Math/Statistics Seminar: Alessandro Selvitella
Part 1: On stationary solutions to the Nonlinear Schrodinger Equation on Hd
In this talk, we discuss some qualitative properties of stationary solutions to the NLS on the hyperbolic space. We prove a variational characterization of the ground state, its uniqueness and nondegeneracy, and give a complete description of the spectrum of the linearized operator around the ground state. We prove some rigidity theorems in weighted spaces too. Most of these results have natural counterparts in the Euclidean setting. Speculating a little, they suggest that, in the presence of a conning potential, there should exist non-trivial waves pushed by the innitesimal generators of the Poincare group, as in the Euclidean space there exist rotating waves following the subgroup of rotations of the Euclidean group.
Part 2: D'Arcy Thompson's Growth and Form and some questions of John Milnor
In 1917, D'Arcy Thompson published \On growth and form", a book on biological shape analysis full of examples and ideas, but with little mathematical formalism. Thompson believed that the changes of form across species and the growth of an individual are conformal trans-formations. More recently, also John Milnor became interested in the same question. Milnor made some calculations on Thompson's data using invariants like the cross-ratio, but he found contradictory results. In this talk, we outline the possible strategies that we are trying to follow to attack the problem.