The Hilbert transform is a central operator in harmonic analysis as it gives access to the harmonic conjugate function. The link between pairs of martingales (X,Y) under differential subordination and the pair (f,Hf) of a function and its Hilbert transform have been known at least since the work of Burkholder and Bourgain in the UMD setting.

During the last 20 years, new and more exact probabilistic interpretations of operators such as the Hilbert transform have been studied extensively. The motivation for this was in part the study of optimal weighted estimates in harmonic analysis. It has been known since the 70s that H:L^2(w dx) to L^2(w dx) if and only if w is a Muckenhoupt weight with its finite Muckenhoupt characteristic. By a sharp estimate we mean the correct growth of the weighted norm in terms of this characteristic. In one particular case, such an estimate solved a long standing borderline regularity problem in complex PDE.

In this lecture, we present the historic development of the probabilistic interpretation in this area, as well as recent results and open questions.

Abstract: Kolmogorov backward equations related to stochastic evolution equations (SEE) in Hilbert space, driven by trace class Gaussian noise have been intensively studied in the literature. In this talk I discuss the extension to non trace class Gaussian noise in the particular case when the leading linear operator generates an analytic semigroup. This natural generalization leads to several complications, requiring new existence and uniqueness results for SEE with initial singularities and a new notion of an extended transition semigroup. This is joint work with Arnulf Jentzen and Ryan Kurniawan (ETH).

We consider the non-linear equation $T^{-1} u+\partial_tu-\partial_x^2\pi(u)=\xi$

driven by space-time white noise $\xi$, which is uniformly parabolic because we assume that $\pi'$ is bounded away from zero and infinity. Under the further assumption of Lipschitz continuity of $\pi'$ we show that the stationary solution is - as for the linear case - almost surely Hölder continuous with exponent $\alpha$ for any $\alpha<\frac{1}{2}$ w. r. t. the parabolic metric. More precisely, we show that the corresponding local Hölder norm has stretched exponential moments.

On the stochastic side, we use a combination of martingale arguments to get second moment estimates with concentration of measure arguments to upgrade to Gaussian moments. On the deterministic side, we first perform a Campanato iteration based on the De Giorgi-Nash Theorem as well as finite and infinitesimal versions of the $H^{-1}$-contraction principle, which yields Gaussian moments for a weaker Hölder norm. In a second step this estimate is improved to the optimal

Hölder exponent at the expense of weakening the integrability to stretched exponential.

This is joint work with Felix Otto.

First order asymptotic of scalar renewal sequences with infinite mean characterized by regular variation has been classified in the 60's (Garsia and Lamperti). In the recent years, the question of higher order asymptotic for renewal sequences with infinite mean was motivated by obtaining 'mixing rates' for dynamical systems with infinite measure. In this talk I will present the recent results we have obtained on higher order expansion for renewal sequences with infinite mean (not necessarily generated by independent processes) in the regime of slow regular variation (with small exponents).  I will also discuss some consequences of these results for error rates in certain limit theorems (such as arcsine law for null recurrent Markov processes).

 

The first reaction-diffusion equation developed and studied is the Fisher-KPP equation.  Introduced in 1937, it accounts for the spatial spreading and growth of a species.  Understanding this population-dynamics model is equivalent to understanding the distribution of the maximum particle in a branching Brownian motion.  Various generalizations of this model have been studied in the eighty years since its introduction, including a model with non-local reaction for the cane toads of Australia introduced by Benichou et. al.  I will begin the talk by giving an extended introduction on the Fisher-KPP equation and the typical behavior of its solutions.  Afterwards, I will describe the model for the cane toads equations and give new results regarding this model.  In particular, I will show how the model may be viewed as a perturbation of a local equation using a new Harnack-type inequality and I will discuss the super-linear in time propagation of the toads.  The talk is based on a joint work with Bouin and Ryzhik.

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We consider several classes of sequences of random variables whose Laplace transform presents the same type of \textit{splitting phenomenon} when suitably rescaled. Answering a question of Kowalski-Nikeghbali, we explain the apparition of a universal term, the \textit{Gamma factor}, by a common feature of each model, the existence of an auxiliary randomisation that reveals an independence structure.
The class of examples that belong to this framework includes random uniform permutations, random polynomials or random matrices with values in a finite field and the classical Sathe-Selberg theorems in probabilistic number theory. We moreover speculate on potential similarities in the Gaussian setting of the celebrated Keating and Snaith's moments conjecture. (Joint work with R. Chhaibi)
 

Motivated by a theorem of Barbour, we revisit some of the classical limit theorems in probability from the viewpoint of the Stein method. We setup the framework to bound Wasserstein distances between some distributions on infinite dimensional spaces. We show that the convergence rate for
the Poisson approximation of the Brownian motion is as expected proportional to λ −1/2 where λ is the intensity of the Poisson process. We also exhibit the speed of convergence for the Donsker Theorem and extend this result to enhanced Brownian motion.

 

We will discuss a result to the effect that the moduli space of log stable maps to a toric variety X is "the same" as the Morse-theoretic moduli space of broken gradient flow lines in the "differentiable realization" Y of the fan for X.  This is joint work with Sam Molcho.

Smooth cubic fourfolds are linked to K3 surfaces via their Hodge structures, due to work of Hassett, and via Kuznetsov's K3 category A. The relation between these two viewpoints has recently been elucidated by Addington and Thomas. 
We study both of these aspects further and extend them to twisted K3 surfaces, which in particular allows us to determine the group of autoequivalences of A for the general cubic fourfold. Furthermore, we prove finiteness results for cubics with equivalent K3 categories and study periods of cubics in terms of generalized K3 surfaces.