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CIRM – Centre International de Rencontres Mathématiques

The research school is an introductory meeting to the thematic program “Representation Theory and Noncommutative Geometry” to be held at IHP from January to March 2025. The trimester is part of an ongoing effort to bridge two fiels of Mathematics : the representation theory of locally compact groups and the theory of operator algebras. These research domains share origins in harmonic analysis, spectral theory and quantum mechanics but grew in separate directions. Recent progress in representation theory, involving especially non-Riemannian symmetric spaces and spherical varieties, and new tools developed in operator algebras, especially those involving K-theory and the other methods of non-commutative geometry, offer exciting prospects for new work at the interface between the two fields.

CIRM – Centre International de Rencontres Mathématiques

The main goal of these Etats de la recherche is to review some of the most recent spectacular developments in algebraic, analytic and arithmetic dynamics. Covered themes will include a variety of problems related to the iteration of rational maps on algebraic varieties as well as the study of their dynamical moduli spaces. Depending on the structure of the ambient field over which these maps are defined, methods may vary drastically. Over the field of complex numbers, one speaks of holomorphic dynamics in which quasi-conformal deformations play a key role. The case of maps defined over general metrized fields have led to the developments of dynamics over Berkovich spaces. One can also consider maps over number fields whose study lies at the core of arithmetic dynamics. Interactions between these various fields have deepened in the recent years. Complex pluripotential theory combined with tools from arithmetic geometry has been used to prove equidistribution results with applications to problems of unlikely intersection, and to explore dynamical moduli spaces and their special varieties. Methods in p-adic analysis and from the minimal model program have revolutionized our understanding of groups of birational transformations. Non-archimedean dynamics has been turned into a very efficient tool to analyze degeneration problems in holomorphic dynamics.

Cours et évènements pour étudiants en mathématiques,    Analyse

CIRM – Centre International de Rencontres Mathématiques

This Thematic Month aims to cover topics related to singularity theory of algebraic or analytic spaces, algebraic study of differential equations, and their applications to questions of transcendence. This 5-week program covers different themes that are often not closely related. One of the main objectives is to make them interact. To encourage participants (especially the youngest ones) to attend the entire month and foster interactions outside each one’s expertise zone, the scientific program of each week of the month will consist of courses accessible to non-experts, as well as more specialized presentations.

CIRM – Centre International de Rencontres Mathématiques

Starting in the 2011 Annals of Mathematics paper of Foreman, Rudolph and Weiss there have many results showing that classical problems in dynamical systems are unsolvable using inherently countable resources. The initial results were for abstract measure preserving systems, but more recent work has focussed on smooth systems on compact manifolds. Gerber and Kunde have made very significant contributions for smooth transformations that are weakly mixing and for Kautomorphisms. They also showed that the Kakutani equivalence relation is not Borel. These results raise more questions than they solve! The proposed Research inResidence is to bring the four researchers together to share knowledge and techniques and hopefully make progress on the open problems.

CIRM – Centre International de Rencontres Mathématiques

K-stability is a central topic in modern complex geometry. It characterises Fano manifolds that admit a Kähler-Einstein metric and provides a good notion of compact moduli space for Fano varieties. The winter school will bring together experts on K-stability who will give 8 mini-courses for PhD students and young researchers with a special focus on explicit problems in dimensions two and three. Along with lecture courses, there will be exercise sessions.

CIRM – Centre International de Rencontres Mathématiques

We plan to work on several explicit problems about K-stability of smooth and singular Fano threefolds described in the proposal together with Hamid Abban (Nottingham, England) and Kento Fujita (Osaka, Japan). Hamid Abban works on K-stability of Fano varieites, and birational geometry of Fano varieties. His recent results on K-stability crucially changed the subject over the last two years. Kento Fujita is an expert in the general theory of K-stability of Fano varieties. One of the most important works of Fujita consisted on finding and proving a valuative criterion for K-stability.

CIRM – Centre International de Rencontres Mathématiques

During the past decades, Teichmüller–Thurston theory and its connections to geometrization of 3-manifolds have been used as a model to understand discrete subgroups of higher rank Lie groups and their associated locally homogeneous manifolds. This lead to spectacular developments in which the notion of Anosov group (a subtle higher rank generalization of convex-cocompactness) plays a central role. This school, aimed primarily at PhD students and young postdocs, will present the fundamental results on which these recent developments build, including both geometric, dynamical and analytic aspects of discrete subgroups of Lie groups.

CIRM – Centre International de Rencontres Mathématiques

Friezes, introduced 50 years ago by Coxeter, are patterns of integers defined by a simple, local condition. They consist of a finite (or infinite) number of rows written in a lattice, starting with a row of 0s and a row of 1s and satisfying a local SL2-rule: for any four neighbours a, b, c, d forming a diamond we require ad − bc = 1. Since the discovery of links to cluster algebras of type A, a plethora of generalisations have been studied in the last decades. With this workshop, we will provide a platform to establish new links between friezes and research areas in geometry, algebra and combinatorics.

Géométrie et topologie,    Théorie des graphes et combinatoire

Centre International de Rencontres Mathématiques (CIRM)