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Activity Funded

Group of Mathematical Physics of the University of Lisbon

Grupo de Física Matemática

Reference
UIDP/00208/2020
Project Start Date
2020-01-01
Project End Date
2024-12-31
Principal Investigator
Funding Program
Concurso de avaliação no âmbito do Programa Plurianual de Financiamento de Unidades de I&D (2017/2018) - Financiamento Programático

Abstract

10.2 Summary in English for general dissemination purposes The Group of Mathematical Physics of the University of Lisbon (GFMUL)will push forward his programs of Stochastic Analysis, where the tools of Classical Mechanics are deformed to accommodate trajectories as irregular as Brownian ones.The whole point of this program is to preserve as much as possible the main geometric structures of the classical theory. We have an original method to do this which also applies, for instance, to Fluid Dynamics, allowing to treat dissipative systems such as the Navier-Stokes equation. Geometric Spectral theory, devoted to the relations between geometry of surfaces and spectrum of differential operators, will be another of GFMUL main research lines, as well as problems of Analysis and Geometry of pseudo differential operators relevant in Quantum Mechanics.Those tools are also quite useful in questions regarding Quantum foundations. An ambitious theoretical approach to Quantum graphs will be elaborated in our center. Quantum graphs are structures of vertices connected by edges, together with a (pseudo) differential Schroedinger operator acting on functions defined on individual edges. Geometric aspects of Quantum Theory in infinite dimension, i.e. of Quantum Field Theory, will remain one of our research topics, with emphasis on general notions of Integrability. Originated in Hamiltonian Mechanics, where Integrability means that the trajectories of the system are confined to submanifolds of the phase space and the solution of dynamical equations can be formulated by explicit integrals, it takes today different forms and relates with Algebraic Geometry, Random Matrix theory and other parts of mathematics. One aspect of this research project will be the study of the consequences of an important result found at GFMUL recently on the correspondence between Ordinary Differential equations and Integrable Models.This will involve understanding its connections with String Theory and some parts of Algebraic Geometry, a classical field of mathematics studying geometric aspects of solutions of polynomial equations. For finite dimensional Quantum theory, the Bethe Ansatz of inverse-scattering method provides a special notion of Integrability allowing to find explicit solutions for some specific quantum systems. For Quantum Field Theory the study of Supersymmetric Gauge Theories will be continued, but in an interdisciplinary perspective. A Gauge Theory is a mathematical approach to symmetries of physical systems .Maxwell equations is an elementary example. Supersymmetry takes into account two kinds of particles, bosons and fermions, with different statistics. The method of random matrices (matrix-valued random variables) will be related with other mathematical fields. Most of the topics will be investigated in international collaborations, through multiple contacts already established or new ones resulting from the intensive participation of GFMUL members in conferences abroad. 10.3 Summary in English for evaluation GFMUL will apply tools of Stochastic Analysis in two fields, deterministic until recently. The first is Geometric Mechanics.There is more than one way to define what could be a Stochastic Geometric Mechanics, where typical trajectories have the irregularities of Brownian ones.Their common aim is to preserve as much as possible the fundamental geometric structures of Classical Mechanics. We have an original and very general way to do this (the method of "Stochastic Deformation") and will develop this approach, valid also for some infinite dimensional systems of Fluid Dynamics.The second deterministic field is Optimal transport theory and our method of Stochastic Deformation will be analysed in PDEs and probabilistic perspectives. Our program of Spectral Geometry will develop along several lines, one devoted to isoperimetric inequalities, asymptotic behaviour of extremal sets, trace inequalities etc. Numerical studies for high dimensional domains, in this fields, are quite important and will be vigorously pursued. The construction of a systematic theory of Quantum Graphs, whose first results were obtained recently at GFMUL, will be another ambitious project along this line. The many consequences of the ODE/IM correspondence established in 2016/17 will be analysed, especially the relations with String Theory and those involving Painlevé equations and Donaldson Thomas invariants on one side as well as connections between Algebraic Geometry and Integrable systems on the other one. Specific models of Quantum Integrable systems with non-periodic boundary conditions, in terms of the Algebraic Bethe Ansatz and the classification of generalised Gaudin algebras will be investigated, using the methods of fusion and twisting. Another aspect of Integrability is the use of the localisation method in Supersymmetric Gauge Theory, allowing to obtain exact results. The analysis and evaluation of random matrix ensembles, in particular for the study of phase transitions will go on but in a more interdisciplinary perspective. Indeed, random matrix theory is now in contact with a number of other fields of Mathematics : Operator theory, reproducing kernels, representation theory, algebraic combinatorics. Toeplitz matrices, in particular, should be carefully studied. The tools of pseudo-differential operators theory will be involved in higher dimensional representations of Quantum Mechanics, i.e. representations where observable operators act on Hilbert space of functions of dimension higher than the traditional configuration space (for instance on the phase-space).These representations provide, in particular, unified Hilbert space formulations of pure and mixed states. It is also part of our research program to look for a canonical quantisation of General Relativity generalising the spin-network quantisation.

Institutions

Main Institutions

  • Universidade de Lisboa Grupo de Física Matemática (ULisboa GFM)
  • FCiênciasID Associação para a Investigação e Desenvolvimento de Ciências (Fciências.ID)

Funding 347.500,00 €

Fundação para a Ciência e a Tecnologia (FCT) - Portugal

347.500,00 €