Warwick

My goal is to estimate unknown parameters in the vector field of a rough differential equation, when the expected signature for the driving force is known and we estimate the expected signature of the response by Monte Carlo averages.

I will introduce the "expected signature matching estimator" which extends the moment matching estimator and I will prove its consistency and asymptomatic normality, under the assumption that the vector field is polynomial.  Finally, I will describe the polynomial system one needs to solve in order to compute this estimatior.

Random polymers are used to model various physical ( Ising inter- faces, wetting, etc.) and biological ( DNA denaturation, etc.) phenomena They are modeled as a one dimensional random walk (Xn), with excursion length distribution

P(E1 = n) = (n)=nc, c > 1, and (n) a slowly varying function. The polymer gets a random reward, whenever it visits or crosses an interface. The random rewards are realised as a sequence of i.i.d. variables (Vn). Depending on the relation be- tween the mean value of the disorder Vn and the temperature, the polymer might prefer to stick on the interface (pinning) or undergo a long excursion away from it (depinning).

In this talk we will review some aspects of random polymer models. We will also discuss in more detail the pinning-depinning transition of the 'Pinning' model and also its relation to other directed polymer models

In the first part of the talk we will introduce the notion of property testing and briefly discuss some results in testing graph properties in the framework of property testing.

Then, we will discuss a recent result about testing expansion in bounded degree graphs. We focus on the notion of vertex-expansion:   \newline an $a$-expander is a graph $G = (V,E)$ in which every subset $U$ of $V$ of at most $|V|/2$ vertices has a neighborhood of size at least $a|U|$. Our main result is that one can distinguish good expanders from graphs that are far from being weak expanders in time approximately $O(n^{1/2})$.

We design a property testing algorithm that accepts every $a$-expander with probability at least 2/3 and rejects every graph that is $\epsilon$-far from an $a^*$-expander with probability at least 2/3, where $a^* = O(a^2/(d^2 log(n/\epsilon)))$, $d$ is the maximum degree of the graphs, and a graph is called $\epsilon$-far from an $a^*$-expander if one has to modify (add or delete) at least $\epsilon d n$ of its edges to obtain an $a^*$-expander. The algorithm assumes the bounded-degree graphs model with adjacency list graph representation and its running time is $O(d^2 n^{1/2} log(n/\epsilon)/(a^2 \epsilon^3))$.

This is a joint work with Christian Sohler.

It is a classical result that the Alexander polynomial of a fibered knot has to be monic. But in general the converse does not hold, i.e. the Alexander polynomial does not detect fibered knots. We will show that the collection of all twisted Alexander polynomials (which are a natural generalization of the ordinary Alexander polynomial) detect fibered 3-manifolds.

As a corollary it follows that given a 3-manifold N the product S1 x N is symplectic if and only if N is fibered.

I will discuss recent results concerning the uniqueness of Lagrangian particle trajectories associated to weak solutions of the Navier-Stokes equations. In two dimensions, for which the weak solutions are unique, I will present a mcuh simpler argument than that of Chemin & Lerner that guarantees the uniqueness of these trajectories (this is joint work with Masoumeh Dashti, Warwick). In three dimensions, given a particular weak solution, Foias, Guillopé, & Temam showed that one can construct at leaset one trajectory mapping that respects the volume-preserving nature of the underlying flow. I will show that under the additional assumption that $u\in L^{6/5}(0,T;L^\infty)$ this trajectory mapping is in fact unique (joint work with Witek Sadowski, Warsaw).