Geometry Seminar
Date: March 4, 2022
Time: 4:00PM - 5:00PM
Location: BLOC 306
Speaker: Igor Zelenko , TAMU
Title: Morse inequalities for eigenvalue branches of generic families of self-adjoint matrices
Abstract: The talk is based on the joint work with Gregory Berkolaiko. The eigenvalue branches of families of self-adjoint matrices are not smooth at points corresponding to repeated eigenvalues (called diabolic points or Dirac points). Generalizing the notion of critical points as points for which the homotopical type of (local) sub-level set changes after the passage through the corresponding value, in the case of the generic family we give an effective criterion for a diabolic point to be critical for those branches and compute the contribution of each such critical point to the Morse polynomial of each branch, getting the appropriate Morse inequalities as a byproduct of the theory. The motivation comes from the Floquet-Bloch theory of Schrodinger equations with periodic potential and other problems in Mathematical Physics.