PRR Project

Summary

CAMGSD centers its scientific activity on the broad areas of Differential Equations and Dynamical Systems, and of Geometry and Topology. These are principal areas of mathematics that are thus strongly represented in the scientific work of the members of CAMGSD and of the members of the Department of Mathematics. The research conducted in Differential Equations and Dynamical Systems by members of the Center includes in particular the study of the qualitative theory of differential equations, bifurcation theory, nonuniform hyperbolicity, and Morse-Smale structures; functional and difference equations, reaction-diffusion equations, and elliptic problems in partial differential equations; geometric mechanics, Hamiltonian systems, and the integrability and nonintegrability of equations of mathematical physics; variational problems and Morse theory, variational methods, and general relativity; ergodic theory, also in connection with nonuniform hyperbolicity, topological dynamics, and zeta functions; as well as stochastic analysis, Markov processes, particle systems.Among the rather varied collection of topics studied by members of CAMGSD in Differential Equations and Dynamical Systems, calculus of variations plays an important role linking mathematical analysis and differential equations, both ordinary and partial, as well as developing links with geometry and topology via geometric variational problems. Calculus of variations was instrumental in the development of Lagrangian mechanics, for instance with the Euler-Lagrange equations, and of optimal control theory. Roughly speaking, one could briefly refer to the topic in layman’s terms as having the purpose of solving optimization problems in the context of Differential Equations and Dynamical Systems, and thus also of many situations that can be modeled by them. However, particularly since the 1970’s the area experienced a considerable development building also on techniques from geometric analysis, measure theory, and more modern theory of PDEs. This development was and still is much indebted to the study of elasticity and microstructure in the context of materials science, in which case certain mathematical hypotheses (for example related to convexity or growth) used in the theory may not be satisfied in potential applications and which thus motivated some of the more recent developments.Various members of CAMGSD actively develop work on nonlinear PDEs, calculus of variations, geometric measure theory and mathematical relativity with some collaborations between them, but none connecting simultaneously all these subjects. The modern theory of the Calculus of variations makes strong connections between all these topics and has a strong potential to attack nonlinear problems that are also relevant for the applications. Unsurprisingly, this requires both breadth and depth in various mathematical fields, that although related are hard to grasp simultaneously. In order to increase this capacity in CAMGSD, we would like to acquire a researcher capable of building bridges between all the above areas, and the topics of geometric PDEs and geometric calculus of variations largely have the potential to bridge this gap. This broad area is also important for different topics of applied mathematics such as mathematical physics, optimal control, variational Bayesian methods, and image processing. In particular, the modern development of Calculus of variations has potential applications to the study of defects, phase transitions and other problems in materials science, since these often lead to optimization problems that can be formulated rigorously in the context of the Calculus of variations, with some departure from the Lagrangian approach to the direct method of establishing the existence and possible regularity of minimizers. This often requires a panoply of mathematical techniques that need to be used effectively, ranging from convex functions to lower semicontinuity properties.The tasks that the hired assistant professor will perform include: (1) to conduct research in the identified areas and to publish its results in high ranking journals; (2) to disseminate the results of the research in international scientific events; (3) to collaborate in the organization of seminars at CAMGSD that promote knowledge around the topics just described, in particular inviting researchers from other institutions, both national and foreign; (4) to organize or host local and international scientific events; (5) to apply for grants of various types, e.g., ERC grants, in the areas identified, both those in pure mathematics and those with emphasis on applications, so as to attract additional funding that will promote the growth of the activities in this area, taking advantage of the Research & Innovation Funding support structure of IST-ID, in particular the ERC accelerator program; (6) to collaborate in the aforementioned opportunities that may arise from the private sector.