Seminars from until

It was observed in the 1980's that the entropy of an action by two or more commuting automorphisms on a compact abelian group is sometimes given by a zeta- or L-value. For example zeta(3) arises in this way. In the talk I will report on joint work with Thu Hà Trieu, which at least for expansive actions explains the deeper reason for these observations by a combination of operator algebra theory, in particular the theory of their determinants, K-theory and cyclic and Deligne cohomology. The talk is addressed to a general audience and will not go into technical details. Instead we will sketch the relevant theories and explain how to fit them together for our purpose.

We consider a class of Riesz transforms $R_V^a$ related to the Schrödinger operator $L = -\frac{1}{2}\Delta + V$ with $V$ being a non-negative potential. We will focus on $L^\infty$ boundedness of $R_V^a$ and give a specific integral condition on $V$ which provides the boundedness. We will also show that potentials with power or exponential growth satisfy this condition.

The Kerr de Sitter geometry models a rotating black hole in an expanding universe. I will review its stability properties in the context of the Einstein vacuum equations with positive cosmological constant, and present recent progress on the non-linear stability problem for the cosmological region.

What really happens when we shuffle a deck of cards — and what mathematics lies behind this seemingly simple act?

Persi Diaconis explores the mathematics of three ways that real people shuffle real cards.

Remarkably, these three methods call for threes quite different kinds of mathematics.

Riffle shuffling leads to elementary combinatorics; overhand shuffling involves sophisticated coupling methods; and smooshing leads all the way to stochastic partial differential equations.

Behind these familiar ways of mixing cards lie challenging questions in analysis and probability.

Yet the problems themselves are easy to understand, making them an engaging gateway to surprisingly deep mathematics.

Along the way, the course will reveal connections to card tricks, gambling, probability, and other areas of mathematics.

The goal of this talk is to present the construction of new families of complete Calabi-Yau metrics with maximal volume growth on the small resolutions of the ordinary double point singularity in dimension 3. These metrics have tangent cone $\mathbb{C} ×(\mathbb{C}^2/\mathbb{Z}_2)$ at infinity and are parametrised by their Kähler class. As the Kähler class degenerates, the metrics converge in the pointed Gromov-Hausdorff sense to a singular Calabi-Yau metric with an isolated conical singularity modelled on the Stenzel metric at the ordinary double point, thereby providing a new metric realisation of the Atiyah flop. As we will explain, this construction is meant to provide a local adiabatic model for certain degenerations of compact Calabi-Yau manifolds.