Analysis as a tool in Mathematical Physics (webinar)

Segundo Encuentro de Análisis Global

Speakers:

Alexander Cardona (Universidad de los Andes)

Alessandra Frabetti (Institut Camille Jordan, Université de Lyon 1 )

Steve Rosenberg (Boston University)

Thomas Schick (Universität Göttingen)

Jorge Zanelli (Centro de Estudios Científicos - CECs )

Organizers:

Leonardo Arturo Cano García (Universidad Nacional de Colombia - Sede Bogotá)

Carolina Neira Jiménez (Universidad Nacional de Colombia - Sede Bogotá)

Sylvie in Páramo de Iguaque, 26th July 2015.

In honour of the 60th birthday of Sylvie Paycha

Thankful of the influence that Sylvie Paycha has played in our professional lives, and as witnesses of the energy she has invested to promote mathematics in Colombia, we are organizing this webinar on 17th July from 9am to 12 (GMT-5) to celebrate her 60th birthday and her contributions to mathematics.

Link to zoom: https://zoom.us/meeting/register/tJctcOygqD0tHNZEtTYdNzGhejy3XT2AlRZW

Schedule:

9:00-9:30am Steve Rosenberg

Title: Diffeomorphism groups of line bundles over symplectic manifolds

Abstract: Global gravitational anomalies are associated to nontrivial topology of diffeomorphism groups of manifolds. There are infinite dimensional Chern-Simons invariants that detect nontrivial loops on diffeomorphism groups of line bundles over symplectic manifolds. This work, with Yoshi Maeda and Fabian Torres-Ardila, is directly related to Sylvie Paycha's focus on infinite dimensional geometry and the importance of the group of bounded invertible pseudo-differential operators.

9:35-10:05am Alessandra Frabetti

Title: Transport maps and direct connections on groupoids

Abstract: In Martin Hairer's theory of Regularity Structures, transport maps relate local solutions of stochastic PDEs and allow to construct global solutions on a flat space. In a work in progress with Sylvie Paycha, Sara Azzali and Youness Boutaib, we interpretate transport maps as direct connections on Lie groupoids on manifolds, following an idea by Nicolae Teleman who generalises the usual parallel transport associated to a connection on a vector bundle.

10:10-10:40am Jorge Zanelli

Title: Will all of Mathematics eventually become Physical Mathematics?

Abstract: Physical Mathematics was defined as Mathematical ideas which are of current interest for their potential application to theoretical physics. As Albert Schwarz noted, this is a time-dependent definition. We all know examples of purely abstract ideas in mathematics that eventually find applications in physics. I have my own particular example of an idea that in fifty years evolved from a bizarre mathematical abstraction into a central figure in theoretical physics, with applications that range from elementary particles to gravitation, to superconductivity, to material science and to a description of the earth's climate. It is as if nature were the testing ground where all of mathematics can be tried out. If that were the case, it would make no sense to ask whether some piece of mathematics is more or less applied, but where or when it will find its application.

10:45-11:15am Thomas Schick

Title: Differential K-theory. What, why, how.

Abstract: Differential K-theory merges global topological information (classification of vector bundles) with local geometric information (connections and their curvature forms). This reflects a paradigm of quantum physics, where we also have an interplay of non-local, local, and topological phenomena. Consequently, differential K-theory and its cousins have been used in certain models of quantum field theories to describe field strengths and quantization effects. Similarly, it has been used as an efficient tool for secondary invariants in geometry and topology.

The talk aims to give a brief introduction to

* what differential K-theory is (using an axiomatic approach)

* why we care about differential K-theory, sketching some applications

* how we construct differential K-theory

11:20-11:50m Alexander Cardona

Title: Pseudo-differential calculus and index theory on homogeneous spaces

Abstract: The global pseudo-differential calculus developped by M. Ruzhansky and collaborators in the case of compact Lie groups has shown to be useful to “discretize” half of the variables in the usual calculus of pseudo-differential operators, defined from local symbols. Among the advantages of this formulation there is the reduction of calculations involving integrals to sums, in particular in the case of spectral invariants. In this talk we consider the pseudo-differential calculus with global symbols in the case of homogeneous spaces, and go back to the computation of topological invariants of such spaces using these methods.

Closing