Vorträge in der Woche 29.06.2015 bis 05.07.2015

Dienstag, 30.06.2015: The PDEs of Mathematical Finance

Gisele R Goldstein (University of Memphis)

We discuss the two most important equations of mathematical finance, the Black-Scholes equation and the Cox-Ingersoll-Ross equation. These are deterministic equations having stochastic backgrounds in the mathematical finance context. These equations are studied on weighted sup norm spaces with various positive weights, Results to be presented include the semigroup generation, the relationship of the Black- Scholes equation to the generalized heat equation, and chaos for the generalized heat and Black-Scholes equations. For the Cox-Ingersoll-Ross equation, we obtain (C0) semigroup generation and a new Feynman-Kac type formula. The results presented are joint work with several co-authors, H. Emamirad, J. A. Goldstein, R. Mininni, Ph. Rogeon, and S. Romanelli.

Dienstag, 30.06.2015: Non-stationary problems and low-rank approximation of matrices and tensors

Ivan Oseledets (Institute of Numerical Mathematics, Moskau)

Dienstag, 30.06.2015: The third moment of symmetric square L-functions

Benjamin Sprung (Goettingen)

Donnerstag, 02.07.2015: Morse index and Geometry

Prof. André Neves (Imperial College)

The relation between the Morse index and topology has been very fruitful over the last 100 years. I will survey some of these developments concerning the theory of geodesics and minimal surfaces and also talk about some new results. This is joint work with Fernando Marques.

Donnerstag, 02.07.2015: In search of a really useful principle of linearized stability

Jerry Goldstein (University of Memphis)

The heat equation on all of N dimensional Euclidean space containing the nonlinear term u to the pth power has a positive equilibrium solution u of the form a constant times r to the q-th power, as long as N is at least three. Is this solution u stable? This difficult question is looked at using the Principle of Linearized Stability. But the corresponding linearized equation involves the inverse square potential and is well know to have complicated, singular behavior. This leads to surprising, dimension dependent results. Moreover, it leads to the realization that the Principle of Linearized Stability has its conclusion hold in many cases when its hypotheses fail to hold. This is related to recent spectacular results in applied mathematics and physics.

Donnerstag, 02.07.2015: Atomic Bose-Einstein Condensates in Optical Ring-Resonators

Prof. Claus Zimmermann (Universität Tübingen)

Freitag, 03.07.2015: The classical entropy of quantum states

Prof. Jan Philip Solovej (University of Copenhagen)

To quantify the inherent uncertainty of quantum states Wehrl ('79) suggested a definition of their classical entropy based on the coherent state transform. He conjectured that this classical entropy is minimized by states that also minimize the Heisenberg uncertainty inequality, i.e., Gaussian coherent states. Lieb ('78) proved this conjecture and conjectured that the same holds when Euclidean Glauber coherent states are replaced by SU(2) Bloch coherent states. This conjecture was settled recently in joint work with Lieb. Recently we simplified the proof and generalized it to SU(N) for general $N$. In proving the conjecture we study the quantum channels known as Universal Quantum Cloning Machine} and determine their minimal output entropy.