Past

: Recently strict local martingales have been used to model

exchange rates. In such models, put-call parity does not hold if one

assumes minimal superreplicating costs as contingent claim prices. I

will illustrate how put-call parity can be restored by changing the

definition of a contingent claim price.

More precisely, I will discuss a change of numeraire technique when the

underlying is only a local martingale. Then, the new measure is not

necessarily equivalent to the old measure. If one now defines the price

of a contingent claim as the minimal superreplicating costs under both

measures, then put-call parity holds. I will discuss properties of this

new pricing operator.

To illustrate this techniques, I will discuss the class of "Quadratic

Normal Volatility" models, which have drawn much attention in the

financial industry due to their analytic tractability and flexibility.

This talk is based on joint work with Peter Carr and Travis Fisher.

We will explain Bridgelands results on the stabiltiy manifold of a K3 surface. As an application we will define the stringy Kaehler moduli space of a K3 surface and comment on the mirror symmetry picture.

I will begin by defining the notion of a characteristic class of surface bundles, and constructing the MMM (Miller-Morita-Mumford) classes as examples. I will then talk about a recent theorem of Church, Farb, and Thibault which shows that the characteristic numbers associated to certain MMM-classes do not depend on how the total space is fibred as a surface bundle - they depend only on the topology of the total space itself. In particular they don't even depend on the genus of the fibre. Hence there are many 'coincidences' between the characteristic numbers of very different-looking surface bundles.

A corollary of this is an obstruction to low-genus fiberings: given a smooth manifold E, the non-vanishing of a certain invariant of E implies that any surface bundle with E as its total space must have a fibre with genus greater than a certain lower bound.

Also, following the paper of Church-Farb-Thibault, I will sketch how to construct examples of 4-manifolds which fibre in two distinct ways as a surface bundle over another surface, thus giving concrete examples to which the theorem applies.

There exist two constructions of families of exotic monotone Lagrangian tori in complex projective spaces and products of spheres, namely the one by Chekanov and Schlenk, and the one via the Lagrangian circle bundle construction of Biran. It was conjectured that these constructions give Hamiltonian isotopic tori. I will explain why this conjecture is true in the complex projective plane and the product of two two-dimensional spheres.

Hex was discovered independently by Piet Hein in Copenhagen in 1942 and byJohn Nash in Princeton in 1948.  The game is interesting because its rules are very simple, yet it is not known how to play best possible.  For example, a winning first move for the first player (who does have  a winning strategy) is still unknown. The talk will tell the history of the game and give simple proofs for basic results about it. Also the reverse game of HEX,sometimes called REX, will be discussed. New results about REX are under publication in Discrete Mathematics in a paper:  How to play Reverse Hex (joint work with Ryan Hayward and Phillip Henderson).

Quaternionic quantum Hamiltonians describing nonrelativistic spin particles

require the ambient physical space to have five dimensions. The quantum

dynamics of a spin-1/2 particle system characterised by a generic such

Hamiltonian is described. There exists, within the structure of quaternionic

quantum mechanics, a canonical reduction to three spatial dimensions upon

which standard quantum theory is retrieved. In this dimensional reduction,

three of the five dynamical variables oscillate around a cylinder, thus

behaving in a quasi one-dimensional manner at large distances. An analogous

mechanism exists in the case of octavic Hamiltonians, where the ambient

physical space has nine dimensions. Possible experimental tests in search

for the signature of extra dimensions at low energies are briefly discussed.

(Talk based on joint work with Eva-Maria Graefe, Imperial.)

This is a joint work with Craig Evans. We study the partial regularity of minimizers for certain functionals in the calculus of variations, namely the modified Landau-de Gennes energy functional in nematic liquid crystal theory introduced by Ball and Majumdar.

We describe various approaches to the problem of expressing a polynomial $f(x) = \sum_{i=0}^{m} a_i x^i$ in terms of a different radix $r(x)$ as $f(x) = \sum_{j=0}^{n} b_j(x) r(x)^j$ with $0 \leq \deg(b_j) < \deg(r)$. Two approaches, the naive repeated division by $r(x)$ and the divide and conquer strategy, are well known. We also describe an approach based on the use of precomputed Newton inverses, which appears to offer significant practical improvements. A potential application of interest to number theorists is the fibration method for point counting, in current implementations of which the runtime is typically dominated by radix conversions.

In this talk I'll explain how to build CAT(0) cube complexes and construct Lipschitz maps between them. The existence of suitable Lipschitz maps is used to prove that the asymptotic dimension of a

CAT(0) cube complex is no more than its dimension.

Planar maps are graphs embedded in the plane, considered up to continuous deformation. They have been studied extensively in combinatorics, and they have also significant geometrical applications. Particular cases of planar maps are p-angulations, where each face (meaning each component of the complement of edges) has exactly p adjacent edges. Random planar maps have been used in theoretical physics, where they serve as models of random geometry.Our goal is to discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces.More precisely, we consider a random planar map M(n) which is uniformly distributed over the set of all p-angulations with n vertices. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power -1/4. Both in the case p=3 and when p>3 is even, we prove that the resulting random metric spaces converge as n tends to infinity to a universal object called the Brownian map. This convergence holds in the sense of the Gromov-Hausdorff distance between compact metric spaces. In the particular case of triangulations (p=3), this solves an open problem stated by Oded Schramm in his 2006 ICM paper. As a key tool, we use bijections between planar maps and various classes of labeled trees

In this talk I will describe Toda's results on deformations of the category Coh(X) of coherent sheaves on a complex manifold X. They come from deformations of X as a complex manifold, non-commutative deformations, and gerby deformations (which can all be interpreted as deformations of X as a generalized complex manifold). Toda also described how to deform Fourier-Mukai equivalences, and I will present some examples coming from mirror SYZ fibrations.

We consider nonlinear stochastic wave equations in dimension d\le 3.

Using Malliavin Calculus, we give upper bounds for the small eigenvalues of the inverse of two point densities.These provide a rate of degeneracy when points go close to each other.  Then, we analyze the consequences of this result on lower estimates for hitting probabilities. 

String compactifications incorporating non-vanishing H-flux have received increased attention over the past decade for their potential relevance to the moduli stabilization problem. Their internal spaces are in general not Kähler and, therefore, not Calabi-Yau. In the heterotic string an important technical problem is to construct gauge bundles on such spaces. I will present a method of how to explicitly construct gauge bundles over homogeneous nearly-Kähler manifolds of dimension six and discuss some of the arising implications for model building.

Studies of the ocean circulation and climate have come to be dominated by the results of complex numerical models encompassing hundreds of thousands of lines of computer code and whose physics may be more difficult to penetrate than the real system. Some insight into the large-scale ocean circulation can perhaps be gained by taking a step back and considering the gross time scales governing oceanic changes. These can derived from a wide variety of simple considerations such as energy flux rates, signal velocities, tracer equilibrium times, and others. At any given time, observed changes are likely a summation of shifts taking place over all of these time scales.

Abstract: In this paper we deal with a utility maximization problem at finite horizon on a continuous-time market with conical (and time varying) constraints (particularly suited to model a currency market with proportional transaction costs). In particular, we extend the results in \cite{CO} to the situation where the agent is initially endowed with a random and possibly unbounded quantity of assets. We start by studying some basic properties of the value function (which is now defined on a space of random variables), then we dualize the problem following some convex analysis techniques which have proven very useful in this field of research. We finally prove the existence of a solution to the dual and (under an additional boundedness assumption on the endowment) to the primal problem. The last section of the paper is devoted to an application of our results to utility indifference pricing. This is a joint work with G. Benedetti (CREST).

After a short introduction to homogeneous relational structures (structures such that all local symmetries are global), I will discuss some different topics relating homogeneity to homomorphisms: a family of notions of 'homomorphism-homogeneity' that generalise homogeneity; generic endomorphisms of homogeneous structures; and constraint satisfaction problems.

Using the spectral multiplicities of the standard torus, we
endow the Laplace eigenspaces with Gaussian probability measures.
This induces a notion of random Gaussian eigenfunctions
on the torus ("arithmetic random waves''.)  We study the
distribution of the nodal length of random Laplace eigenfunctions for high
eigenvalues,and our primary result is that the asymptotics for the variance is
non-universal, and is intimately related to the arithmetic of
lattice points lying on a circle with radius corresponding to the
energy. This work is joint with Manjunath Krishnapur and Par Kurlberg

Three-wave interactions form the basis of our understanding of many

nonlinear pattern forming systems because they encapsulate the most basic

nonlinear interactions. In problems with two comparable length scales, such

as the Faraday wave experiment with multi-frequency forcing, consideration

of three-wave interactions can explain the presence of the spatio-temporal

chaos found in some experiments, enabling some previously unexplained

results to be interpreted in a new light. The predictions are illustrated

with numerical simulations of a model partial differential equation.

When modelling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation behaviour.

The stochastic description of this problem gives rise to a multi-dimensional Markov jump process, which can be approximated by a system of stochastic differential equations. Long-time behaviour of the process can be better understood by looking at the steady-state solution of the corresponding Fokker-Planck equation.

In this talk we consider a new finite element method which uses simulated trajectories of the Markov-jump process to inform the choice of mesh in order to approximate this invariant distribution. The method has been implemented for systems in 3 dimensions, but we shall also consider systems of higher dimension.

We will describe the space of Bridgeland stability conditions

of the derived category of some CY3 algebras of quivers drawn on the

Riemann sphere. We give a biholomorphic map from the upper-half plane to

the space of stability conditions lifting the period map of a meromorphic

differential on a 1-dimensional family of elliptic curves. The map is

equivariant with respect to the actions of a subgroup of $\mathrm{PSL}(2,\mathbb Z)$ on the

left by monodromy of the rational elliptic surface and on the right by

autoequivalences of the derived category.

The complement of a divisor in the rational elliptic surface can be

identified with Hitchin's moduli space of connections on the projective

line with prescribed poles of a certain order at marked points. This is

the space of initial conditions of one of the Painleve equations whose

solutions describe isomonodromic deformations of these connections.

This is joint work with Uri Onn. We use motivic integration to get the growth rate of the sequence consisting of the number of conjugacy classes in quotients of G(O) by congruence subgroups, where $G$ is suitable algebraic group over the rationals and $O$ the ring of integers of a number field.

The proof uses tools from the work of Nir Avni on representation growth of arithmetic groups and results of Cluckers and Loeser on motivic rationality and motivic specialization.

Successful navigation through a complicated and evolving environment is a fundamental task carried out by an enormous range of organisms, with migration paths staggering in their length and intricacy. Selecting a path requires the detection, processing and integration of a myriad of cues drawn from the surrounding environment and in many instances it is the intrinsic orientation of the environment that provides a valuable navigational aid.

In this talk I will describe the use of transport models to describe migration in oriented environments, and demonstrate the scaling approaches that allow us to derive macroscopic models for movement.

I will illustrate the methods through a number of apposite examples, including the migration of cells in the extracellular matrix, the macroscopic growth of brain tumours and the movement of wolves in boreal forest.

Motivated by the study of micro-vascular disease, we have been investigating the relationship between the structure of capillary networks and the resulting blood perfusion through the muscular walls of the heart. In order to derive equations describing effective fluid transport, we employ an averaging technique called homogenisation, based on a separation of length scales. We find that the tissue-scale flow is governed by Darcy's Law, whose coefficients we are able to explicitly calculate by averaging the solution of the microscopic capillary-scale equations. By sampling from available data acquired via high-resolution imaging of the coronary capillaries, we automatically construct physiologically-realistic vessel networks on which we then numerically solve our capillary-scale equations. By validating against the explicit solution of Poiseuille flow in a discrete network of vessels, we show that our homogenisation method is indeed able to efficiently capture the averaged flow properties.

We prove existence of a global semigroup of conservative solutions of the nonlinear variational wave equation $u_{tt}-c(u) (c(u)u_x)_x=0$. The equation was derived by Saxton as a model for liquid crystals. This equation shares many of the peculiarities of the Hunter–Saxton and the Camassa–Holm equations. In particular, the equation possesses two distinct classes of solutions denoted conservative and dissipative. In order to solve the Cauchy problem uniquely it is necessary to augment the equation properly. In this talk we describe how this is done for conservative solutions. The talk is based on joint work with X. Raynaud.

The theory of modular forms owes in many ways lots of its results to the existence of the Hecke operators and their nice properties. However, even acting on modular forms of level 1, lots of basic questions remain unresolved. We will describe and prove some known properties of the Hecke operators, and state Maeda's conjecture. This conjecture, if true, has many deep consequences in the theory. In particular, we will indicate how it implies the nonvanishing of certain L-functions.

The notion of an E-infinity ring spectrum arose about thirty years ago,

and was studied in depth by Peter May et al, then later reinterpreted

in the framework of EKMM as equivalent to that of a commutative S-algebra.

A great deal of work on the existence of E-infinity structures using

various obstruction theories has led to a considerable enlargement of

the body of known examples. Despite this, there are some gaps in our

knowledge. The question that is a major motivation for this talk is

`Does the Brown-Peterson spectrum BP for a prime p admit an E-infinity

ring structure?'. This has been an important outstanding problem for

almost four decades, despite various attempts to answer it.

I will explain what BP is and give a brief history of the above problem.

Then I will discuss a construction that gives a new E-infinity ring spectrum

which agrees with BP if the latter has an E-infinity structure. However,

I do not know how to prove this without assuming such a structure!

We define a notion of discrete Ricci curvature for a metric measure space by looking at whether "small balls are closer than their centers are". In a Riemannian manifolds this gives back usual Ricci curvature up to scaling. This definition is very easy to apply in a series of examples such as graphs (eg the discrete cube has positive curvature). We are able to generalize several Riemannian theorems in positive curvature, such as concentration of measure and the log-Sobolev inequality. This definition also allows to prove new theorems both in the Riemannian and discrete case: for example improved bounds on spectral gap of the Laplace-Beltrami operator, and fast convergence results for some Markov Chain Monte Carlo methods

In this talk we show how the computation of the group of components of Prym varieties of spectral covers leads to cohomological results on the moduli space of stable bundles originally due to Harder-Narasimhan. This is joint work with Christian Pauly.

We establish a large deviations principle for the block sizes of a uniformly random non-crossing partition. As an application we obtain a variational formula for the maximum of the support of a compactly supported probability measure in terms of its free cumulants, provided these are all non-negative. This is useful in free probability theory, where sometimes the R-transform is known but cannot be inverted explicitly to yield the density.

Four-dimensional (4d) supergravities with non-geometric terms in their potential are very promising models for phenomenology. Indeed, these terms, generated by so-called non-geometric fluxes, generically help to obtain de Sitter vacua, or to stabilise moduli. Unfortunately, deriving these theories from a compactified ten-dimensional (10d) supergravity has not been achieved so far. One reason is that non-geometric fluxes do not seem to match any 10d field, and another reason is the appearance of global issues in 10d non-geometric configurations.

After reviewing some background material, we present in this talk a solution to the two previous issues. Thanks to a field redefinition, we make the non-geometric Q-flux appear in a 10d action, which only differs from the NSNS action by a total derivative. In addition, this new action is globally well-defined, at least in some examples, and one can then perform the dimensional reduction to recover the 4d non-geometric potential. We also mention an application to the heterotic string.

Based on 1106.4015.