GTA_en

Director : Cédric BONNAFÉ

Research Themes :

The GTA team covers a broad field of fundamental mathematics, including geometry (algebraic, differential, non-commutative, discrete, and convex), topology (algebraic or differential), and algebra in the broad sense (algebraic combinatorics, group theory, number theory, and representations). Since 2018, the team has also developed joint working groups with the EPS and ACSIOM teams. It is structured in 4 strongly interconnected topic areas: algebraic geometry and number theory; differential geometry and group theory; topology, quantum algebra, and non-commutative geometry; and combinatorial algebra and geometry. It is facilitated by two weekly seminars (AGATA and Darboux).

Algebraic geometry at IMAG is divided into three sections. The first section is close to arithmetic geometry, and concerns, for example, the study of torsors, fundamental group pattern, and more generally Tannakien forms. While remaining in the context of arithmetic, we also study algebraic fields, their Picards groups, and intersection theory. More recently, commutative algebra has developed applications in number theory. In complex algebraic geometry, we study the birational geometry of algebraic varieties and spaces of modules – their invariants and theories of intersection. Our interests also touch complex geometry with research areas dedicated to Kähler-Einstean metrics and to K-stability. 

The questions of the representation of algebraic groups is also at the core of this topic: in number theory, representations of the p-adic and adelic groups are studied (Langlands program, formula traces), while representations of finite or complex groups are often combined in geometric methods (fiber cohomology, l-adic cohomology). 

Differential geometry pays particular attention to metric points of view in the broad sense (Finsler, Riemannan, pseudo-Riemannan, tropical, etc.). Today, it includes spaces known as singular. Interests cover a large spectrum from global to macroscopic (e.g. asymptotic volumes, entropy, operator properties, flows) to local (e.g. notions of curvature and volume). The diverse minimization questions studied situate us at the intersection of convex analysis, optimization, and control. The action of unavoidable groups (diffeomorphism, isometry, but also projective structures) directs us frequently to the geometric theory of groups. This is a developing topic. It is carried out particularly by geometric methods (volumic entropy, systole, quasi-isometric rigidity, hyperbolicity, actions on median spaces), combinatoric methods (actions on complexes: simplicity, cubic CAT(0), etc.) and probabilistic methods (random steps). These methods are illustrated by many classical families: modular groups of surfaces, Artin groups, Thompson groups, lamplighter groups, automaton groups, etc. 

The team’s work in quantum topology revolves around quantifications of the representations of the spaces of fundamental group surfaces and varieties in the third dimension. In particular, the representation of these spaces are studied (skein algebras and generalizations, uniqueness conjectures), along with their categorizations, and their geometric properties in relation to Chern-Simons gauge theory (volume conjecture).

The Non-Commutative Geometry topic focuses on the topological and differential study of groupoids associated with certain singular geometries. The spectral invariants for Dirac-type operators are generated with the help of couplings between diverse homologies and range from analytical indices to eta-type secondary classes. The homologies of algebra associated with groupoids (k-theory topology, Hochschild homologies and cyclics) are the main tool that allows us to produce invariants on singular underlying spaces. The principal examples also appear in the study of the space of the leaves of foliation, dual unitary of a discrete group or its actions, and the transverse space of a quasi-crystal. The applications studied concern the topology of PSC metric space (linked with the topic of Riemannian geometry and the Gromov-Lawson conjecture), the harmonic analysis through transversally elliptical operator (linked with the topic of harmonic analysis), the homotopical invariance of high signatures and Borel and Novikov conjectures, and the problem of labeling holes in spectrum for quasi-crystals (linked with the solid physics).