Past

We consider random walks on two classes of random graphs and explore the likely structure of the the set of unvisited vertices or vacant set. In both cases, the size of the vacant set $N(t)$ can be obtained explicitly as a function of $t$. Let $\Gamma(t)$ be the subgraph induced by the vacant set. We show that, for random graphs $G_{n,p}$ above the connectivity threshold, and for random regular graphs $G_r$, for constant $r\geq 3$, there is a phase transition in the sense of the well-known Erdos-Renyi phase transition. Thus for $t\leq (1-\epsilon)t^*$ we have a unique giant plus components of  size $O(\log n)$ and for $t\geq (1+\epsilon)t^*$ we have only components of  size $O(\log n)$.

In the case of $G_r$ we describe the likely degree sequence, size of the giant component and structure of the small ($O(\log n)$) size components.

Please note that this seminar has been cancelled due to unforeseen circumstances.

In this talk we show that there exists a smooth classical solution to the HJB equation for a large class of constrained problems with utility functions that are not necessarily differentiable or strictly concave.

The value function is smooth if admissible controls satisfy an integrability condition or if it is continuous on the closure of its domain.

The key idea is to work on the dual control problem and the dual HJB equation. We construct a smooth, strictly convex solution to the dual HJB equation and show that its conjugate function is a smooth, strictly concave solution to the primal HJB equation satisfying the terminal and boundary conditions

The derived category of a variety has (relatively) recently come into play as an invariant of the variety, useful as a tool for classification. As the derived category contains cohomological information about the variety, it is perhaps a natural question to ask how close the derived category is to the motive of a variety.

We will begin by briefly recalling Grothendieck's category of Chow motives of smooth projective varieties, recall the definition of Fourier-Mukai transforms, and state some theorems and examples. We will then discuss some conjectures of Orlov http://arxiv.org/abs/math/0512620, the most general of which is: does an equivalence of derived categories imply an isomorphism of motives?

Due to illness the speaker has been forced to postpone at short notice. A new date will be announced as soon as possible.

"We will talk on stability, simplicity, nip, etc of partial types. We will review some known results and we will discuss some open problems."

Flows involving immiscible liquids are encountered in a variety of industrial and natural processes. Recent applications in micro- and nano-fluidics have led to a significant scientific effort whose aim (among other aspects) is to enable theoretical predictions of the spatiotemporal dynamics of the interface(s) separating different flowing liquids. In such applications the scale of the system is small, and forces such as surface tension or externally imposed electrostatic forces compete and can, in many cases, surpass those of gravity and inertia. This talk will begin with a brief survey of applications where electrohydrodynamics have been used experimentally in micro-lithography, and experiments will be presented that demonstrate the use of electric fields in producing controlled encapsulated droplet formation in microchannels.

The main thrust of the talk will be theoretical and will mostly focus on the paradigm problem of the dynamics of electrified falling liquid films over topographically structured substrates.

Evolution equations will be developed asymptotically and their solutions will be compared to direct simulations in order to identify their practicality. The equations are rich mathematically and yield novel examples of dissipative evolutionary systems with additional effects (typically these are pseudo-differential operators) due to dispersion and external fields.

The models will be analysed (we have rigorous results concerning global existence of solutions, the existence of dissipative dynamics and an absorbing set, and analyticity), and accurate numerical solutions will be presented to describe the large time dynamics. It is found that electric fields and topography can be used to control the flow.Time permitting, I will present some recent results on transitions between convective to absolute instabilities for film flows over periodic topography.

he Iwasawa theory of elliptic curves over the rationals, and more
generally of modular forms, has mostly been studied with the
assumption that the form is "ordinary" at p -- i.e. that the Hecke
eigenvalue is a p-adic unit. When this is the case, the dual of the
p-Selmer group over the cyclotomic tower is a torsion module over the
Iwasawa algebra, and it is known in most cases (by work of Kato and
Skinner-Urban) that the characteristic ideal of this module is
generated by the p-adic L-function of the modular form.

I'll talk about the supersingular (good non-ordinary) case, where
things are slightly more complicated: the dual Selmer group has
positive rank, so its characteristic ideal is zero; and the p-adic
L-function is unbounded and hence doesn't lie in the Iwasawa algebra.
Under the rather restrictive hypothesis that the Hecke eigenvalue is
actually zero, Kobayashi, Pollack and Lei have shown how to decompose
the L-function as a linear combination of Iwasawa functions and
explicit "logarithm-like" series, and to modify the definition of the
Selmer group correspondingly, in order to formulate a main conjecture
(and prove one inclusion). I will describe joint work with Antonio Lei
and Sarah Zerbes where we extend this to general supersingular modular
forms, using methods from p-adic Hodge theory. Our work also gives
rise to new phenomena in the ordinary case: a somewhat mysterious
second Selmer group and L-function, which is related to the
"critical-slope L-function" studied by Pollack-Stevens and Bellaiche.


This talk is about the Induced Dimension Reduction (IDR) methods developed by Peter Sonneveld and, more recently, Martin van Gijzen. We sketch the history, outline the underlying principle, and give a few details about different points of view on this class of Krylov subspace methods. If time permits, we briefly outline some recent developments in this field and the benefits and drawbacks of these and IDR methods in general.

In this talk I want to ask how to create a coherent mathematical framework for pricing and hedging which starts with the information available in the market and does not assume a given probabilistic setup. This calls for re-definition of notions of arbitrage and trading and, subsequently, for a ``probability-free first fundamental theorem of asset pricing". The new setup should also link with a classical approach if our uncertainty about the model vanishes and we are convinced a particular probabilistic structure holds. I explore some recent results but, predominantly, I present the resulting open questions and problems. It is an ``internal talk" which does not necessarily present one paper but rather wants to engage into a discussion. Ideas for the talk come in particular from joint works with Alex Cox and Mark Davis.

Solutions which are time-bounded in L^3 up to time T can be continued

past this time, by a landmark result of Escauriaza-Seregin-Sverak,

extending Serrin's criterion. On the other hand, the local Cauchy

theory holds up to solutions in BMO^-1; we aim at describing how one

can obtain intermediate regularity results, assuming a priori bounds

in negative regularity Besov spaces.

This is joint work with J.-Y. Chemin, Isabelle Gallagher and Gabriel

Koch.

First of all, we are going to recall some basic facts and definitions about homogeneous Riemannian manifolds. Then we are going to talk about existence and non-existence of invariant Einstein metrics on compact homogeneous manifolds. In this context, we have that it is possible to associate to every homogeneous space a graph. Then, the graph theorem of Bohm, Wang and Ziller gives an existence result of invariant Einstein metrics on a compact homogeneous space, based on properties of its graph. We are going to discuss this theorem and sketch its proof.

Given a deterministically time-changed Brownian motion Z starting from 1, whose time-change V (t) satisfies V (t) > t for all t > 0, we perform an explicit construction of a process X which is Brownian motion in its own filtration and that hits zero for the first time at V (S), where S := inf {t > 0 : Z_t = 0}. We also provide the semimartingale decomposition of X under the filtration jointly generated by X and Z. Our construction relies on a combination of enlargement of filtration and filtering techniques. The resulting process X may be viewed as the analogue of a 3-dimensional Bessel bridge starting from 1 at time 0 and ending at 0 at the random time V (S). We call this a dynamic Bessel bridge since S is not known at time 0 but is slowly revealed in time by observing Z. Our study is motivated by insider trading models with default risk. (this is a joint work with Luciano Campi and Albina Danilova)

In 1968, Dixmier posed six problems for the algebra of polynomial

  differential operators, i.e. the Weyl algebra. In 1975, Joseph

solved the third and sixth problems and, in 2005, I solved the

  fifth problem and gave a positive solution to the fourth problem

  but only for homogeneous differential operators. The remaining three problems are still open. The first problem/conjecture of Dixmier (which is equivalent to the Jacobian Conjecture as was shown in 2005-07 by Tsuchimito, Belov and Kontsevich) claims that the Weyl algebra `behaves'

like a finite field. The first problem/conjecture of

  Dixmier:   is it true that an algebra endomorphism of the Weyl

  algebra an automorphism? In 2010, I proved that this question has

  an affirmative answer for the algebra of polynomial

  integro-differential operators. In my talk, I will explain the main

  ideas, the structure of the proof and recent progress on the first problem/conjecture of Dixmier.

A random planar graph $P_n$ is a graph drawn uniformly at random from the class of all (labelled) planar graphs on $n$ vertices. In this talk we show that with probability $1-o(1)$ the number of vertices of degree $k$ in $P_n$ is very close to a quantity $d_k n$ that we determine explicitly. Here $k=k(n) \le c \log n$. In the talk our main emphasis will be on the techniques for proving such results. (Joint work with Kosta Panagiotou.)