Project 1. Geometry and non-linear analyses in singular spaces and applications
The project brings together the graduate programs in Mathematics (grade 7), Teleinformatics Engineering (grade 6) and Economics (grade 5) and has as a guiding principle the use of geometric and analytical methods in the Theory of Information and signal processing, with applications in Telecommunications and Finances. The PPG (Graduate Program) in Mathematics is one of the pioneers, nationally, in terms of international collaborations, formalized, in the last years, in many agreements and projects, for example PRONEX, COFECLUB, Science Without Borders, among others. The Mathematics major was promoted to grades 6 and 7 in consecutive CAPES assessments, whereas the GP in Teleinformatics Engineering, despite its recent installation, has been awarded with international quality status. In the last years, researchers of both GPs have participated in joint research projects, financed by funding agencies, some of which with counterpart from foreign institutions. Recently, the interest of research, originally focused in geometric, dynamical or analytical aspects of signal processing and optimization in the presence of curvature, derived to possibilities of application to Finances of machine learning methods adapted to non-linear or singular contexts. With these developments, the projects started to involve researchers of the GP in Economics, a program with intense history of collaboration with administration and public finances bodies in the State of Ceará. The project seeks to integrate these internal collaborations to the foreign cooperation network maintained by the most internationally consolidated majors, involving, in the present proposal, prestigious American and French universities. Such integration is fully aligned with the UFC Internationalization Agenda. In general terms, the mathematical models defined in the project are elaborated in terms of partial differential equations, deterministic or stochastic. Complex phenomenon, involving nonlinearities, phase transitions and structure emergencies which optimize energy can be geometrically understood as manifestations of a space curvature, object of study of differential geometry and its analytical counterpart, Geometrical Analysis. The theme involves the use of geometric, dynamical and analytical techniques in singular or stratified spaces to the study of the relation between stochastic processes, phase transitions and geometry of information applied to large data sets about financial assets.
