Past

The $C_\ell$-free process starts with the empty graph on $n$ vertices and adds edges chosen uniformly at random, one at a time, subject to the condition that no copy of $C_\ell$ is created. For every $\ell \geq 4$ we show that, with high probability as $n \to \infty$, the maximum degree is $O((n \log n)^{1/(\ell-1)})$, which confirms a conjecture of Bohman and Keevash and improves on bounds of Osthus and Taraz. Combined with previous results this implies that the $C_\ell$-free process typically terminates with $\Theta(n^{\ell/(\ell-1)}(\log n)^{1/(\ell-1)})$ edges, which answers a question of Erd\H{o}s, Suen and Winkler. This is the first result that determines the final number of edges of the more general $H$-free process for a non-trivial \emph{class} of graphs $H$. We also verify a conjecture of Osthus and Taraz concerning the average degree, and obtain a new lower bound on the independence number. Our proof combines the differential equation method with a tool that might be of independent interest: we establish a rigorous way to `transfer' certain decreasing properties from the binomial random graph to the $H$-free process.

If $X$ is a Fano variety with canonical bundle $O(-k)$, its derived category

has a semi-orthogonal decomposition (I will say what that means)

\[ D(X) = \langle O(-k+1), ..., O(-1), O, A \rangle, \]

where the subcategory $A$ is the "interesting piece" of $D(X)$. In the previous talk we saw that $A$ can have very rich geometry. In this talk we will see a less well-understood example of this: when $X$ is a smooth cubic in $P^5$, $A$ looks like the derived category of a K3 surface. We will discuss Kuznetsov's conjecture that $X$ is rational if and only if $A$ is geometric, relate it to Hassett's earlier work on the Hodge theory of $X$, and mention an autoequivalence of $D(Hilb^2(K3))$ that I came across while studying the problem.

There is a long-studied correspondence between intersections of two quadrics and hyperelliptic curves, first noticed by Weil and since used

as a testbed for many fashionable theories: Hodge theory, motives, and moduli of vector bundles in the '70s and '80s, derived categories in the '90s, non-commutative geometry and mirror symmetry today. The story generalizes to three, four, and more quadrics, exhibiting new geometric behaviour at each step. The case of four quadrics nicely illustrates the modern theory of flops and derivced categories and, as a special case, gives a pair of derived-equivalent Calabi-Yau 3-folds.

In this talk we will present some recent results on the long time

asymptotics of the generalized (non-Markovian) Langevin equation (gLE). In particular,

we will discuss about the ergodic properties of the gLE and present estimates on the rate of convergence to equilibrium, we will present

a homogenization result (invariance principle) and we will discuss

about the convergence of the gLE dynamics to the (Markovian) Langevin

dynamics, in some appropriate asymptotic limit. The analysis is based on the approximation of the gLE by a

high (and possibly infinite) dimensional degenerate Markovian system,

and on the analysis of the spectrum of the generator of this Markov

process. This is joint work with M. Ottobre and K. Pravda-Starov.

In this talk I describe some results obtained in collaboration with

J.F. Lafont and A. Sisto, which concern rigidity theorems for a class of

manifolds which are ``mostly'' non-positively curved, but may not support

any actual non-positively curved metric.

More precisely, we define a class of manifolds which contains

non-positively curved examples.

Building on techniques coming from geometric group theory, we show

that smooth rigidity holds within our class of manifolds

(in fact, they are also topologically rigid - i.e. they satisfy the Borel

conjecture - but this fact won't be discussed in my talk).

We also discuss some results concerning the quasi-isometry type of the

fundamental groups

of mostly non-positively curved manifolds.

A very general model of evolving graphs was introduced by Cooper and Frieze in 2003, and further analysed by Cooper. At each stage of the process, either a new edge is added
between existing vertices, or a new vertex is added and joined to some number of existing vertices. Each vertex gaining a new neighbour may be chosen either uniformly, or by preferential attachment, i.e., with probability proportional to the current degree.
It is known that the degrees of vertices in any such model follow a ``power law''. Here we study in detail the degree sequence of a graph obtained from such a procedure, looking at the vertices of large degree as well as the numbers of vertices of each fixed degree.
This is joint work with Graham Brightwell.

The talk will explain how a geometric principle gave rise to a new variational description of information-theoretic entropy and how this led to the solution of a problem dating back to the 50's: whether the the central limit theorem is driven by an analogue of the second law of thermodynamics.

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.