Events

Extracting the square root of a number and its reciprocal play fundamental role in 3D graphics (e.g. normalizing vectors). In this talk, we will discuss the fast (and brilliant) inverse square root algorithm from the game Quake III and analyze its performance.

More information: https://zucmap.ethz.ch/call_made

The notion of a John domain was initially introduced in 1961 by Fritz John, and later named after him by Martio and Sarvas. Typically, its study is motivated by its connections to the properties of quasiconformal and quasisymmetric mappings. Moreover, John domains find extensive applications in the theory of Sobolev functions in metric measure spaces and functional analysis, as they represent essentially the sole class of domains that uphold the Sobolev-Poincaré inequality. In this presentation, I will introduce several recent applications of John domains in the theory of the calculus of variations.

''As the term suggests - Adelic torus orbits - are nothing but the orbit of an algebraic torus over the ring of Adeles. They provide a powerful tool to collectively study the behavior of collections of geometric data given by arithmetic data. In this talk we will motivate the use of adelic torus orbits by looking at a concrete example: > Already in the 19th century Gauss studied integral binary quadratic forms. He observed that there are essentially only finitely many different integral binary quadratic forms with fixed discriminant. In more modern terms, these different forms arise through a natural action of the ideal class group of a quadratic number field. To study the properties of different forms at the same time it is convenient to consider the Adelic extension of the modular curve. We will see that forms who are not equivalent over the integers might be equivalent over the Adeles. After introducing the necessary concepts and motivating the idea behind Adelic torus orbits we will discuss how they can be used to prove equidistribution results on (real) homogeneous spaces.

Since their introduction in the early 2000s, cluster algebras have shown up all over mathematics, and notably also in geometry. This talk aims at giving a gentle introduction to cluster algebras, while staying close to (hyperbolic) geometry. Namely, we will take a look at (decorated) Teichmüller spaces for punctured surfaces, and describe coordinates on these that are endowed with a cluster structure. This exhibits a beautiful interplay of geometry, combinatorics and cluster theory. Time permitting, we will generalize this approach and give a rough overview of cluster algebras showing up in higher Teichmüller theory.

Coarse correlated equilibria are generalizations of Nash equilibria which have first been introduced in Moulin et Vial (1978). They include a correlation device which can be interpreted as a mediator recommending strategies to the players, which makes it particularly relevant in a context of market failure. After establishing an existence and approximation results result in a fairly general setting, we develop a methodology to compute mean-field coarse correlated equilibria (CCEs) in a linear-quadratic framework. We identify cases in which CCEs outperform Nash equilibria in terms of both social utility and control levels. Finally, we apply such a methodology to a CO2 abatement game between countries (a slightly modified version of Barrett (1994)). We show that in that model CCEs allow to reach higher abatement levels than the NE, with higher global utility. The talk is based on joint works with F. Cannerozzi (Milan University), F. Cartellier (ENSAE) and M. Fischer (Padua University).

I will explain how to compute the integral Chow ring of the moduli stack R_2 of Prym pairs of genus 2.