Past

I will discuss new rigidity and rationality phenomena

(related to the phenomenon of Arnold tongues) in the theory of

nonabelian group actions on the circle. I will introduce tools that

can translate questions about the existence of actions with prescribed

dynamics, into finite combinatorial questions that can be answered

effectively. There are connections with the theory of Diophantine

approximation, and with the bounded cohomology of free groups. A

special case of this theory gives a very short new proof of Naimi’s

theorem (i.e. the conjecture of Jankins-Neumann) which was the last

step in the classification of taut foliations of Seifert fibered

spaces. This is joint work with Alden Walker.

The notion quantization originates from information theory, where it refers to the approximation of a continuous signal on a discrete set. Our research on quantization is mainly motivated by applications in quadrature problems. In that context, one aims at finding for a given probability measure $\mu$ on a metric space a discrete approximation that is supported on a finite number of points, say $N$, and is close to $\mu$ in a Wasserstein metric.

In general it is a hard problem to find close to optimal quantizations, if  $N$ is large and/or  $\mu$ is given implicitly, e.g. being the marginal distribution of a stochastic differential equation. In this talk we analyse the efficiency of empirical measures in the constructive quantization problem. That means the random approximating measure is the uniform distribution on $N$ independent $\mu$-distributed elements.

We show that this approach is order order optimal in many cases. Further, we give fine asymptotic estimates for the quantization error that involve moments of the density of the absolutely continuous part of $\mu$, so called high resolution formulas. The talk ends with an outlook on possible applications and open problems.

The talk is based on joint work with Michael Scheutzow (TU Berlin) and Reik Schottstedt (U Marburg).

I'll recall the quasi-Hamiltonian approach to moduli spaces of flat connections on Riemann surfaces, as a nice finite dimensional algebraic version of operations with loop groups such as fusion. Recently, whilst extending this approach to meromorphic connections, a new operation arose, which we will call "fission". As will be explained, this operation enables the construction of many new algebraic symplectic manifolds, going beyond those we were trying to construct.

We present some recent results on the metastability of continuous time Markov chains on finite sets using potential theory. This approach is applied to the case of supercritical zero range processes.

I will discuss the dynamical emergence of IR conformal invariance describing the low energy excitations of near-extremal R-charged global AdS${}_5$ black holes. To keep some non-trivial dynamics in the sector of ${\cal N}=4$ SYM captured by the near horizon limits describing these IR physics, we are lead to study large N limits in the UV theory involving near vanishing horizon black holes. I will consider both near-BPS and non-BPS regimes, emphasising the differences in the local AdS${}_3$ throats emerging in both cases. I will compare these results with the predictions obtained by Kerr/CFT, obtaining a natural quantisation for the central charge of the near-BPS emergent IR CFT describing the open strings stretched between giant gravitons.

In this work, we consider the hedging error due to discrete trading in models with jumps. We propose a framework enabling to

(asymptotically) optimize the discretization times. More precisely, a strategy is said to be optimal if for a given cost function, no strategy has

(asymptotically) a lower mean square error for a smaller cost. We focus on strategies based on hitting times and give explicit expressions for

the optimal strategies. This is joint work with Peter Tankov.

PLEASE NOTE THAT THIS SEMINAR HAS BEEN CANCELLED DUE TO ILLNESS.

Initial stage of the flow with a free surface generated by a vertical

wall moving from a liquid of finite depth in a gravitational field is

studied. The liquid is inviscid and incompressible, and its flow is

irrotational. Initially the liquid is at rest. The wall starts to move

from the liquid with a constant acceleration.

It is shown that, if the acceleration of the plate is small, then the

liquid free surface separates from the wall only along an

exponentially small interval. The interval on the wall, along which

the free surface instantly separates for moderate acceleration of the

wall, is determined by using the condition that the displacements of

liquid particles are finite. During the initial stage the original

problem of hydrodynamics is reduced to a mixed boundary-value problem

with respect to the velocity field with unknown in advance position of

the separation point. The solution of this

problem is derived in terms of complete elliptic integrals. The

initial shape of the separated free surface is calculated and compared

with that predicted by the small-time solution of the dam break

problem. It is shown that the free surface at the separation point is

orthogonal to the moving plate.

Initial acceleration of a dam, which is suddenly released, is calculated.

• Bifurcation from isolated eigenvalues of finite multiplicity of the linearisation.

• Pseudo-inverses and parametrices for paths of Fredholm operators of index zero.

• Detecting a change of orientation along such a path.

• Lyapunov-Schmidt reduction

I will discuss some of new concepts and objects of two-dimensional number theory: 

how the same object can be studied via low dimensional noncommutative theories or higher dimensional commutative ones, 

what is higher Haar measure and harmonic analysis and how they can be used in representation theory of non locally compact groups such as loop groups and Kac-Moody groups, 

how classical notions split into two different notions on surfaces on the example of adelic structures, 

what is the analogue of the double quotient of adeles on surfaces and how one

could approach automorphic functions in geometric dimension two.

The liquid crystal (LC) flow model is a coupling between

orientation (director field) of LC molecules and a flow field.

The model may probably be one of simplest complex fluids and

is very similar to a Allen-Cahn phase field model for

multiphase flows if the orientation variable is replaced by a

phase function. There are a few large or small parameters

involved in the model (e.g. the small penalty parameter for

the unit length LC molecule or the small phase-change

parameter, possibly large Reynolds number of the flow field,

etc.). We propose a C^0 finite element formulation in space

and a modified midpoint scheme in time which accurately

preserves the inherent energy law of the model. We use C^0

elements because they are simpler than existing C^1 element

and mixed element methods. We emphasise the energy law

preservation because from the PDE analysis point of view the

energy law is very important to correctly catch the evolution

of singularities in the LC molecule orientation. In addition

we will see numerical examples that the energy law preserving

scheme performs better under some choices of parameters. We

shall apply the same idea to a Cahn-Hilliard phase field model

where the biharmonic operator is decomposed into two Laplacian

operators. But we find that under our scheme non-physical

oscillation near the interface occurs. We figure out the

reason from the viewpoint of differential algebraic equations

and then remove the non-physical oscillation by doing only one

step of a modified backward Euler scheme at the initial time.

A number of numerical examples demonstrate the good

performance of the method. At the end of the talk we will show

how to apply the method to compute a superconductivity model,

especially at the regime of Hc2 or beyond. The talk is based

on a few joint papers with Chun Liu, Qi Wang, Xingbin Pan and

Roland Glowinski, etc.

We analyse the effect of a natural change to the time variable on the convergence of the Crank-Nicholson scheme when applied to the solution of the heat equation with Dirac delta function initial conditions. In the original variables, the scheme is known to diverge as the time step is reduced with the ratio (lambda) of the time step to space step held constant - the value of lambda controls how fast the divergence occurs. After introducing the square root of time variable we prove that the numerical scheme for the transformed PDE now always converges and that lambda controls the order of convergence, quadratic convergence being achieved for lambda below a critical value. Numerical results indicate that the time change used with an appropriate value of lambda also results in quadratic convergence for the calculation of gamma for a European call option without the need for Rannacher start-up steps. Finally, some results and analysis are presented for the effect of the time change on the calculation of the option value and greeks for the American put calculated by the penalty method with Crank-Nicholson time-stepping.

In the first part of the talk I will introduce a notion of convexity ($\mathcal{X}$-convexity) which applies to any given family of vector fields: the main model which we have in mind is the case of vector fields satisfying the H\"ormander condition.

Then I will give a PDE-characterization for $\mathcal{X}$-convex functions using a viscosity inequality for the intrinsic Hessian and I will derive bounds for the intrinsic gradient and intrinsic local Lipschitz-continuity for this class of functions.\\

In the second part of the talk I will introduce a notion of subdifferential for any given family of vector fields (namely $\mathcal{X}$-subdifferential) and show that a non empty $\mathcal{X}$-subdifferential at any point characterizes the class of $\mathcal{X}$-convex functions.

As application I will prove a Jensen-type inequality for $\mathcal{X}$-convex functions in the case of Carnot-type vector fields. {\em (Joint work with Martino Bardi)}.

More perspectives on spectra.

We present recent results of Dani Wise which tie together many of the themes of this term's jGGT meetings: hyperbolic and relatively hyperbolic groups, (in particular limit groups), graphs of spaces, 3-manifolds and right-angled Artin groups.
Following this, we make an attempt at explaining some of the methods, beginning with special non-positively curved cube complexes.

It is known that the minimum number of generators d(G^n) of the n-th direct power G^n of a non-trivial finite group G tends to infinity with n. This prompts the question: in which ways can the sequence {d(G^n)} tend to infinity? This question was first asked by Wiegold who almost completely answered it for finitely generated groups during the 70's. The question can then be generalised to any algebraic structure and this is still an open problem currently being researched. I will talk about some of the results obtained so far and will try to explain some of the methods used to obtain them, both for groups and for the more general algebraic structure setting.

Fractional differential equations are becoming increasingly used as a modelling tool for processes associated with anomalous diffusion or spatial heterogeneity. However, the presence of a fractional differential operator causes memory (time fractional) or nonlocality (space fractional) issues that impose a number of computational constraints. In this talk we discuss efficient, scalable techniques for solving fractional-in-space reaction diffusion equations combining the finite element method with robust techniques for computing the fractional power of a matrix times a vector. We shall demonstrate the methods on a number examples which show the qualitative difference in solution profiles between standard and fractional diffusion models.

In the talk I plan to overview several constructions for finite dimensional represenations of DAHA: construction via quantization of Hilbert scheme of points in the plane (after Gordon, Stafford), construction via quantum Hamiltonian reduction (after Gan, Ginzburg), monodromic construction (after Calaque, Enriquez, Etingof). I will discuss the relations of the constructions to the conjectures from the first lecture.

We shall discuss how the algebra norm can be used to identify structure in groups. No prior familiarity with the area will be assumed.