UCL

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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Coisotropic submanifolds arise naturally in symplectic geometry as level sets of moment maps and in algebraic geometry in the context of normal crossing divisors. In examples, the Marsden-Weinstein quotient or (Fano) complete intersections are often uniruled. 
We show that under natural geometric assumptions on a coisotropic submanifold, the symplectic quotient of the coisotropic is always geometrically uniruled. 
I will explain how to assign a Lagrangian and a hypersurface to a fibered, stable coisotropic C. The Lagrangian inherits a fibre bundle structure from C, the hypersurface captures the generalised Reeb dynamics on C. To derive the result, we then adapt and apply techniques from Lagrangian Floer theory and symplectic field theory.
This is joint work with Jonny Evans.
 

The so-called Ax-Lindemann theorem asserts that the Zariski closure of a certain subset of a homogeneous variety (typically abelian or Shimura) is itself a homogeneous variety. This theorem has recently been proven in full generality by Klingler-Ullmo-Yafaev and Gao. This statement leads to a variety of questions about topological and Zariski closures of certain sets in  homogeneous varieties which can be approached by both ergodic and o-minimal techniques.  In a series of recent papers with E. Ullmo, we formulate conjectures and prove a certain number of results  of this type.  In this talk I will present these conjectures and results and explain the ideas of proofs
 

Since Donaldson-Thomas proposed a programme for studying gauge theory in higher dimensions, there has
been significant interest in understanding special Yang-Mills connections in Ricci-flat 7-manifolds with holonomy
G_2 called G_2-instantons.  However, still relatively little is known about these connections, so we begin the
systematic study of G_2-instantons in the SU(2)^2-invariant setting.  We provide existence, non-existence and
classification results, and exhibit explicit sequences of G_2-instantons where “bubbling" and "removable
singularity" phenomena occur in the limit.  This is joint work with Goncalo Oliveira (Duke).