Abstract: I will describe some recent results related to some simplified models of Liquid Crystals, with a view towards geometric free boundaries. A simplified version of the Ericksen-Leslie system has been introduced by FH Lin in the 80's. After describing the state of the art for this system, I will introduce a new one involving a free boundary. Though the mathematical analysis of the system is still very preliminary, some results are still available in 2 dimensions and I will mainly motivate the introduction of geometric variational problems with free boundaries and how one can deal with them, thanks to recent advances in compensated-compactness in odd dimension. I will mention several open questions and possible further generalizations.

Abstract: I will discuss some recent results obtained in collaboration with A. Figalli, S. Kim and H. Shahgholian. We consider minimizers of the Dirichlet energy among maps constrained to take values outside a smooth domain O in ℝ^m. These minimizers can be thought of as harmonic maps into the manifold-with-boundary given by the complement of O, while O acts as an obstacle. I will discuss results concerning the regularity of the minimizers, the location of their singularities, and the structure of the free boundary.

Abstract: Keller-Segel type cross-diffusion systems have been playing an outstanding role in the understanding of various patterning phenomena in biology. Concentrating on issues of predominant application relevance, the description of taxis-driven explosions has been among the most challenging topics in their analysis, and a natural focus of the literature in this regard is on the characterization of solution behavior near collapse. The presentation aims at reporting both on classical and on some recent developments, with a particular focus on the identification of circumstances under which solutions either must blow up at single points only, or alternatively may form ingularities throughout larger regions in space.

Abstract: Collision-induced breakage equations describe the fragmentation of clusters as a result of pairwise collisions with possible transfer of matter. Accounting or not for the latter leads to different dynamical properties but also requires different analytical arguments to study the well-posedness, which will be described when the size of the clusters ranges in the set of positive integers.

Joint works with M. Ali (O.P. Jindal Global University) and A.K. Giri (IIT Roorkee).

Meeting point: Campo S.Mamede, Caminho do Castelo (GPS Coordinates: 41.4483, -8.2899) 

The tour ends next to the wall with the inscription "Portugal was born here."

R. de Valdonas 4, 4800-476 Guimarães (GPS coordinates: 41.4433, -8.2945)