In this talk we aim to study large-time asymptotics of solutions of the KFP equation with a short-range potential in dimension one. First we summarise several known results about the free operator and then demonstrate the influence of the potential. Representation formula of the semigroup in terms of the resolvent is employed to obtain the asymptotics of solutions. In these calculations we have to find the expansion of the resolvent near the threshold of the essential spectrum.

The notion of Quillen cohomology, first defined by Quillen in his seminal book and later extended to include coefficients in spectra, is a universal form of cohomology which enjoys a wide range of applications: it features in obstruction theories and spectral sequences abutting to mapping spaces, controls deformations of objects in the relevant contexts and tends to appear in moduli problems classifying objects realizing a given cohomological invariant. In this talk we will describe an approach to the computation of Quillen cohomology of categories enriched in a well behaved model category S. When S is the category of simplicial sets one recovers a model for the theory of oo-categories, in which case our computation identifies the Quillen cohomology of an oo-category C as the cohomology of spectrum valued functors on the twisted arrow category of C. When S is the category of marked simplicial sets one recovers a model for the theory of (oo,2)-categories, in which case our computation identifies the Quillen cohomology of an (oo,2)-category C with the cohomology of spectrum valued functors indexed by what we call the twisted 2-cell category of C. These results can be used to give explicit and computable obstruction theories for problems such as lifting homotopy commutative diagrams (in either a 1-categorical or a 2-categorical setting). This is joint work with Joost Nuiten and Matan Prasma.

Let $X$ be a simply connected, complete Riemannian manifold with pinched negative sectional curvatures. The critical exponent $\delta\Gamma$ of a discrete group of isometries $\Gamma$ of $X$ is equal to the abscissa of convergence of the Poincaré series of $\Gamma$; and for a cocompact $\Gamma0$, this is given by the volume growth of $X$. (Moreover, if $X$ is a symmetric space then there is a relationship between $\delta\Gamma$ and the bottom of the spectrum of the Laplacian on $X/\Gamma$.) Using a more dynamical approach, we characterise the existence of a uniform gap $\delta\Gamma<\delta{\Gamma0}$ for a family of (infinite index) normal subgroups $\Gamma$ of $\Gamma0$, in terms of permutation representations given by the quotients $\Gamma0/\Gamma$.

Certaines théories des champs classiques bidimensionnelles, où les champs prennent leurs valeurs dans un groupe ou un espace symétrique, possèdent la propriété d’intégrabilité. Les équations du mouvement prennent une forme particulière, dite forme de Lax, et l’on peut définir une infinité de charges conservées en involution par rapport au crochet de Poisson. C. Klimcik a proposé en 2002 une déformation de certains de ces modèles qui préserve l’intégrabilité. On donnera une description de ces modèles et de leurs généralisations, en particulier la déformation d’un modèle associé à la propagation d’une supercorde.