One of the classical tools of number theory is the so-called local-global

principle, or Hasse principle, going back to Hasse's work in the 1920's.

His first results concern quadratic forms, and norms of number fields.

Over the years, many positive and negative results were proved, and there

is now a huge number of results in this topic.

This talk will present some old and new results, in particular in the

continuation of Hasse's cyclic norm theorem. These have been obtained

jointly with Parimala and Tingyu Lee.

which associate certain A infinity category to a symplectic manifold. This construction has various application to symplectic

geometry Mirror symmetry and Gauge theory. I will explain some of them in this talk.

In this talk we shall review some recent progress in Hilbert's sixth problem, consisting in justifying Fluid Mechanics equations

from deterministic systems of interacting particles obeying Newton's laws.

This corresponds to joint works with Thierry Bodineau and Laure Saint-Raymond.

Langlands' automorphic transfer from reductive groups to linear

groups is equivalent to the existence of local and global Fourier

transform operators induced by representations of the dual groups, of

local and global functional spaces that should be stabilized by these

operators and of a Poisson linear form on the global functional spaces

that should be fixed by Fourier transform.

Looking for a definition of non-additive Fourier transform

operators leads to the crucial question of determining the Fourier

transform of the operator of point-wise multiplication of functions. It

has to be a generalisation of ordinary additive convolution operators.

of a Weyl group can be regarded as the value at 1 of a family of

algebras depending on a non-zero complex parameter (the Hecke

algebra introduced by Iwahori in the 1960's). It turns out that

this family can be extended naturally for the parameter 0 or infinity

(the asymptotic Hecke algebra). Remarkably the asymptotic Hecke

algebra contains much information about the corresponding algebraic

group. Several examples of this will be presented in the talk.

**Noam Elkies
**

In this talk I will present a joint work with N.S. Daleo, J.D.

Hauenstein and B. Mourrain where we we develop computational methods for

computing ranks and border ranks of tensors along with decompositions.

We shall discuss the arc connectedness of the space of

metrics of positive scalar curvature on a complete open 3-manifold. It

is a joined work with L. Bessières, F. Coda Marques and S. Maillot.

We show that all members of a large class of normal subgroups of the mapping class group of a closed surface are

A celebrated result of Hirzebruch and Zagier states that the

generating series of Hirzebruch-Zagier divisors on a Hilbert modular

surface is an elliptic modular form with values in the cohomology.

We discuss some generalizations and applications of this result.

In particular, we prove an analogue for special divisors on integral

models of ball quotients. In this setting the generating series takes

values in an arithmetic Chow group in the setting of Arakelov geometry. If

time permits, we address some applications to arithmetic theta lifts and

the Colmez conjecture.

This is joint work with B. Howard, S. Kudla, M. Rapoport, and T. Yang.

The talk will present an abstract topos-theoretic framework for building Galois-type theories in a variety of different

mathematical contexts : this unifies and generalises Grothendieck's theory of ‘Galoisian categories’ and Fraïssé's

construction in model theory.

This theory allows one to construct fundamental groups in many classical contexts such as finite groups, finite

graphs, motives and many more.

We will in particular present an approach based on it for investigating the independence from l of l-adic cohomology.

Gunther Cornelissen

Reconstructing global fields from L-series

I will present a historical survey of which invariants of a global field do and which don't determine the field up to isomorphism,

and then present two new contributions to the field: one using Dirichlet L-series, and another (related) using a dynamical system

which I would like to call an "abelian anabelian space". Proofs use nothing beyond class field theory.

The heat kernel of a Riemannian manifold is the minimal positive

fundamental solution of the heat equation associated with the

Laplace-Beltrami operator. Upper and lower estimates of heat kernels

play important role in Analysis on manifolds. A celebrated theorem of Li

and Yau provides two sided Gaussian estimates of the heat kernel on a

complete Riemannian manifold of non-negative Ricci curvature. In this

talk we present heat kernel estimates on a complete manifold with ends,

assuming that the heat kernel on each end satisfies the Li-Yau estimate.

It turns out that the behaviour of the heat kernel on the entire

manifold depends on the property of the ends to be parabolic or not (a

manifold is called parabolic if Brownian motion on it is recurrent, or,

equivalently, if any positive superharmonic function is constant)."

Periodic tilings of R^n have been extensively studied since the end of the nineteenth century in relation to crystallography. They include Euclidean tilings but also, more generally, tilings whose symmetry group consists of affine transformations. When the tiles are noncompact, the symmetry group may no longer be a group of translations up to finite index: Margulis constructed famous examples with nonabelian free groups. We will explain that any right-angled Coxeter group may be realized as a symmetry group of a periodic affine tiling of R^n, where n depends quadratically on the number of generators. As a consequence, many other groups may be realized as such symmetry groups, including all right-angled Artin groups, all Coxeter groups, and all fundamental groups of compact hyperbolic surfaces or 3-manifolds. This is joint work with J. Danciger and F. Guéritaud.

The `conventional' representation zeta function of a compact p-adic Lie group G is the Dirichlet generating function enumerating (finite dimensional) irreducible complex representations of G. I will briefly motivate the study of these zeta functions and survey some of the key results in the subject. More generally, we can attach a zeta function to every `suitable' infinite-dimensional representation of G, the `conventional' zeta function being essentially the zeta function associated to the regular representation of G. In my talk I will report on recent results obtained with Steffen Kionke, focussing on zeta functions attached to induced representations. One simple and beautiful source of explicit examples arises from distance-transitive actions of profinite groups on rooted trees, further concrete and instructive examples can be obtained by means of the Kirillov orbit method and techniques of p-adic integration.

Groebner bases are a fundamental tool in symbolic computation that

allow effective computations with ideals in a polynomial ring. These

depend on a choice of a total order on the monomials in the polynomial

ring. However while there are an infinite number of such orders,

there are in fact only a finite number of different (reduced) Groebner

bases for an ideal. These were shown in the 80s by Mora/Robbiano and

Bayer/Morrison to correspond to cones in a polyhedral fan. After

reviewing these facts, and applications of this Groebner fan and its

variants, I will discuss joint work with Felipe Rincon, motivated by

problems in tropical geometry, that highlights the role of matroids

(from combinatorics) in this subject.

A central object of interest in the study of moduli
spaces of Riemann surfaces is their cohomology, which
describes invariants---characteristic classes---for families
of Riemann surfaces. In 1985 Harer proved the remarkable
theorem that the cohomology of the moduli space of genus g
surfaces is independent of the genus in a range of degrees
tending to infinity with g: Madsen and Weiss' 2007 proof of
the Mumford Conjecture completely described the cohomology
in this stable range.

I will describe joint work with S. Galatius and A. Kupers in
which we show there is a larger range of degrees, the
metastable range, in which the cohomology is no longer stable
but becomes periodic in a certain sense. The main technical
tool is a theory of cellular E_2-algebras, which I shall
explain. James Robinson

There are many distinct definitions of dimension, often tailored towards

particular problems.

One definition, due to Assouad, is particularly interesting in the

context of determining when a metric space $(X,d)$ can be embedded into

a finite-dimensional Euclidean space using a bi-Lipschitz map.

Finite "Assouad dimension" is necessary, but not sufficient, for the

existence of such an embedding.

I will compare the Assouad dimension with more familiar definitions, and

illustrate some of its unique properties with a simple proof of a

variant of the Kakeya Conjecture: any subset $K$ of $R^n$ that contains

a half-line in every direction must have maximal Assouad dimension

($d_A(K)=n$).

I will then discuss classical results and counterexamples realted to the

bi-Lipschitz embedding problem, and more recent results on the "almost

bi-Lipschitz" embedding via linear maps of subsets $X$ of Hilbert and

Banach spaces when $X-X$ has finite Assouad dimension.

The talk relies in part on joint work with Jonathan Fraser (St Andrews)

and Eric Olson (Reno, Nevada).

Alex Fink, Queen Mary University

Milena Hering, University of Edinburgh

Thomas Kahle, OvGU Magdeburg and CDS

Gregor Kemper, TU München

Dimitra Kosta, University of Edinburgh

Fatemeh Mohammadi, Bristol

Nina Otter, University of Oxford

Joseph Cook, Leeds

Alexandra Enblom, Linköping

Katie Gittins, Neuchatel

Simon Larson, KTH, Stockholm

Liangpan Li, Loughborough

Bernhard Pfirsch, UCL

Christian Rose, Chemnitz

Leonardo Tolomeo, Edinburgh

Shinpei Baba, Heidelberg

Daniel Ballesteros, Durham

Michela Egidi, Dortmund

Selim Ghazouani, ENS

Irene Pasquinelli, Durham

Daniel Pomerleano, Imperial College London

Katie Vokes, Warwick

Tobias Berger, Sheffield

Steven Charlton, Tübingen

Jolanta Marzec, Durham

Rachel Newton, Reading

Jeanine van Order, Bielefeld

Anke Pohl, Jena

Larry Rolen, Trinity College Dublin

Yumi Boote, Manchester

Brent Everitt, York

Richard Hepworth, Aberdeen

Brendan Owens, Glasgow

Simona Paoli, Leicester

Constanze Roitzheim, Kent

Jonathan Woolf, Liverpool