The signature of a path has many properties that make it an excellent feature to be used in machine learning. We exploit this properties to analyse a stream of data that arises from a psychiatric study whose objective is to analyse bipolar and borderline personality disorders. We build a machine learning model based on signatures that tries to answer two clinically relevant questions, based on observations of their reported state over a short period of time: is it possible to predict if a person is healthy, has bipolar disorder or has borderline personality disorder? And given a person or borderline personality disorder, it is possible to predict his or her future mood? Signatures proved to be very effective to tackle these two problems.

Abstract. As the first  step for approaching the uniqueness and blowup properties of the solutions of the stochastic wave equations with multi-plicative noise, we analyze the conditions for the uniqueness and blowup properties of the solution (Xt; Yt) of the equations dXt = Ytdt, dYt = jXtj_dBt, (X0; Y0) = (x0; y0). In particular, we prove that solutions arenonunique if 0 < _ < 1 and (x0; y0) = (0; 0) and unique if 1=2 < _ and (x0; y0) 6= (0; 0). We also show that blowup in _nite time holds if _ > 1 and (x0; y0) 6= (0; 0).

This is a joint work with A. Gomez, J.J. Lee, C. Mueller and M. Salins.

Inverting the signature of a path with ideas from linear algebra with implementations.

In this talk, a novel discontinuous Galerkin (DG) method is introduced by utilising the principle of discrete least squares. The key idea is to build polynomial approximations by the method of  (weighted) discrete least squares instead of usual interpolation or (discrete) $L^2$ projections. The resulting method hence uses more information of the underlying function and provides a more robust alternative to common DG methods. As a result, we are able to construct high-order schemes which are conservative as well as linear stable on any set of collocation points. Several numerical tests highlight the new discontinuous Galerkin discrete least squares (DG-DLS) method to significantly outperform present-day DG methods.

A generic surface homeomorphism (up to isotopy) is what we call it pseudo-Anosov. These maps come equipped with an algebraic integer that measures
how much the map stretches/shrinks in different direction, called the stretch factor. Given a surface homeomorsphism, one can ask if it is the lift (by a branched or unbranched cover) of another homeomorphism on a simpler surface possibly of small genus. Farb conjectured that if the algebraic degree of the stretch factor is bounded above, then the map can be obtained by lifting another homeomorphism on a surface of bounded genus.
This was known to be true for quadratic algebraic integers by a Theorem of Franks-Rykken. We construct counterexamples to Farb's conjecture.

Let \Gamma_{g,1} denote the mapping class group of a genus g surface with one parametrized boundary component.  The group homology H_i(\Gamma_{g,1}) is independent of g, as long as g is large compared to i, by a famous theorem of Harer known as homological stability, now known to hold when 2g > 3i.  Outside that range, the relative homology groups H_i(\Gamma_{g,1},\Gamma_{g-1,1}) contain interesting information about the failure of homological stability.  In this talk, I will discuss a metastability result; the relative groups depend only on the number k = 2g-3i, as long as g is large compared to k.  This is joint work with Alexander Kupers and Oscar Randal-Williams.

The smooth homotopy category is a simultaneous enlargement of the usual homotopy category and of the category of smooth manifolds. Its structure can be described very simply and explicitly by a version of van Est's theorem.  It provides us with an  interpolation between topology and geometry (and with a toy model of derived algebraic geometry and motivic homotopy theory, though I shall not pursue those directions).  My talk will list some situations which the category seems to illuminate: one will be Kapranov's beautiful description of the Lie algebra of the 'group' of based loops in a manifold.
 

We show that a closed almost Kähler 4-manifold of globally constant holomorphic sectional curvature k<=0 with respect to the canonical Hermitian connection is automatically Kähler. The same result holds for k < 0 if we require in addition that the Ricci curvature is J-invariant. The proofs are based on the observation that such manifolds are self-dual, so that Chern–Weil theory implies useful integral formulas, which are then combined with results from Seiberg–Witten theory.