Termine

On complex symmetric spaces of rank one, Stenzel constructed explicit examples of Calabi-Yau metrics with smooth cross-section asymptotic cone. The classification of such metrics has been achieved by Conlon-Hein, but such a general result remains unsolved when the cone has singular cross-section. A new feature in higher rank symmetric spaces is that the possible candidates for asymptotic cones pratically have singular cross-section. After an introduction and survey of known results, I will present an existence theorem of Calabi-Yau metrics on symmetric spaces of rank two with all asymptotic cone having singular cross-section. This covers the rank two symmetric spaces left by Biquard-Delcroix, achieving the classification of exact Calabi-Yau metrics with maximal volume growth on these spaces. If time allows, I will also try to explain why some special symmetric spaces of rank two don't have any invariant Calabi-Yau metrics with a given asymptotic cone, using an obstruction on the valuation induced by such metric if exists.

Let $X$ be a smooth projective curve over a perfect field $k$ of positive characteristic $p$. The differential (also called stratified) fundamental group of $X$ (at a base point $x\in X(k)$ is defined as the Tannakian dual of the category of stratified bundles on $X$. It is the pro-algebraic analogue of the etale fundamental group of $X$. In this talk we discuss the group cohomology of the stratified fundamental group of $X$ by comparing it with the stratified cohomology introduced by Ogus. We obtain some vanishing and non-vanishing results and make some suggestions on the structure of the fundamental group itself.

We consider the long-time behavior of a passive scalar advected by an incompressible velocity field. When the flow is uniformly hyperbolic, it is well known that it is possible to construct special anisotropic Sobolev spaces where the solution operator becomes quasi-compact with a spectral gap, yielding exponential decay in these spaces. In the non-autonomous and non-uniformly hyperbolic case this approach breaks down. In this talk, I will discuss how in the setting of stochastic velocity fields one can recover certain averaged decay estimates using pseudo differential operators to obtain exponential decay of solutions to the transport equation from H^{-\delta} to H^{-\delta} -- a property we call annealed mixing. As a result, we show that (under certain conditions on the velocity) the Markov process obtained by the advection equation with a random source term has a unique stationary measure describing the statistics of "ideal" scalar turbulence.

Kernel methods are a class of versatile tools for machine learning, numerical approximation and scientific computing. I this talk we discuss novel advancements of deep and greedy kernel methods and presents several applications thereof.

Abstract: Understanding (co-)homology operations has shown to be very useful in algebraic topology and algebraic geometry. In this talk we focus on operations of an invariant called Milnor-Witt K-theory, where the latter is an invariant of smooth algebraic varieties which arises naturally in motivic homotopy theory. The structure of the talk is as follows. We start with a short introduction to motivic homotopy theory and in particular to Milnor-Witt K-theory. Afterwards we explain results on additive, stable and non-additive operations on Milnor-Witt K-theory. Finally, we indicate how these operations lead to motivic knot invariants. The last part is ongoing joint work with Matthias Wendt based on unpublished ideas of Aravind Asok and Matthias Wendt.

Originating in statistical physics as a model of a porous medium, Bernoulli percolation has become a fundamental model in probability theory. In classical Bernoulli percolation, edges (or vertices) of \(\mathbb Z^d\) are deleted independently of each other and with fixed survival probability \(p\in[0,1]\). Despite significant progress in understanding this model, basic questions remain, constituting some of the most perplexing problems in probability theory. In the late 1990s, Benjamini, Lyons, Peres, and Schramm initiated a program to study Bernoulli percolation and other invariant percolation models (i.e., random subgraphs whose distribution is invariant under some natural group action) on graphs beyond \(\mathbb Z^d\). Cayley graphs of infinite groups provide a rich class of examples, and the behavior of percolations turns out to be closely related to the geometric properties of the underlying group This talk will start with a brief introduction to Bernoulli percolation, highlighting what is known as well as open questions. I will then give a gentle introduction to the aforementioned program, focussing on its main questions and the different motivations behind. I will provide a glimpse into some of the fascinating mathematics involved, primarily by reviewing the case of amenable groups. Finally, I will present recent progress beyond amenability based on joint work with Chiranjib Mukherjee. In the late 1990s, Benjamini, Lyons, Peres, and Schramm initiated a program to study Bernoulli percolation and other invariant percolation models (i.e., random subgraphs whose distribution is invariant under some natural group action) on graphs beyond $\mathbb Z^d$. Cayley graphs of infinite groups provide a rich class of examples, and the behavior of percolations turns out to be closely related to the geometric properties of the underlying group This talk will start with a brief introduction to Bernoulli percolation, highlighting what is known as well as open questions. I will then give a gentle introduction to the aforementioned program, focussing on its main questions and the different motivations behind. I will provide a glimpse into some of the fascinating mathematics involved, primarily by reviewing the case of amenable groups. Finally, I will present recent progress beyond amenability based on joint work with Chiranjib Mukherjee.