Past

A Kähler group is a finitely presented group that can be realized as fundamental group of a compact Kähler manifold. It is known that every finitely presented group can be realized as fundamental group of a compact real and even symplectic manifold of dimension greater equal than 4 and of a complex manifold of complex dimension greater equal than 2. In contrast, the question which groups are Kähler groups is surprisingly harder and there are large classes of examples for both, Kähler, and non-Kähler groups. This talk will give a brief introduction to the theory of Kähler manifolds and then discuss some basic examples and properties of Kähler groups. It is aimed at a general audience and no prior knowledge of the field will be required.

For a finitely presented group, the Word Problem asks for an algorithm

which declares whether or not words on the generators represent the

identity. The Dehn function is the time-complexity of a direct attack

on the Word Problem by applying the defining relations.

A "hydra phenomenon" gives rise to novel groups with extremely fast

growing (Ackermannian) Dehn functions. I will explain why,

nevertheless, there are efficient (polynomial time) solutions to the

Word Problems of these groups. The main innovation is a means of

computing efficiently with compressed forms of enormous integers.

This is joint work with Will Dison and Eduard Einstein.

Quantifying the uncertainty in computational simulations is one of the central challenges confronting the field of computational science and engineering today. The uncertainty quantification of inverse problems is neatly addressed in the Bayesian framework, where instead of seeking one unique minimiser of a regularised misfit functional, the entire posterior probability distribution is to be characterised. In this talk I review the deep connection between deterministic PDE-constrained optimisation techniques and Bayesian inference for inverse problems, discuss some recent advances made in the Bayesian viewpoint by adapting deterministic techniques, and mention directions for future research.

Composite dilation wavelets are affine systems which extend the notion of wavelets by incorporating a second set of dilations.  The addition of a second set of dilations allows the composite system to capture directional information in addition to time and frequency information.  We classify admissible dilation groups at two extremes: frequency localization through minimally supported frequency composite dilation wavelets and time localization through crystallographic Haar-type composite dilation wavelets. 

The instanton corrections to the hyperkähler metric on moduli spaces of meromorphic flat SL(2,C)-connections on a Riemann surface with prescribed singularities have recently been studied by Gaiotto, Moore and Neitzke. The instantons are given by certain special trajectories of the meromorphic quadratic differentials which form the base of Hitchin's integrable system structure on the moduli space. Bridgeland and Smith interpret such quadratic differentials as defining stability conditions on an associated 3-Calabi-Yau triangulated category whose stable objects correspond to these special trajectories.

The smallest non-trivial examples are provided by the moduli spaces of quaternionic dimension one. In these cases it is possible to study explicitly the periods of the Seiberg-Witten differential on the fibres of the Hitchin system which define the central charge of the stability condition and lift the period map to the space of stability conditions. This provides in particular a new categorical perspective on the original Seiberg-Witten gauge theories.

The talk is based on my paper with E. Beggs appearing in Class. Quantum

Gravity.

Working within a bimodule approach to noncommutative geometry, we show that

even a small amount of noncommutativity drastically constrains the moduli

space of

noncommutative metrics. In particular, the algebra [x,t]=x is forced to have

a geometry

corresponding to a gravitational source at x=0 so strong that even light

cannot

escape. This provides a non-trivial example of noncommutative Riemannian

geometry

and also serves as an introduction to some general results.

One dimensional analysis of Euler-Poisson system shows that when incoming supersonic flow is fixed,

transonic shock can be represented as a monotone function of exit pressure.

From this observation, we expect well-posedness of transonic shock problem for Euler-Poisson system

when exit pressure is prescribed in a proper range.

In this talk, I will present recent progress on transonic shock problem for Euler-Poisson system,

which is formulated as a free boundary problem with mixed type PDE system.

This talk is based on collaboration with Ben Duan, Chujing Xie and Jingjing Xiao

In 1997 V. Bentkus and F. Götze introduced a technique for estimating $L^p$ norms of certain exponential sums without needing an explicit estimate for the exponential sum itself. One uses instead a kind of estimate I call a "moat lemma". I explain this term, and discuss the implications for several kinds of point-counting problem which we all know and love.

The'signature', from the theory of differential equations driven by rough paths,
provides a very efficient way of characterizing curves. From a machine learning
perspective, the elements of the signature can be used as a set of features for
consumption by a classification algorithm.

Using datasets of letters, digits, Indian characters and Chinese characters, we
see that this improves the accuracy of online character recognition---that is
the task of reading characters represented as a collection of pen strokes.

A "bordism representation" (*) is a representation of the abstract

structure formed by manifolds and bordisms between them, and hence of

fundamental interest in topology. I will give an overview of joint work

establishing a simple generators-and-relations presentation of the

3-dimensional oriented bordism bicategory, and also its "signature" central

extension. A representation of this bicategory corresponds in a 2-1 fashion

to a modular category, which must be anomaly-free in the oriented case. J/w

Chris Douglas, Chris Schommer-Pries, Jamie Vicary.

(*) These are also known as "topological quantum field theories".

A folded symplectic form on a manifold is a closed 2-form with the mildest possible degeneracy along a hypersurface. A special class of folded symplectic manifolds are the origami manifolds. In the classical case, toric symplectic manifolds can classified by their moment polytope, and their topology (equivariant cohomology) can be read directly from the polytope. In this talk we examine the toric origami case: we will recall how toric origami manifolds can also be classified by their combinatorial moment data, and present some theorems, almost-theorems, and conjectures about the topology of toric origami manifolds.

The surface subgroup problem asks whether a given group contains a subgroup that is isomorphic to the fundamental group of a closed surface. In this talk I will survey the role that the surface subgroup problem plays in some important solved and unsolved problems in the theory of 3-manifolds, the geometric group theory, and the theory of arithmetic manifolds.

Jakobshavn Isbrae and many other fast flowing outlet glaciers of present

and past ice sheets lie in deep troughs which often have several

overdeepened sections. To make their fast flow possible their bed needs

to be slippery which in turn means high basal water pressures. I will

present a model of subglacial water flow and its application to

Jakobshavn. I find that, somewhat surprisingly, the reason for

Jakobshavn's fast flow might be the pressure dependence of the melting

point of ice. The model itself describes the unusual fluid dynamics occurring underneath the ice; it has an interesting mathematical structure that presents computational challenges.

We consider the pricing of American put options in a model-independent setting: that is, we do not assume that asset prices behave according to a given model, but aim to draw conclusions that hold in any model. We incorporate market information by supposing that the prices of European options are known. In this setting, we are able to provide conditions on the American Put prices which are necessary for the absence of arbitrage. Moreover, if we further assume that there are finitely many European and American options traded, then we are able to show that these conditions are also sufficient. To show sufficiency, we construct a model under which both American and European options are correctly priced at all strikes simultaneously. In particular, we need to carefully consider the optimal stopping strategy in the construction of our process. (Joint with Christoph Hoeggerl).

A pseudofinite group is an infinite model of the theory of finite groups. I will discuss what can be said about pseudofinite groups under various tameness assumptions on the theory (e.g. NIP, supersimplicity), structural results on pseudofinite permutation groups, and connections to word maps and generalisations.

Ricci solitons were introduced by Richard Hamilton in the 80's and they are a generalization of the better know Einstein metrics. During this talk we will define the notion of Ricci soliton and I will try to convince you that these metrics arise "naturally" in a number of different settings. I will also present various examples and talk a bit about some symmetry properties that Ricci solitons have.

Note: This talk is meant to be introductory and no prior knowledge about Einstein metrics will be assumed (or necessary).

After a short survey of the notion of level of distribution for
arithmetic functions, and its importance in analytic number theory, we
will explain how our recent studies of twists of Fourier coefficients of
modular forms (and especially Eisenstein series) by "trace functions"
lead to an improvement of the results of Friedlander-Iwaniec and
Heath-Brown for the ternary divisor function in arithmetic progressions
to prime moduli.

This is joint work with É. Fouvry and Ph. Michel.

Many real-life complex systems arise as a network of simple interconnected individual agents. A central question is to determine how network topology and individual agent dynamics combine to create the global dynamics.

In this talk we focus on the case of continuous-time random walks on networks, with a waiting time of the walker on each node assuming arbitrary probability distributions. Such random walks are useful to model diffusion processes over complex temporal networks representing human interactions, often characterized by non-Poissonian contact patterns.

We find that the mixing time of the random walker, i.e. the relaxation time for the process to reach stationarity, is determined by a combination of three factors: the spectral gap, associated to bottlenecks in the underlying topology, burstiness, related to the second moment of the waiting time distribution, and the characteristic time of its exponential tail, which is an indicator of the tail `fatness'. We show

theoretically that a strong modular structure dampens the importance of burstiness, and empirically that either of the three factors may be dominant in real-life data.

These results are available in arXiv:1309.4155