Differential Geometry
Members
Ernani de Sousa Ribeiro Júnior
Fernanda Ester Camillo Camargo
Gregório Pacelli Feitosa Bessa
Luquésio Petrola de Melo Jorge
Description
Differential Geometry has its roots in the use of the techniques of Calculus to handle geometrical problems. However, it only became an established research area in the nineteenth century, with the discoveries of non-Euclidean geometries by J. Bolyai, N. Lobatchevski and, mainly, through the works of J. F. K. Gauss and B. Riemann. At this time, two revolutionary ideas flourished: that of the existence of a great many geometries, beyond that of Euclid, as well as that of the possibility of intrinsic measurement of the curvature of a surface. In spite of having abstract roots, these geometries in curved spaces came out to be the natural scenario for General Relativity and, more recently, to Field Theory, a classical topic in contemporary Physics. On the other hand, throughout the last century - and following a general trend in Mathematics - Differential Geometry became a research area with several specializations, all of which presenting extreme theoretical complexity. At the same time, it spectacularly broadened its scope of applications, to such diverse fields as Computer Graphics, Economics, Engineering, Medicine and Statistics, among others.
Research in Differential Geometry at UFC is intimately linked to the global development of the area, for it has emerged concurrently to other major groups in the field, all of which were profoundly influenced by S. S. Chern - one of the greatest geometers of the twentieth century - and Manfredo P. do Carmo, the founder of the so-called brazilian school of Differential Geometry. From this initial birth, we developed into a team of researchers which, presently, amounts to fourteen geometers, who have awarded PhD degrees to a great many students - who, in turn, settled as professors in several universities throughout Brazil -, cooperate in many international research projects and concentrate their publications in international specialized journals of the highest standards.
Among our main lines of research, we highlight investigations on Geometry of Submanifolds and Geometric Analysis. As the name suggests, in the first theme we focus on the investigation of the geometry of submanifolds - objects that derive from a more general notion than that of surfaces or membranes - whose shape have least area or dispend less energy, in some sense; therefore, two of the main subjacent questions are those of stability and rigidity of submanifolds of ambient spaces possessing special geometric structures. Geometric Analysis, in turn, is a recent specialization, whose prominence derives from the fact that its techniques allowed geometers to solve a number of problems that remained open for decades, as is the case of the celebrated Poincaré Conjecture. Within Geometric Analysis, we have focused on problems related to the existence and uniqueness of solutions of geometric partial differential equations, in several curved spaces.
