Past

We will give an introduction to the theory of d-manifolds, a new class of geometric objects recently/currently invented by Joyce (see http://people.maths.ox.ac.uk/joyce/dmanifolds.html). We will start from scratch, by recalling the definition of a 2-category and talking a bit about $C^\infty$-rings, $C^\infty$-schemes and d-spaces before giving the definition of what a d-manifold should be. We will then discuss some properties of d-manifolds, and say some words about d-manifold bordism and its applications.

We will start off with a crash course in General relativity, and then I'll describe a 'recipe' for a time machine. This will lead us to the question whether or not the topology of the universe can change. We will see that, in some sense, this is topologically allowed. However, the Einstein equation gives a certain condition on the Ricci tensor (which is violated by certain quantum effects) and meeting this condition is a more delicate problem.

In biological cells, molecules are transported actively or by diffusion and react with each other when they are close.

The reactions occur with certain probability and there are few molecules of some chemical species. Therefore, a stochastic model is more accurate compared to a deterministic, macroscopic model for the concentrations based on partial differential equations.

At the mesoscopic level, the domain is partitioned into voxels or compartments. The molecules may react with other molecules in the same voxel and move between voxels by diffusion or active transport. At a finer, microscopic level, each individual molecule is tracked, it moves by Brownian motion and reacts with other molecules according to the Smoluchowski equation. The accuracy and efficiency of the simulations are improved by coupling the two levels and only using the micro model when it is necessary for the accuracy or when a meso description is unknown.

Algorithms for simulations with the mesoscopic, microscopic and meso-micro models will be described and applied to systems in molecular biology in three space dimensions.

Cell motility is a crucial part of many biological processes including wound healing, immunity and embryonic development. The interplay between mechanical forces and biochemical control mechanisms make understanding cell motility a rich and exciting challenge for mathematical modelling. We consider the two-phase, poroviscous, reactive flow framework used in the literature to describe crawling cells and present a stripped down version. Linear stability analysis and numerical simulations provide insight into the onset of polarization of a stationary cell and reveal qualitatively distinct families of travelling wave solutions. The numerical solutions also capture the experimentally observed behaviour that cells crawl fastest when the surface they crawl over is neither too sticky nor too slippy.

We introduce the notion of a real cubing. Roughly speaking, real cubings are to CAT(0) cube complexes what real trees are to simplicial trees. We develop an analogue of the Rips’ machine and establish the structure of groups acting nicely on real cubings.

To identify the Martin boundary for a transient Markov chain with Green's function G(x,y), one has to identify all possible limits Lim G(x,y_n)/G(0,y_n) with y_n "tending to infinity". For homogeneous random walks, these limits are usually obtained from the exact asymptotics of Green's function G(x,y_n). For non-homogeneous random walks, the exact asymptotics af Green's function is an extremely difficult problem. We discuss several examples where Martin boundary can beidentified by using large deviation technique. The minimal Martin boundary is in general not homeomorphic to the "radial"  compactification obtained by Ney and Spitzer for homogeneous random walks in Z^d : convergence of a sequence of points y_n toa point on the Martin boundary does not imply convergence of the sequence y_n/|y_n| on the unit sphere. Such a phenomenon is a consequence of non-linear optimal large deviation trajectories.

Symplectic implosion is a construction in symplectic geometry due to Guillemin, Jeffrey and Sjamaar, which is related to geometric invariant theory for non-reductive group actions in algebraic geometry. This talk (based on joint work in progress with Andrew Dancer and Andrew Swann) is concerned with an analogous construction in hyperkahler geometry.

 In the recent series of papers Kleban, Simmons, and Ziff gave a non-rigorous computation  (base on Conformal Field Theory) of probabilities of several connectivity events for critical percolation. In particular they showed that the probability that there is a percolation cluster connecting two points on the boundary and a point inside the domain can be factorized in therms of pairwise connection probabilities. We are going to use SLE techniques to rigorously compute probabilities of several connectivity events and prove the factorization formula.

We compute three-point functions of single trace operators in planar N = 4 SYM. We consider the limit where one of the operators is much smaller than the other two. We find a precise match between weak and strong coupling in the Frolov-Tseytlin classical limit for a very general class of classical solutions. To achieve this match we clarify the issue of back-reaction and identify precisely which three-point functions are captured by a classical computation.

The standard approach to continuous-time finance starts from postulating a

statistical model for the prices of securities (such as the Black-Scholes

model). Since such models are often difficult to justify, it is

interesting to explore what can be done without any stochastic

assumptions. There are quite a few results of this kind (starting from

Cover 1991 and Hobson 1998), but in this talk I will discuss

probability-type properties emerging without a statistical model. I will

only consider the simplest case of one security, and instead of stochastic

assumptions will make some analytic assumptions. If the price path is

known to be cadlag without huge jumps, its quadratic variation exists

unless a predefined trading strategy earns infinite capital without

risking more than one monetary unit. This makes it possible to apply the

known results of Ito calculus without probability (Follmer 1981, Norvaisa)

in the context of idealized financial markets. If, moreover, the price

path is known to be continuous, it becomes Brownian motion when physical

time is replaced by quadratic variation; this is a probability-free

version of the Dubins-Schwarz theorem.

(Joint with Ronnie Nagloo.) I investigate algebraic relations between sets of solutions (and their derivatives) of the "generic" Painlev\'e equations I-VI, proving a somewhat weaker version of ``there are NO algebraic relations".

In this talk I shall discuss special polynomials associated with rational solutions of the Painlevé equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations. Further I shall illustrate applications of these polynomials to vortex dynamics and rogue waves.

The Painlevé equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, and have arisen in a variety of physical applications. Further the Painlevé equations may be thought of as nonlinear special functions. Rational solutions of the Painlevé equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the fourth Painlevé equation these polynomials are known as the generalized Hermite polynomials and generalized Okamoto polynomials. The locations of the roots of these polynomials have a highly symmetric (and intriguing) structure in the complex plane.

It is well known that soliton equations have symmetry reductions which reduce them to the Painlevé equations, e.g. scaling reductions of the Boussinesq and nonlinear Schrödinger equations are expressible in terms of the fourth Painlevé equation. Hence rational solutions of these equations can be expressed in terms of the generalized Hermite and generalized Okamoto polynomials.

I will also discuss the relationship between vortex dynamics and properties of polynomials with roots at the vortex positions. Classical polynomials such as the Hermite and Laguerre polynomials have roots which describe vortex equilibria. Stationary vortex configurations with vortices of the same strength and positive or negative configurations are located at the roots of the Adler-Moser polynomials, which are associated with rational solutions of the Kortweg-de Vries equation.

Further, I shall also describe some additional rational solutions of the Boussinesq equation and rational-oscillatory solutions of the focusing nonlinear Schrödinger equation which have applications to rogue waves.

The class of finite dimensional algebras over an algebraically closed field K

may be divided into two disjoint subclasses (tame and wild dichotomy).

One class

consists of the tame algebras for which the indecomposable modules

occur, in each dimension d, in a finite number of discrete and a

finite number of one-parameter families. The second class is formed by

the wild algebras whose representation theory comprises the

representation theories of all finite dimensional algebras over K.

Hence, the classification of the finite dimensional modules is

feasible only for the tame algebras. Frequently, applying deformations

and covering techniques, we may reduce the study of modules over tame

algebras to that for the corresponding simply connected tame algebras.

We shall discuss the problem concerning connection between the

tameness of simply connected algebras and the weak nonnegativity of

the associated Tits quadratic forms, raised in 1975 by Sheila Brenner.

We will study the question of whether the adjoint of a given matrix can be written as a rational function in the matrix. After showing necessary and sufficient conditions, rational interpolation theory will help to characterize the most important existing cases. Several topics related to our question will be explored. They range from short recurrence Krylov subspace methods to the roots of harmonic polynomials and harmonic rational functions. The latter have recently found interesting applications in astrophysics, which will briefly be discussed as well.