I shall report about a new direct construction of Nori motives, discovered independently by Barbieri-Viale and Prest on the one hand, by Joyal and myself on the other hand. Unlike previous constructions, one uses only standard constructions in category theory, like Frey free abelian category on a given additive category, and Serre’s construction of quotient categories.

The statistical measure Evolutionary Entropy characterizes Darwinian Fitness and predicts the outcome of competition for limited resources between related entities at various levels of organization: metabolic, cellular, organismic and social. I will discuss the mathematical basis of the selection principle and describe its application to the evolution of aging and the origin of age-related diseases.

 

Hodge theory, as developed by Deligne and Griffiths, is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties. It is an essential fact that at heart, Hodge theory is NOT algebraic. On the other hand, according to both the Hodge conjecture and the Grothendieck period conjecture, this transcendence is severely constrained.

 

Tame geometry, whose idea was introduced by Grothendieck in the 80s, seems a natural setting for understanding these constraints. Tame geometry, developed by model theorists as o-minimal geometry, has for prototype real semi-algebraic geometry, but is much richer. It studies structures where every definable set has a finite geometric complexity.

 

The aim of this course is to present a number of recent applications of tame geometry to several problems related to Hodge theory and periods. After recalling basics on o-minimal structures and their tameness properties, I will discuss:
– the use of tame geometry in proving algebraization results (Pila-Wilkie theorem; o-minimal Chow and GAGA theorems in definable complex analytic geometry);
– the tameness of period maps; algebraicity of images of period maps;
– functional transcendence results: Ax-Schanuel conjecture from abelian varieties to Shimura varieties and variations of Hodge structures. Applications to atypical intersections (André-Oort conjecture and Zilber-Pink conjecture);
– the geometry of Hodge loci and their closures.

 

 

Hodge theory, as developed by Deligne and Griffiths, is the main tool for analyzing the geometry and arithmetic of complex algebraic varieties. It is an essential fact that at heart, Hodge theory is NOT algebraic. On the other hand, according to both the Hodge conjecture and the Grothendieck period conjecture, this transcendence is severely constrained.

 

Tame geometry, whose idea was introduced by Grothendieck in the 80s, seems a natural setting for understanding these constraints. Tame geometry, developed by model theorists as o-minimal geometry, has for prototype real semi-algebraic geometry, but is much richer. It studies structures where every definable set has a finite geometric complexity.

 

The aim of this course is to present a number of recent applications of tame geometry to several problems related to Hodge theory and periods. After recalling basics on o-minimal structures and their tameness properties, I will discuss:
– the use of tame geometry in proving algebraization results (Pila-Wilkie theorem; o-minimal Chow and GAGA theorems in definable complex analytic geometry);
– the tameness of period maps; algebraicity of images of period maps;
– functional transcendence results: Ax-Schanuel conjecture from abelian varieties to Shimura varieties and variations of Hodge structures. Applications to atypical intersections (André-Oort conjecture and Zilber-Pink conjecture);
– the geometry of Hodge loci and their closures.

 

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

In physics, the renormalisation group provides a powerful point of view to understand random systems with strong correlations. Despite advances in a number of particular problems, in general its mathematical justification remains a holy grail. I will give an introduction to the main concepts from the point of view of a mathematician and illustrate its use in some examples.

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

In physics, the renormalisation group provides a powerful point of view to understand random systems with strong correlations. Despite advances in a number of particular problems, in general its mathematical justification remains a holy grail. I will give an introduction to the main concepts from the point of view of a mathematician and illustrate its use in some examples.

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

In physics, the renormalisation group provides a powerful point of view to understand random systems with strong correlations. Despite advances in a number of particular problems, in general its mathematical justification remains a holy grail. I will give an introduction to the main concepts from the point of view of a mathematician and illustrate its use in some examples.

——–  IMPORTANT INFORMATION  ——–

Due to the health situation related to the Coronavirus epidemic, the course has been cancelled and postponed at a later date to be confirmed.

In physics, the renormalisation group provides a powerful point of view to understand random systems with strong correlations. Despite advances in a number of particular problems, in general its mathematical justification remains a holy grail. I will give an introduction to the main concepts from the point of view of a mathematician and illustrate its use in some examples.

In these lectures we will present the mathematical proofs of conformal invariance of a number of models coming from planar statistical mechanics, including the Ising and dimer models. In particular, we will explain how discrete notions of holomorphicity can be used to solve discrete versions of classical Boundary Value Problems, and how this analysis is related to conformal invariance of certain observables in planar statistical mechanics.

In these lectures we will present the mathematical proofs of conformal invariance of a number of models coming from planar statistical mechanics, including the Ising and dimer models. In particular, we will explain how discrete notions of holomorphicity can be used to solve discrete versions of classical Boundary Value Problems, and how this analysis is related to conformal invariance of certain observables in planar statistical mechanics.

In these lectures we will present the mathematical proofs of conformal invariance of a number of models coming from planar statistical mechanics, including the Ising and dimer models. In particular, we will explain how discrete notions of holomorphicity can be used to solve discrete versions of classical Boundary Value Problems, and how this analysis is related to conformal invariance of certain observables in planar statistical mechanics.

Ergodic Theory is a powerful tool in the study of linear groups. When trying to crystallize its role, emerges the theory of AREAs, that is Algebraic Representations of Ergodic Actions, which provides a categorical framework for various previously studied concepts and methods. Roughly, this theory extends the focus of Representation Theory from Groups to Group Actions, exploiting the tension between Ergodic Theory and Algebraic Geometry. In this series of talks I will introduce this theory and survey some of its applications, focusing on Superrigidity and Arithmeticity results.

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IHES Covid-19 regulations:

– all the participants who will attend the event in person will have to keep their mask on in indoor spaces
and where the social distancing is not possible;
– speakers will be free to wear their mask or not at the moment of their talk if they feel more comfortable
without it;
– Up to 25 persons in the conference room, every participant will be asked to be able to provide a health pass
– Over 25 persons in the conference room, every participant will be asked to provide a health pass which will
be checked at the entrance of the conference room.

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