L6

Given two probability distributions $P_R$ and $P_B$ on the positive reals with finite means, colour the real line alternately with red and blue intervals so that the lengths of the red intervals have distribution $P_R$, the lengths of the blue intervals have distribution $P_B$, and distinct intervals have independent lengths. Now iteratively update this colouring of the line by coalescing intervals: change the colour of any interval that is surrounded by longer intervals so that these three consecutive intervals subsequently form a single monochromatic interval. Say that a colour (either red or blue) `wins' if every point of the line is eventually of that colour. I will attempt to answer the following question: under what natural conditions on the distributions is one of the colours almost surely guaranteed to win?

If G is a semi-simple Lie group, it is known that all lattices
are arithmetic unless (up to finite index) G=SO(n,1) or SU(n,1).
Non-arithmetic lattices have been constructed in SO(n,1) for
all n and there are infinitely many non-arithmetic lattices in
SU(1,1). Mostow and Deligne-Mostow constructed 9 commensurability
classes of non-arithmetic lattices in SU(2,1) and a single
example in SU(3,1). The problem is open for n at least 4.
I will survey the history of this problem, and then describe
recent joint work with Martin Deraux and Julien Paupert, where
we construct 10 new commensurability classes of non-arithmetic
lattices in SU(2,1). These are the first examples to be constructed
since the work of Deligne and Mostow in 1986.

Building a suitable family of walls in the Cayley complex of a finitely
presented group G leads to a nontrivial action of G on a CAT(0) cube
complex, which shows that G does not have Kazhdan's property (T).  I
will discuss how this can be done for certain random groups in Gromov's
density model.  Ollivier and Wise (building on earlier work of Wise on
small-cancellation groups) have built suitable walls at densities <1/5,
but their method fails at higher densities.  In recent joint work with
Piotr Przytycki we give a new construction which finds walls at densites
<5/24.

Networks are a convenient way to represent systems of interacting entities. Many networks contain "communities" of nodes that are more densely connected to each other than to nodes in the rest of the network.

Most methods for detecting communities are designed for static networks. However, in many applications, entities and/or interactions between entities evolve in time.

We investigate "multilayer modularity maximization", a method for detecting communities in temporal networks. The main difference between this method and most previous methods for detecting communities in temporal networks is that communities identified in one temporal snapshot are not independent of connectivity patterns in other snapshots.  We show how the resulting partition reflects a trade-off between static community structure within snapshots and persistence of community structure between snapshots. As a focal example in our numerical experiments, we study time-dependent financial asset correlation networks.

Our study addresses the question of how an arbitrage-free semimartingale model is affected when the knowledge about a random time is added. Precisely, we focus on the No-Unbounded-Profit-with-Bounded-Risk condition, which is also known in the literature as the first kind of no arbitrage. In the general semimartingale setting, we provide a sufficient condition on the random time and price process for which the no arbitrage is preserved under filtration enlargement. Moreover we study the condition on the random time for which the no arbitrage is preserved for any process. This talk is based on a joint work with Tahir Choulli, Jun Deng and Monique Jeanblanc.

We consider a controlled stochastic system in presence of state-constraints. Under the assumption of exponential stabilizability of the system near a target set, we aim to characterize the set of points which can be asymptotically driven by an admissible control to the target with positive probability. We show that this set can be characterized as a level set of the optimal value function of a suitable unconstrained optimal control problem which in turn is the unique viscosity solution of a second order PDE which can thus be interpreted as a generalized Zubov equation.

We study the problem of maximizing expected utility from terminal wealth in a semi-static market composed of derivative securities, which we assume can be traded only at time zero, and of stocks, which can be
traded continuously in time and are modeled as locally-bounded semi-martingales.

Using a general utility function defined on the positive real line, we first study existence and uniqueness of the solution, and then we consider the dependence of the outputs of the utility maximization problem on the price of the derivatives, investigating not only stability but also differentiability, monotonicity, convexity and limiting properties.

A permutation group is called sharply n-transitive if it acts 

freely and transitively on the set of ordered n-tuples of distinct 

points. The investigation of such permutation groups is a classical 

branch of group theory; it led Emile Mathieu to the discovery of the 

smallest finite simple sporadic groups in the 1860's. In this talk I 

will discuss the case where the permutation group is assumed to be a 

locally compact transformation group, and explain how this set-up is 

related to Gromov hyperbolicity and to arithmetic lattices in products 

of trees.

Let $G$ be a reductive group such as $SL_n$ over the field $k((t))$, where $k$ is an algebraic closure of a finite field, and let $W$ be the affine Weyl group of $G$.  The associated affine Deligne-Lusztig varieties $X_x(b)$ were introduced by Rapoport.  These are indexed by elements $x$ in $G$ and $b$ in $W$, and are related to many important concepts in algebraic geometry over fields of positive characteristic.  Basic questions about the varieties $X_x(b)$ which have remained largely open include when they are nonempty, and if nonempty, their dimension.  We use techniques inspired by geometric group theory and representation theory to address these questions in the case that $b$ is a translation.  Our approach is constructive and type-free, sheds new light on the reasons for existing results and conjectures, and reveals new patterns.  Since we work only in the standard apartment of the building for $G$, which is just the tessellation of Euclidean space induced by the action of the reflection group $W$, our results also hold over the p-adics.  This is joint work with Elizabeth Milicevic (Haverford) and Petra Schwer (Karlsruhe).

We consider the layer potentials associated with operators $L=-\mathrm{div}A \nabla$ acting in the upper half-space $\mathbb{R}^{n+1}_+$, $n\geq 2$, where the coefficient matrix $A$ is complex, elliptic, bounded, measurable, and $t$-independent. A "Calder\'{o}n--Zygmund" theory is developed for the boundedness of the layer potentials under the assumption that solutions of the equation $Lu=0$ satisfy interior De Giorgi-Nash-Moser type estimates. In particular, we prove that $L^2$ estimates for the layer potentials imply sharp $L^p$ and endpoint space estimates. The method of layer potentials is then used to obtain solvability of boundary value problems. This is joint work with Steve Hofmann and Marius Mitrea.