Voevodsky asked what the topology of the universe is in a 
continuous interpretation of type theory, such as Johnstone's 
topological topos. We can actually give a model-independent answer: it 
is indiscrete. I will briefly introduce "intensional Martin-Loef type 
theory" (MLTT) and formulate and prove this in type theory (as opposed 
to as a meta-theorem about type theory). As an application or corollary, 
I will also deduce an analogue of Rice's Theorem for the universe: the 
universe (the large type of all small types) has no non-trivial 
extensional, decidable properties. Topologically this is the fact that 
it doesn't have any clopens other than the trivial ones.

It is well-known that a matrix $A$ is Hurwitz stable if and only if there exists a positive definite solution to the Lyapunov matrix equation $A X + X A^* = B$, where $B$ is Hermitian negative definite. We present a verified numerical algorithm to rigorously prove the stability of a given matrix $A$ in the presence of rounding errors.  The computational cost of the algorithm is cubic and it is fast since we can cast almost all operations in level 3 BLAS for which interval arithmetic can be implemented very efficiently.  This is a joint work with Andreas Frommer and the results are already published in ETNA in 2013.

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.