focus on the case of C having bad reduction, and acquiring potentially good reduction over a wildly ramified extension of K. We will show that, if C is an elliptic curve, the Galois representation can be completely determined in this case, thus allowing one to fully classify Galois representations attached to elliptic curves. Furthermore, the same can be done for a special family of hyperelliptic curves, obtaining a result which is surprisingly similar to that for the corresponding elliptic curves case.