Responsable : Geoffrey Powell
Ce séminaire a lieu généralement le mardi à 14h en Salle I001.
Séminaires à venir
We give a survey of various avatars of the trisecant Fay identity which appear in the context of Integrable systems (as forms of the Associative Yang-Baxter Equation) and as conditions on generating functions for period polynomials of (quasi-)modular forms and group cocycle conditions for some multiparametric modular groups.
Two different approaches to finding of invariants will be discussed. All ideas, methods and results will be illustrated by the classical problem of GL-classification of binary and n-ary forms. If time allows, the more general case of invariants for irreducible representations will be considered.
This talk is meant as an introduction to, or rather as an overview of, material published in several papers and books, through the last 20 years. The rather bold idea was, to use the emergent non-commutative algebraic geometry and its deformation theory, to look for a purely mathematical model that would make sense of the present theoretical physics, and hopefully help us unifying general relativity and quantum theory. There are, of course, lots of research groups of mathematical physicists, working on theories with the same aim; establishing a ”Theory of Everything”. As for most of them, the starting point is to take a critical new look at some of the basic notions in physics. In my work, these notions include: Objects, Points,States, Moduli Space of such objects, their Dynamical Structures, giving rise to Space, Time and Evolution. The technology I am using for this, is ”Non-commutative Deformation Theory” (see the book with the same title, by Eriksen, Laudal and Siqveland, on Chapman and Hall (2017)), and most of what I know about this, can be found in an accepted paper ”Deformation Theory and Mathematical Models in Science” that should have been published in ”Pure and Applied Mathematics Quarterly”, October this year. Singularities comes in as fundamental ”Bags of Information”, and the theory of deformation, applied to singularities, is taking care of the Creation of the Moduli Spaces we want to study. In western religious thinking, in particular in the Greek creation myths, the ”gap”, created by the original separation of heaven and earth, became the primordial state before the Creation, combining the notions of primordial water and darkness. This has become the Big Bang of modern Cosmology, and the Bag of Information-Energy, responsible for this event had to be found somewhere. It turned out that it had been hidden in the notion of Deformation Theory, in particular in the deformations of Singularities! One such singularity, the affine scheme, consisting of a point with a 3 dimensional tangent space, is the Origin of the Big Bang! Of course, what I mean above is that, starting with this singularity, the mathematical tools we now have can be used to make a very ”convincing” ”Toy Model” for the Start, and subsequent Evolution, of the Universe, as we know it today. This Toy Model is the subject of the talk.