The seminar meets regularly on wednesday at 4pm in Eckhart 202. We also have special seminars during other days.
Abstract: We classify all solutions to the $SU(N+1)$ Toda system. As a consequence, we construct non-topological solutions to $SU(3)$ and $B_2$ Chern-Simons system.
A variational approach, consisting in minimizing a certain error functional, will be set up to study the Navier-Stokes system. The functional-analytical scenario is chosen so that feasible vector fields, if they turn out to be weak solutions in a suitable sense, are automatically smooth. The error functional measures ``how far" feasible fields are from being a solution. Once the basic ingredients are described, the task is two-fold: 1. Prove existence of minimizers for the error. 2. Show that the minimum value vanishes. Non-standard tools are required to cover these two steps. On the one hand, coercivity fails both in dimension 2 and 3. For dimension 2, a substitute can be found taking advantage of the natural scaling. In dimension 3, this is not possible due to supercriticality. The job of showing that the minimum vanishes requires to look at optimality conditions and the linearized system.
This is a survey of some recent development in the study of Ricci solitons. These manifolds are very important in the study of singularities of the Ricci flow and are natural extensions of Einstein manifolds. Among other things, we will discuss curvature estimates, volume growth and topology at infinity. Understanding certain partial differential equations on Ricci solitons is very important for this study.
This work is concerned with the study of a model case of first order Hamilton-Jacobi equations posed on a ``junction'', that is to say the union of a finite number of half-lines with a unique common point. The main result is a comparison principle. We also prove existence and stability of solutions. The two challenging difficulties are the singular geometry of the domain and the discontinuity of the Hamiltonian. As far as discontinuous Hamiltonians are concerned, these results seem to be new. They are applied to the study of some models arising in traffic flows. The techniques developed in the present work provide new powerful tools for the analysis of such problems.
There will be special activities during these two weeks.
Locomotion due to body undulations is observed across the entire spectrum of swimming organisms, from microorganisms to fish. The internal force generating mechanisms range from the action of dynein molecular motors within a mammalian sperm to muscle activation in lamprey. We will present recent progress in building multiscale computational models that couple biochemistry, passive elastic properties and active force generation with a surrounding fluid for these two swimmers.