L6

Answering a question of Milnor, Grigorchuk constructed in the early eighties the
first examples of groups of intermediate growth, that is, finitely generated
groups with growth strictly between polynomial and exponential.
In  joint work with Laurent Bartholdi, we show that under a mild regularity assumption, any function greater than exp(n^a), where `a' is a solution of the equation
  2^(3-3/x)+ 2^(2-2/x)+2^(1-1/x)=2,
is a growth function of some group. These are the first examples of groups
of intermediate growth where the asymptotic of  the growth function is known.
Among applications of our results is the fact that any group of locally subexponential growth
can be embedded as a subgroup of some group of intermediate growth (some of these latter groups cannot be  subgroups in Grigorchuk groups).

In a recent work with Tianyi Zheng, we  provide  near optimal lower bounds
for Grigorchuk torsion groups, including the first Grigorchuk group. Our argument is by a construction of random walks with non-trivial Poisson boundary, defined by 
a measure with power law decay.

Recent tools make it possible to partition the space of rational Dehn 
surgery slopes for a knot (or in some cases a link) in a 3-manifold into 
domains over which the Heegaard Floer homology of the surgered manifolds 
behaves continuously as a function of slope. I will describe some 
techniques for determining the walls of discontinuity separating these 
domains, along with efforts to interpret some aspects of this structure 
in terms of the behaviour of co-oriented taut foliations. This talk 
draws on a combination of independent work, previous joint work with 
Jake Rasmussen, and work in progress with Rachel Roberts.

I will prove that the knot Floer homology group
HFK-hat(K, g-1) for a genus g fibered knot K is isomorphic to a
variant of the fixed points Floer homology of an area-preserving
representative of its monodromy. This is a joint work with Gilberto
Spano.
 

(joint work with Françoise Dal'Bo and Andrea Sambusetti)

Given a finitely generated group G acting properly on a metric space X, the exponential growth rate of G with respect to X measures "how big" the orbits of G are. If H is a subgroup of G, its exponential growth rate is bounded above by the one of G. In this work we are interested in the following question: what can we say if H and G have the same exponential growth rate? This problem has both a combinatorial and a geometric origin. For the combinatorial part, Grigorchuck and Cohen proved in the 80's that a group Q = F/N (written as a quotient of the free group) is amenable if and only if N and F have the same exponential growth rate (with respect to the word length). About the same time, Brooks gave a geometric interpretation of Kesten's amenability criterion in terms of the bottom of the spectrum of the Laplace operator. He obtained in this way a statement analogue to the one of Grigorchuck and Cohen for the deck automorphism group of the cover of certain compact hyperbolic manifolds. These works initiated many fruitful developments in geometry, dynamics and group theory. We focus here one the class of Gromov hyperbolic groups and propose a framework that encompasses both the combinatorial and the geometric point of view. More precisely we prove that if G is a hyperbolic group acting properly co-compactly on a metric space X which is either a Cayley graph of G or a CAT(-1) space, then the growth rate of H and G coincide if and only if H is co-amenable in G.  In addition if G has Kazhdan property (T) we prove that there is a gap between the growth rate of G and the one of its infinite index subgroups.

The Fujita equation $u_{t}=\Delta u+u^{p}$, $p>1$, has been a canonical blow-up model for more than half a century. A great deal is known about the singularity formation under a variety of conditions. In particular we know that blow-up behaviour falls broadly into two categories, namely Type I and Type II. The former is generic and stable while the latter is rare and highly unstable. One of the central results in the field states that in the Sobolev subcritical regime, $1<p<\frac{n+2}{n-2}$, $n\geq 3$, only type I is possible whenever the domain is \emph{convex} in $\mathbb{R}^n$. Despite considerable effort the requirement of convexity has not been lifted and it is not clear whether this is an artefact of the methodology or whether the geometry of the domain may actually affect the blow-up type. In my talk I will discuss how the question of the blow-up type for non-convex domains is intimately related to the validity of some Li-Yau-Hamilton inequalities.

We would like to introduce a few different setups of martingale optimal transport problem in dimension greater than one. Optimal correlations of martingales become an interesting issue, and each problem has its own unique feature in this regard

Joint work with Abdul-Lateef Haji-Ali

This talk will discuss efficient numerical methods for estimating the probability of a large portfolio loss, and associated risk measures such as VaR and CVaR. These involve nested expectations, and following Bujok, Hambly & Reisinger (2015) we use the number of samples for the inner conditional expectation as the key approximation parameter in the Multilevel Monte Carlo formulation. The main difference in this case is the indicator function in the definition of the probability. Here we build on previous work by Gordy & Juneja (2010) who analyse the use of a fixed number of inner samples, and Broadie, Du & Moallemi (2011) who develop and analyse an adaptive algorithm. I will present the algorithm, outline the main theoretical results and give the numerical results for a representative model problem. I will also discuss the extension to real portfolios with a large number of options based on multiple underlying assets.

The local volatility model is a celebrated model widely used for pricing and hedging financial derivatives. While the model’s main appeal is its capability of reproducing any given surface of observed option prices—it provides a perfect fit—the essential component of the model is a latent function which can only be unambiguously determined in the limit of infinite data. To (re)construct this function, numerous calibration methods have been suggested involving steps of interpolation and extrapolation, most often of parametric form and with point-estimates as result. We seek to look at the calibration problem in a probabilistic framework with a nonparametric approach based on Gaussian process priors. This immediately gives a way of encoding prior believes about the local volatility function, and a hypothesis model which is highly flexible whilst being prone to overfitting. Besides providing a method for calibrating a (range of) point-estimate, we seek to draw posterior inference on the distribution over local volatility to better understand the uncertainty attached with the calibration. Further, we seek to understand dynamical properties of local volatility by augmenting the hypothesis space with a time dimension. Ideally, this gives us means of inferring predictive distributions not only locally, but also for entire surfaces forward in time.