15:00 – Anton Alekseev (Geneva): The Horn problem and planar networks
The Horn problem is a classical problem of Linear Algebra which establishes a complete set of inequalities on eigenvalues of a sum of two Hermitian matrices with given spectra. It was solved by Klyachko and Knutson-Tao in the end of 1990s. Surprizingly, the same set of inequalities comes up in the problem of maximal multi-paths in a planar network equipped with Boltzmann weights on its edges. The link between the two problems is via the tropical limit. In order to control this limit we are using the Liouville volume on the space of solutions of the Horn problem.
The talk is based on a joint work with M. Podkopaeva and A. Szenes.
16:30 – Frédéric Klopp (Paris-Jussieu): Stark-Wannier ladders and cubic exponential sums
The talk is devoted to Stark-Wannier ladders i.e. the resonances of a one dimensional periodic operator in a constant electric field. The periodic sequences of points in the lower half of the complex plane have been conjecture to be very sensitive to the nulmber theoretical properties of the electric field. Computing the asymptotics of the reflection coefficients in the case of simple 1-periodic potential, we related the resonances to cubic exponential sums in which the frequency is computed from the electric field. In the case of rational frequency, we derive “large imarginary part” asymptotics for the resonances.
The talk is based on joint work with A. Fedotov (St Petersburg).