Past

In the Heston stochastic volatility model, the variance process is given by a mean-reverting square-root process. It is known that its transition probability density can be represented by a non-central chi-square density. There are fundamental differences in the behaviour of the variance process depending on the number of degrees of freedom: if the number of degrees of freedom is larger or equal to 2, the zero boundary is unattainable; if it is smaller than 2, the zero boundary is attracting and attainable.

We focus on the attainable zero boundary case and in particular the case when the number of degrees of freedom is smaller than 1, typical in foreign exchange markets. We prove a new representation for the density based on powers of generalized Gaussian random variables. Further we prove that Marsaglia's polar method extends to the generalized Gaussian distribution, providing an exact and efficient method for generalized Gaussian sampling. Thus, we establish a new exact and efficient method for simulating the Cox-Ingersoll-Ross process for an attracting and attainable zero boundary, and thus establish a new simple method for simulating the Heston model.

We demonstrate our method in the computation of option prices for parameter cases that are described in the literature as challenging and practically relevant.

Mathematical modelling of the movement of animals, micro-organisms and cells is of great relevance in the fields of biology, ecology and medicine. Movement models can take many different forms, but the most widely used are based on extensions of simple random walk processes. In this talk I will review some of the basic ideas behind the theory of random walks and diffusion processes and discuss how these models are used in the context of modelling animal movement. I will present several case studies, each of which is an extension or application of some of the simple random walk ideas discussed previously. Specifically, I will consider problems related to biased and correlated movements, path analysis of movement data, sampling and processing issues and the problem of determining movement processes from observed patterns. I will also discuss some biological examples of how these models can be used, including chemosensory movements and interactions between zooplankton and the movements of fish.

A $C${\em -relation} is the ternary relation induced by an ultrametric distance, in particular a valuation on a field, when we only remember the relation:

$C(x;y,z)$

iff $d(x,y)

We discuss the following question of Nick Katz and Frans Oort: Given an

Algebraically closed field K , is there an Abelian variety over K of

dimension g which is not isogenous to a Jacobian? For K the complex

numbers

its easy to see that the answer is yes for g>3 using measure theory, but

over a countable field like $\overline{\mbthbb{Q}}$ new methods are required. Building on

work

of Chai-Oort, we show that, as expected, such Abelian varieties exist for

$K=\overline{\mbthbb{Q}}$ and g>3 . We will explain the proof as well as its connection to

the

Andre Oort conjecture.

Consider the action of a complex reductive group on a complex projective variety X embedded in projective space. Geometric Invariant Theory allows us to construct a 'categorical' quotient of an open subset of X, called the semistable set. If in addition X is smooth then it is a symplectic manifold and in nice cases we can construct a moment map for the action and the Marsden-Weinstein reduction gives a symplectic quotient of the group action on an open subset of X. We will discuss both of these constructions and the relationship between the GIT quotient and the Marsden-Weinstein reduction. The quotients we have discussed provide a quotient for only an open subset of X and so we then go on to discuss how we can construct quotients of certain subvarieties contained in the complement of the semistable locus.

Optimisation problems involving objective functions defined on function spaces often have a natural interpretation as a variational problem, leading to a solution approach via calculus of variations. An equally natural alternative approach is to approximate the function space by a finite-dimensional subspace and use standard nonlinear optimisation techniques. The second approach is often easier to use, as the occurrence of absolute value terms and inequality constraints poses no technical problem, while the calculus of variations approach becomes very involved. We argue our case by example of two applications in mathematical finance: the computation of optimal execution rates, and pre-computed trade volume curves for high frequency trading.

I will give an introduction to Kashiwara's theory of crystal bases. Crystals are combinatorial objects associated to integrable modules for quantum groups that, together with the related notion of crystal bases, capture several combinatorial aspects of their representation theory.

I define what it means for a quantum

field theory to be asymptotically safe and

discuss possible applications to theories

of gravity and matter.

This will be a discussion on Stochastic Parameterisation, led by Hannah.

I will explain how to embed arbitrary RAAGs (Right Angled

Artin Groups) in Ham (the group of hamiltonian symplectomorphisms of

the 2-sphere). The proof is combination of topology, geometry and

analysis: We will start with embeddings of RAAGs in the mapping class

groups of hyperbolic surfaces (topology), then will promote these

embeddings to faithful hamiltonian actions on the 2-sphere (hyperbolic

geometry and analysis).

A basic example of  shear flow wasintroduced  by DiPerna and Majda to study the weaklimit of oscillatory solutions of the Eulerequations of incompressible ideal fluids. Inparticular, they proved by means of this examplethat weak limit of solutions of Euler equationsmay, in some cases, fail to be a solution of Eulerequations. We use this shear flow example toprovide non-generic, yet nontrivial, examplesconcerning the immediate loss of smoothness andill-posedness of solutions of the three-dimensionalEuler equations, for initial data that do notbelong to $C^{1,\alpha}$. Moreover, we show bymeans of this shear flow example the existence ofweak solutions for the three-dimensional Eulerequations with vorticity that is  having anontrivial density concentrated on non-smoothsurface. This is very different from what has beenproven for the two-dimensional Kelvin-Helmholtzproblem where a minimal regularity implies the realanalyticity of the interface. Eventually, we usethis shear flow to provide explicit examples ofnon-regular solutions of the three-dimensionalEuler equations that conserve the energy, an issuewhich is related to the Onsager conjecture.

This is a joint work with Claude Bardos.

The main aim of this talk will be to present a proof of the Tsen-Lang theorem on the existence of points on complete intersections and sketch a proof of the Grabber-Harris-Starr theorem giving the existence of a section to a fibration of a rationally connected variety over a curve. Time permitting, recent work of de Jong and Starr on rationally simply connected varieties will be discussed with applications to the number theory of hypersurfaces.

Surfaces of large genus are intriguing objects. Their geometry

has been studied by finding geometric properties that hold for all

surfaces of the same genus, and by finding families of surfaces with

unexpected or extreme geometric behavior. A classical example of this is

the size of systoles where on the one hand Gromov showed that there exists

a universal constant $C$ such that any (orientable) surface of genus $g$

with area normalized to $g$ has a homotopically non-trivial loop (a

systole) of length less than $C log(g)$. On the other hand, Buser and

Sarnak constructed a family of hyperbolic surfaces where the systole

roughly grows like $log(g)$. Another important example, in particular for

the study of hyperbolic surfaces and the related study of Teichmüller

spaces, is the study of short pants decompositions, first studied by Bers.

The talk will discuss two ideas on how to further the understanding of

surfaces of large genus. The first part will be about joint results with

F. Balacheff and S. Sabourau on upper bounds on the sums of lengths of

pants decompositions and related questions. In particular we investigate

how to find short pants decompositions on punctured spheres, and how to

find families of homologically independent short curves. The second part,

joint with L. Guth and R. Young, will be about how to construct surfaces

with large pants decompositions using random constructions.

Abstract: A random polytope $K_n$ is, by definition, the convex hull of $n$ random independent, uniform points from a convex body $K subset R^d$. The investigation of random polytopes started with Sylvester in 1864. Hundred years later R\'enyi and Sulanke began studying the expectation of various functionals of $K_n$, for instance number of vertices, volume, surface area, etc. Since then many papers have been devoted to deriving precise asymptotic formulae for the expectation of the volume of $K \setminus K_n$, for instance. But with few notable exceptions, very little has been known about the distribution of this functional. In the last couple of years, however, two breakthrough results have been proved: Van Vu has given tail estimates for the random variables in question, and M. Reitzner has obtained a central limit theorem in the case when $K$ is a smooth convex body. In this talk I will explain these new results and some of the subsequent development: upper and lower bounds for the variance, central limit theorems when $K$ is a polytope. Time permitting, I will indicate some connections lattice polytopes.

Abstract: We consider the inverse problem of finding the diffusion coefficient of a linear uniformly elliptic partial differential equation in divergence form, from noisy measurements of the forward solution in the interior. We adopt a Bayesian approach to the problem. We consider the prior measure on the diffusion coefficient to be either a Besov or Gaussian measure. We show that if the functions drawn from the prior are regular enough, the posterior measure is well-defined and Lipschitz continuous with respect to the data in the Hellinger metric. We also quantify the errors incurred by approximating the posterior measure in a finite dimensional space. This is joint work with Stephen Harris and Andrew Stuart.

There are nontrivial solutions of the incompressible Euler equations which are compactly supported in space and time. If they were to model the motion of a real fluid, we would see it suddenly start moving after staying at rest for a while, without any action by an external force. There are C1 isometric embeddings of a fixed flat rectangle in arbitrarily small balls of the three dimensional space. You should therefore be able to put a fairly large piece of paper in a pocket of your jacket without folding it or crumpling it. I will discuss the corresponding mathematical theorems, point out some surprising relations and give evidences that, maybe, they are not merely a mathematical game.

The static two price economy of conic finance is first employed to

define capital, profit, and subsequently return and leverage. Examples

illustrate how profits are negative on claims taking exposure to loss

and positive on claims taking gain exposure. It is argued that though

markets do not have preferences or objectives of their own, competitive

pressures lead markets to become capital minimizers or leverage

maximizers. Yet within a static context one observes that hedging

strategies must then depart from delta hedging and incorporate gamma

adjustments. Finally these ideas are generalized to a dynamic context

where for dynamic conic finance, the bid and ask price sequences are

seen as nonlinear expectation operators associated with the solution of

particular backward stochastic difference equations (BSDE) solved in

discrete time at particular tenors leading to tenor specific or

equivalently liquidity contingent pricing. The drivers of the associated

BSDEs are exhibited in complete detail.

   I will present a decidability result for theories of large fields of algebraic numbers, for example certain subfields of the field of totally real algebraic numbers. This result has as special cases classical theorems of Jarden-Kiehne, Fried-Haran-Völklein, and Ershov.

   The theories in question are axiomatized by Galois theoretic properties and geometric local-global principles, and I will point out the connections with the seminal work of Ax on the theory of finite fields.

Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this talk, two commonly used SSAs will be studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. The connections between SSAs and the deterministic models (based on reaction-diffusion PDEs) will be presented. I will consider chemical reactions both at a surface and in the bulk. I will show how the "microscopic" parameters should be chosen to achieve the correct "macroscopic" reaction rate. This choice is found to depend on which SSA is used. I will also present multiscale algorithms which use models with a different level of detail in different parts of the computational domain.

We consider the iterative solution of large sparse linear least squares (LS) problems. Specifically, we focus on the design and implementation of reliable stopping criteria for the widely-used algorithm LSQR of Paige and Saunders. First we perform a backward perturbation analysis of the LS problem. We show why certain projections of the residual vector are good measures of convergence, and we propose stopping criteria that use these quantities. These projections are too expensive to compute to be used directly in practice. We show how to estimate them efficiently at every iteration of the algorithm LSQR. Our proposed stopping criteria can therefore be used in practice.

This talk is based on joint work with Xiao-Wen Chang, Chris Paige, Pavel Jiranek, and Serge Gratton.

I will first introduce and motivate the notion of 'homological stability' for a sequence of spaces and maps. I will then describe a method of proving homological stability for configuration spaces of n unordered points in a manifold as n goes to infinity (and applications of this to sequences of braid groups). This method also generalises to the situation where the configuration has some additional local data: a continuous parameter attached to each point.

However, the method says nothing about the case of adding global data to the configurations, and indeed such configuration spaces rarely do have homological stability. I will sketch a proof -- using an entirely different method -- which shows that in some cases, spaces of configurations with additional global data do have homological stability. Specifically, this holds for the simplest possible global datum for a configuration: an ordering of the points up to even permutations. As a corollary, for example, this proves homological stability for the sequence of alternating groups.

Many problems in computer science can be modelled as metric spaces, whereas for mathematicians they are more likely to appear as the opening question of a second year examination. However, recent interesting results on the geometry of finite metric spaces have led to a rethink of this position. I will describe some of the work done and some (hopefully) interesting and difficult open questions in the area.

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

We first present the localized virtual cycles by cosections of obstruction sheaves constructed by Kiem and Li. This construction has two kinds of applications: one is define invariants for non-proper moduli spaces; the other is to reduce the obstruction classes. We will present two recent applications of this construction: one is the Gromov-Witten invariants of stable maps with fields (joint work with Chang); the other is studying Donaldson-Thomas invariants of Calabi-Yau threefolds (joint work with Kiem).

Abstract: Flagella and cilia are ubiquitous in biology as a means of motility and critical for male gametes migration in reproduction, to mucociliary clearance in the lung, to the virulence of devastating parasitic pathogens such as the Trypanosomatids, to the filter feeding of the choanoflagellates, which are constitute a critical link in the global food chain. Despite this ubiquity and importance, the details of how the ciliary or flagellar waveform emerges from the underlying mechanics and how the cell, or the environs, may control the beating pattern by regulating the axoneme is far from fully understood. We demonstrate in this talk that mechanics and modelling can be utilised to interpret observations of axonemal dynamics, swimming trajectories and beat patterns for flagellated motility impacts on the science underlying numerous areas of reproductive health, disease and marine ecology. It also highlights that this is a fertile and challenging area of inter-disciplinary research for applied mathematicians and demonstrates the importance of future observational and theoretical studies in understanding the underlying mechanics of these motile cell appendages.

The purpose of this talk is to provide a large class of examples of large initial data which gives rise to a global smooth solution. We shall explain what we mean by large initial data. Then we shall explain the concept of slowly varying function and give some flavor of the proofs of global existence.

The signature of the path is an essential object in rough path theory which takes value in tensor algebra and it is anticipated that the expected signature of Brownian motion might characterize the rough path measure of Brownian path itself. In this presentation we study the expected signature of a Brownian path in a Bananch space E stopped at the first exit time of an arbitrary regular domain, although we will focus on the case E=R^{2}. We prove that such expected signature of Brownian motion should satisfy one particular PDE and using the PDE for the expected signature and the boundary condition we can derive each term of expected signature recursively. We expect our method to be generalized to higher dimensional case in R^{d}, where d is an integer and d >= 2.

All this is related to my ongoing research projects with Chris Douglas and Arthur Bartels, in which we investigate conformal nets from a category
theoretical
perspective.

Abstract: We construct solutions to Burgers type equations perturbed by a multiplicative

space-time white noise in one space dimension. Due to the roughness of the driving noise, solutions are not regular enough to be amenable to classical methods. We use the theory of controlled rough paths to give a meaning to the spatial integrals involved in the definition of a weak solution. Subject to the choice of the correct reference rough path, we prove unique solvability for the equation. We show that our solutions are stable under smooth approximations of the driving noise. A more general class of approximations will also be discussed. This is joint work with Martin Hairer and Jan Maas.

We present theory and numerics of affine processes and several of their applications in finance. The theory is appealing due to methods from probability theory, analysis and geometry. Applications are diverse since affine processes combine analytical tractability with a high flexibility to model stylized facts like heavy tails or stochastic volatility.