C6

The extension of standard stochastic models (SDEs, SPDEs) to general fractional noises is known to be a tricky issue, which cannot be studied within the classical martingale setting. We will see how the recently-introduced theory of regularity structures allows us to overcome these difficulties, in the case of a heat equation model with non-linear perturbation driven by a space-time fractional Brownian motion.

The analysis relies in particular on the exhibition of an explicit process at the core of the dynamics, the so-called K-rough path, the definition of which shows strong similarities with that of a classical rough path.

In this talk we will give an overview of the recent research on global quantizations on spaces of different types: compact and nilpotent Lie groups, general locally compact groups, compact manifolds with boundary.

The talk will focus on continuous time random walk with unbounded i.i.d. random conductances on the grid $\mathbb{Z}^d$  In the first place, in a joint work with Kumagai and Mathieu, we obtain Gaussian heat kernel bounds and also local CLT for bounded from above and not bounded from below conductances. The proof is given at first in a general framework, then it is specified in the case of plynomial lower tail conductances. It is essentially based on percolation and spectral analysis arguments, and Harnack inequalities. Then we will discuss the same questions for the same model with i.i.d. random conductances, bounded from below and with finite expectation.

I shall discuss joint work with Mladen Bestvina in which we prove that the group of simplicial automorphisms of the complex of free factors for a
free group $F$ is exactly $Aut(F)$, provided that $F$ has rank at least $3$. I shall begin by sketching the fruitful analogy between automorphism groups of free groups, mapping class groups, and arithmetic lattices, particularly $SL_n({\mathbb{Z}})$. In this analogy, the free factor complex, introduced by Hatcher and Vogtmann, appears as the natural analogue in the $Aut(F)$ setting of the spherical Tits building associated to $SL_n $ and of the curve complex associated to a mapping class group. If $n>2$, Tits' generalisation of the Fundamental Theorem of Projective Geometry (FTPG) assures us that the automorphism group of the building is $PGL_n({\mathbb{Q}})$. Ivanov proved an analogous theorem for the curve complex, and our theorem complements this. These theorems allow one to identify the abstract commensurators of $GL_n({\mathbb{Z}})$, mapping class groups, and $Out(F)$, as I shall explain.

By Thurston's geometrisation theorem, the complement of any knot admits a unique hyperbolic structure, provided that the knot is not the unknot, a torus knot or a satellite knot. However, this is purely an existence result, and does not give any information about important geometric quantities, such as volume, cusp volume or the length and location of short geodesics. In my talk, I will explain how some of this information may be computed easily, in the case of alternating knots. The arguments involve a detailed analysis of the geometry of certain subsurfaces.

The  study of closed geodesics on a Riemannian manifold is a classical and important part of differential geometry. In 1969 Gromoll and Meyer used Morse - Bott theory to give a topological condition on the loop space of compact manifold M which ensures that any Riemannian metric on M has an infinite number of closed geodesics.  This makes a very close connection between closed geodesics and the topology of loop spaces.  

Nowadays it is known that there is a rich algebraic structure associated to the topology of loop spaces — this is the theory of string homology initiated by Chas and Sullivan in 1999.  In recent work, in collaboration with John McCleary, we have used the ideas of string homology to give new results on the existence of an infinite number of closed  geodesics. I will explain some of the key ideas in our approach to what has come to be known as the closed geodesics problem.

It is well known that there are topological obstructions to a manifold $M$ admitting a Riemannian metric of everywhere positive scalar curvature (psc): if $M$ is Spin and admits a psc metric, the Lichnerowicz–Weitzenböck formula implies that the Dirac operator of $M$ is invertible, so the vanishing of the $\hat{A}$ genus is a necessary topological condition for such a manifold to admit a psc metric. If $M$ is simply-connected as well as Spin, then deep work of Gromov--Lawson, Schoen--Yau, and Stolz implies that the vanishing of (a small refinement of) the $\hat{A}$ genus is a sufficient condition for admitting a psc metric. For non-simply-connected manifolds, sufficient conditions for a manifold to admit a psc metric are not yet understood, and are a topic of much current research.

I will discuss a related but somewhat different problem: if $M$ does admit a psc metric, what is the topology of the space $\mathcal{R}^+(M)$ of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows that this problem is unchanged when modifying $M$ by certain surgeries, and I will explain how this can be used along with work of Galatius and myself to show that the algebraic topology of $\mathcal{R}^+(M)$ for $M$  of dimension at least 6 is "as complicated as can possibly be detected by index-theory". This is joint work with Boris Botvinnik and Johannes Ebert.