UCL

Let E be an elliptic curve over the rationals and p a prime such that E admits a rational p-isogeny satisfying some assumptions. In a joint work with J. Lee and C. Skinner, we prove the anticyclotomic Iwasawa main conjecture for E/K for some suitable quadratic imaginary field K. I will explain our strategy and how this, combined with complex and p-adic Gross-Zagier formulae, allows us to prove that if E has rank one, then the p-part of the Birch and Swinnerton-Dyer formula for E/Q holds true.
 

Stevenhagen conjectured that the density of d such that the negative Pell equation x^2-dy^2=-1 is solvable over the integers is 58.1% (to the nearest tenth of a percent), in the set of positive squarefree integers having no prime factors congruent to 3 modulo 4. In joint work with Peter Koymans, Djordjo Milovic, and Carlo Pagano, we use a recent breakthrough of Smith to prove that the infimum of this density is at least 53.8%, improving previous results of Fouvry and Klüners, by studying the distribution of the 8-rank of narrow class groups of quadratic number fields.

In the 1920s Weyl proved the first non-trivial estimate for the Riemann zeta function on the critical line: \zeta(1/2+it) << (1+|t|)^{1/6+\epsilon}. The analogous bound for a Dirichlet L-function L(1/2,\chi) of conductor q as q tends to infinity is still unknown in full generality. In a breakthrough around 2000, Conrey and Iwaniec proved the analogue of the Weyl bound for L(1/2,\chi) when \chi is assumed to be quadratic of conductor q.  Building on the work of Conrey and Iwaniec, we show (joint work with Matt Young) that the Weyl bound for L(1/2,\chi) holds for all primitive Dirichlet characters \chi. The extension to all moduli q is based on aLindelöf-on-average upper bound for the fourth moment of Dirichlet L-functions of conductor q along a coset of the subgroup of characters modulo d when q^*|d, where q^* is the least positive integer such that q^2|(q^*)^3.

Boundary integral equations are an elegant tool to model and simulate a range of physical phenomena in bounded and unbounded domains.

While mathematically well understood, the numerical implementation (e.g. via boundary element methods) still poses a number of computational challenges, from the efficient assembly of the underlying linear systems up to the fast preconditioned solution in complex applications. In this talk we provide an overview of some of these challenges and demonstrate the efficient implementation of boundary element methods on modern CPU and GPU architectures. As part of the talk we will present a number of practical examples using the Bempp-cl boundary element software, our next generation boundary element package, that has been developed in Python and supports modern vectorized CPU instruction sets and a number of GPU types.

Berry's random wave conjecture is a heuristic that the eigenfunctions of a classically ergodic system ought to display Gaussian random behaviour, as though they were random waves, in the large eigenvalue limit. We discuss two manifestations of this conjecture for eigenfunctions of the Laplacian on the modular surface: Planck scale mass equidistribution, and an asymptotic for the fourth moment. We will highlight how the resolution of these two problems in this number-theoretic setting involves a delicate understanding of the behaviour of certain families of L-functions.

A stacking is a lift of an immersion of graphs $A\to B$ to an embedding of $A$ into the product of $B$ with the real line; their existence relates to orderability properties of groups. I will describe how Louder and Wilton used them to prove Wise's "$w$-cycles" conjecture: given a primitive word $w$ in a free group $F$, and a subgroup $H < F$, the number of conjugates of $H$ which intersect $<w>$ nontrivially is at most rank($H$). I will also discuss applications of the result to questions of coherence, and possible extensions of it.

Euler sums and p-adic L-functions

In 1973, Serre observed that the Hecke eigenvalues of Eisenstein series can be p-adically interpolated. In other words, Eisenstein series can be viewed as specializations of a p-adic family parametrized by the weight. The notion of p-adic variations of modular forms was later generalized by Hida to include families of ordinary cuspforms. In 1998, Coleman and Mazur defined the eigencurve, a rigid analytic space classifying much more general p-adic families of Hecke eigenforms parametrized by the weight. The local nature of the eigencurve is well-understood at points corresponding to cuspforms of weight k ≥ 2, while the weight one case is far more intricate.

In this talk, we discuss the geometry of the eigencurve at weight one Eisenstein points. Our approach consists in studying the deformation rings of certain (deceptively simple!) Artin representations. Via this Galois-theoretic method, we obtain the q-expansion of some non-classical overconvergent forms in terms of p-adic logarithms of p-units in certain number field. Finally, we will explain how these calculations suggest a different approach to the Gross-Stark conjecture.

Lagrangian Floer cohomology can be enriched by using local coefficients to record some homotopy data about the boundaries of the holomorphic disks counted by the theory. In this talk I will explain how one can do this under the monotonicity assumption and when the Lagrangians are equipped with local systems of rank higher than one. The presence of holomorphic discs of Maslov index 2 poses a potential obstruction to such an extension. However, for an appropriate choice of local systems the obstruction might vanish and, if not,
one can always restrict to some natural unobstructed subcomplexes. I will showcase these constructions with some explicit calculations for the Chiang Lagrangian in CP^3 showing that it cannot be disjoined from RP^3 by a Hamiltonian isotopy, answering a question of Evans-Lekili. Time permitting, I will also discuss some work-in-progress on the topology of monotone Lagrangians in CP^3, part of which follows from more general joint work with Jack Smith on the topology of monotone Lagrangians of maximal Maslov number in
projective spaces.

Hannah Fry takes us on a tour of the good, the bad and the downright ugly of the algorithms that surround us. Are they really an improvement on the humans they are replacing?

Hannah Fry is a lecturer in the Mathematics of Cities at the Centre for Advanced Spatial Analysis at UCL. She is also a well-respected broadcaster and the author of several books including the recently published 'Hello World: How to be Human in the Age of the Machine.'

5.00pm-6.00pm, Mathematical Institute, Oxford

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