Forthcoming events in this series


Thu, 05 May 2016
16:00
L6

Eigenvarieties for non-cuspidal Siegel modular forms

Giovanni Rosso
(University of Cambridge)
Abstract

In a recent work Andreata, Iovita, and Pilloni constructed the eigenvariety for cuspidal Siegel modular forms. This eigenvariety has the expected dimension (the genus of the Siegel forms) but it parametrizes only cuspidal forms. We explain how to generalize the construction to the non-cuspidal case. To be precise, we introduce the notion of "degree of cuspidality" and we construct an eigenvariety that parametrizes forms of a given degree of cuspidability. The dimension of these eigenvarieties depends on the degree of cuspidality we want to consider: the more non-cuspidal the forms, the smaller the dimension. This is a joint work with Riccardo Brasca.

Thu, 28 Apr 2016
16:00
L6

From Sturm, Sylvester, Witt and Wall to the present day

Andrew Ranicki
(University of Edinburgh)
Abstract

The talk will be based on some of the material in the joint survey with Etienne Ghys

"Signatures in algebra, topology and dynamics"

http://arxiv.org/abs/1512.092582

In the 19th century Sturm's theorem on the number of roots of a real polynomial motivated Sylvester to define the signature of a quadratic form. In the 20th century the classification of quadratic forms over algebraic number fields motivated Witt to introduce the "Witt groups" of stable isomorphism classes of quadratic forms over arbitrary fields. Still in the 20th century the study of high-dimensional topological manifolds with nontrivial fundamental group motivated Wall to introduce the "Wall groups" of stable isomorphism classes of quadratic forms over arbitrary rings with involution. In our survey we interpreted Sturm's theorem in terms of the Witt-Wall groups of function fields. The talk will emphasize the common thread running through this developments, namely the notion of the localization of a ring inverting elements. More recently, the Cohn localization of inverting matrices over a noncommutative ring has been applied to topology in the 21st century, in the context of the speaker's algebraic theory of surgery.

 

Thu, 10 Mar 2016

16:00 - 17:00
L5

On the number of nodal domains of toral eigenfunctions

Igor Wigman
(King's College London)
Abstract

We study the number of nodal domains of toral Laplace eigenfunctions. Following Nazarov-Sodin's results for random fields and Bourgain's de-randomisation procedure we establish a precise asymptotic result for "generic" eigenfunctions. Our main results in particular imply an optimal lower bound for the number of nodal domains of generic toral eigenfunctions.

Thu, 03 Mar 2016

16:00 - 17:00
L2

Hecke eigenvalue congruences and experiments with degree-8 L-functions

Neil Dummigan
(University of Sheffield)
Abstract

I will describe how the moduli of various congruences between Hecke eigenvalues of automorphic forms ought to show up in ratios of critical values of $\text{GSP}_2 \times \text{GL}_2$ L-functions. To test this experimentally requires the full force of Farmer and Ryan's technique for approximating L-values given few coefficients in the Dirichlet series.

Thu, 25 Feb 2016

16:00 - 17:00
L2

Badly approximable points

Victor Beresnevich
(University of York)
Abstract

I will discuss the notion of badly approximable points and recent progress and problems in this area, including Schmidt's conjecture, badly approximable points on manifolds and real numbers badly approximable by algebraic numbers.

Thu, 18 Feb 2016

16:00 - 17:00
L5

(Joint Number Theory and Logic) On a modular Fermat equation

Jonathan Pila
(University of Oxford)
Abstract

I will describe some diophantine problems and results motivated by the analogy between powers of the modular curve and powers of the multiplicative group in the context of the Zilber-Pink conjecture.

Thu, 04 Feb 2016

16:00 - 17:00
L5

Strongly semistable sheaves and the Mordell-Lang conjecture over function fields

Damian Rössler
(University of Oxford)
Abstract

We shall describe a new proof of the Mordell-Lang conjecture in positive characteristic, in the situation where the variety under scrutiny is a smooth subvariety of an abelian variety. 
Our proof is based on the theory of semistable sheaves in positive characteristic, in particular on  Langer's theorem that the Harder-Narasimhan filtration of sheaves becomes strongly semistable after a finite number of iterations of Frobenius pull-backs. Our proof produces a numerical upper-bound for the degree of the finite morphism from an isotrivial variety appearing in the statement of the Mordell-Lang conjecture. This upper-bound is given in terms of the Frobenius-stabilised slopes of the cotangent bundle of the variety.

Thu, 28 Jan 2016

16:00 - 17:00
L5

Iwasawa theory for the symmetric square of a modular form

David Loeffler
(University of Warwick)
Abstract

Iwasawa theory is a powerful technique for relating the behaviour of arithmetic objects to the special values of L-functions. Iwasawa originally developed this theory in order to study the class groups of number fields, but it has since been generalised to many other settings. In this talk, I will discuss some new results in the Iwasawa theory of the symmetric square of a modular form. This is a joint project with Sarah Zerbes, and the main tool in this work is the Euler system of Beilinson-Flach elements, constructed in our earlier works with Kings and Lei.

Thu, 21 Jan 2016

16:00 - 17:00
L5

Height of rational points on elliptic curves in families

Pierre Le Boudec
(EPFL (Ecole Polytechnique Federale de Lausanne))
Abstract

Given a family $F$ of elliptic curves defined over $Q$, we are interested in the set $H(Y)$ of curves $E$ in $F$, of positive rank, and for which the minimum of the canonical heights of non-torsion rational points on $E$ is bounded by some parameter $Y$. When one can show that this set is finite, it is natural to investigate statistical properties of arithmetic objects attached to elliptic curves in the set $H(Y)$. We will describe some problems related to this, and will state some results in the case of families of quadratic twists of a fixed elliptic curve.

Mon, 18 Jan 2016

16:00 - 17:00
L3

4th moment of quadratic Dirichlet L-functions in function fields

Alexandra Florea
(Stanford University)
Abstract

We discuss moments of $L$-functions in function fields, in the hyperelliptic ensemble, focusing on the fourth moment of quadratic Dirichlet $L$-functions at the critical point. We explain how to obtain an asymptotic formula with some of the secondary main terms.

Thu, 03 Dec 2015

16:00 - 17:00
L5

Galois theory of periods and applications

Francis Brown
(University of Oxford)
Abstract

A period is a certain type of number obtained by integrating algebraic differential forms over algebraic domains. Examples include pi, algebraic numbers, values of the Riemann zeta function at integers, and other classical constants.
Difficult transcendence conjectures due to Grothendieck suggest that there should be a Galois theory of periods.
I will explain these notions in very introductory terms and show how to set up such a Galois theory in certain situations.
I will then discuss some applications, in particular to Kim's method for bounding $S$-integral solutions to the equation $u+v=1$, and possibly to high-energy physics.

Thu, 26 Nov 2015

16:00 - 17:00
L5

On the Central Limit Theorem for the number of steps in the Euclidean algorithm

Ian Morris
(University of Surrey)
Abstract

The number of steps required by the Euclidean algorithm to find the greatest common divisor of a pair of integers $u,v$ with $1<u<v<n$ has been investigated since at least the 16th century, with an asymptotic for the mean number of steps being found independently by H. Heilbronn and J.D. Dixon in around 1970. It was subsequently shown by D. Hensley in 1994 that the number of steps asymptotically follows a normal distribution about this mean. Existing proofs of this fact rely on extensive effective estimates on the Gauss-Kuzman-Wirsing operator which run to many dozens of pages. I will describe how this central limit theorem can be obtained instead by a much shorter Tauberian argument. If time permits, I will discuss some related work on the number of steps for the binary Euclidean algorithm.

Thu, 19 Nov 2015

16:00 - 17:00
L5

Prime number races with very many competitors

Adam Harper
(University of Cambridge)
Abstract

The prime number race is the competition between different coprime residue classes mod $q$ to contain the most primes, up to a point $x$ . Rubinstein and Sarnak showed, assuming two $L$-function conjectures, that as $x$ varies the problem is equivalent to a problem about orderings of certain random variables, having weak correlations coming from number theory. In particular, as $q \rightarrow \infty$ the number of primes in any fixed set of $r$ coprime classes will achieve any given ordering for $\sim 1/r!$ values of $x$. In this talk I will try to explain what happens when $r$ is allowed to grow as a function of $q$. It turns out that one still sees uniformity of orderings in many situations, but not always. The proofs involve various probabilistic ideas, and also some harmonic analysis related to the circle method. This is joint work with Youness Lamzouri.

Thu, 12 Nov 2015

16:00 - 17:00
L5

Iwasawa theory for the symmetric square of a modular form - Cancelled

Sarah Zerbes
(University College London)
Abstract

I will discuss some new results on the Iwasawa theory for the $3$-dimensional symmetric square Galois representation of a modular form, using the Euler system of Beilinson-Flach elements I constructed in joint work with Kings, Lei and Loeffler.

Thu, 05 Nov 2015

16:00 - 17:00
L5

Around the Möbius function

Kaisa Matomäki
(University of Turku)
Abstract

The Möbius function plays a central role in number theory; both the prime number theorem and the Riemann Hypothesis are naturally formulated in terms of the amount of cancellations one gets when summing the Möbius function. In a recent joint work with Maksym Radziwill we have shown that the sum of the Möbius function exhibits cancellation in "almost all intervals" of arbitrarily slowly increasing length. This goes beyond what was previously known conditionally on the Riemann Hypothesis. Our result holds in fact in much greater generality, and has several further applications, some of which I will discuss in the talk. For instance the general result implies that between a fixed number of consecutive squares there is always an integer composed of only "small" prime factors. This settles a conjecture on "smooth" or "friable" numbers and is related to the running time of Lenstra's factoring algorithm.

Thu, 29 Oct 2015

17:30 - 18:30
L6

A minimalistic p-adic Artin-Schreier (Joint Number Theroy/Logic Seminar)

Florian Pop
(University of Pennsylvania)
Abstract

In contrast to the Artin-Schreier Theorem, its $p$-adic analog(s) involve infinite Galois theory, e.g., the absolute Galois group of $p$-adic fields.  We plan to give a characterization of $p$-adic $p$-Henselian valuations in an essentially finite way. This relates to the $Z/p$ metabelian form of the birational $p$-adic Grothendieck section conjecture.

Thu, 29 Oct 2015

16:00 - 17:00
L5

Arthur's multiplicity formula for automorphic representations of certain inner forms of special orthogonal and symplectic groups

Olivier Taibi
(Imperial College)
Abstract

I will explain the formulation and proof of Arthur's multiplicity formula for automorphic representations of special orthogonal groups and certain inner forms of symplectic groups $G$ over a number field $F$. I work under an assumption that substantially simplifies the use of the stabilisation of the trace formula, namely that there exists a non-empty set $S$ of real places of $F$ such that $G$ has discrete series at places in $S$ and is quasi-split at places outside $S$, and restricting to automorphic representations of $G(A_{F})$ which have algebraic regular infinitesimal character at the places in $S$. In particular, this proves the general multiplicity formula for groups $G$ such that $F$ is totally real, $G$ is compact at all real places of $F$ and quasi-split at all finite places of $F$. Crucially, the formulation of Arthur's multiplicity formula is made possible by Kaletha's recent work on local and global Galois
gerbes and their application to the normalisation of Kottwitz-Langlands-Shelstad transfer factors.

Thu, 22 Oct 2015

16:00 - 17:00
L5

Linear Algebra with Errors, Coding Theory, Cryptography and Fourier Analysis on Finite Groups

Steven Galbraith
(University of Auckland)
Abstract

Solving systems of linear equations $Ax=b$ is easy, but how can we solve such a system when given a "noisy" version of $b$? Over the reals one can use the least squares method, but the problem is harder when working over a finite field. Recently this subject has become very important in cryptography, due to the introduction of new cryptosystems with interesting properties.

The talk will survey work in this area. I will discuss connections with coding theory and cryptography. I will also explain how Fourier analysis in finite groups can be used to solve variants of this problem, and will briefly describe some other applications of Fourier analysis in cryptography. The talk will be accessible to a general mathematical audience.

Thu, 15 Oct 2015

16:00 - 17:00
L5

Sums of seven cubes

Samir Siksek
(University of Warwick)
Abstract

In 1851, Carl Jacobi made the experimental observation that all integers are sums of seven non-negative cubes, with precisely 17 exceptions, the largest of which is 454. Building on previous work by Maillet, Landau, Dickson, Linnik, Watson, Bombieri, Ramaré, Elkies and many others, we complete the proof of Jacobi's observation.

Thu, 11 Jun 2015

16:00 - 17:00
L6

Moduli stacks of potentially Barsotti-Tate Galois representations

Toby Gee
(Imperial College)
Abstract

I will discuss joint work with Ana Caraiani, Matthew Emerton and David Savitt, in which we construct moduli stacks of two-dimensional potentially Barsotti-Tate Galois representations, and study the relationship of their geometry to the weight part of Serre's conjecture.

Thu, 04 Jun 2015

16:00 - 17:00
L5

Bounded intervals containing many primes

Roger Baker
(Brigham Young University)
Abstract

I describe joint work with Alastair Irving in which we improve a result of
D.H.J. Polymath on the length of intervals in $[N,2N]$ that can be shown to
contain $m$ primes. Here $m$ should be thought of as large but fixed, while $N$
tends to infinity.
The Harman sieve is the key to the improvement. The preprint will be
available on the Math ArXiv before the date of the talk.

Thu, 28 May 2015

16:00 - 17:00
L5

Cubic hypersurfaces over global fields

Pankaj Vishe
(University of York)
Abstract

 Let $X$ be a smooth cubic hypersurface of dimension $m$ defined over a global field $K$. A conjecture of Colliot-Thelene(02) states that $X$ satisfies the Hasse Principle and Weak approximation as long as $m\geq 3$. We use a global version of Hardy-Littlewood circle method along with the theory of global $L$-functions to establish this for $m\geq 6$, in the case $K=\mathbb{F}_q(t)$, where $\text{char}(\mathbb{F}_{q})> 3$.

Thu, 21 May 2015

16:00 - 17:00
L5

Anabelian Geometry with étale homotopy types

Jakob Stix
(University of Heidelberg)
Abstract

Classical anabelian geometry shows that for hyperbolic curves the etale fundamental group encodes the curve provided the base field is sufficiently arithmetic. In higher dimensions it is natural to replace the etale fundamental group by the etale homotopy type. We will report on progress obtained in this direction in a recent joint work with Alexander Schmidt.

 

**Joint seminar with Logic.