Chern-Simons Lectures: Jean-Bernard Zuber (Laboratoire de Physique Théorique et Hautes Energies, Sorbonne Université, Paris) Tuesday, April 2-Thursday, April 4

Lecture 1.   Counting curves and knots and links
Tuesday April 2, 11:00-12.30 am, Le Conte 402

 This talk will be devoted to the problem of counting curves, knots and links, a classical mathematical problem in which physics may bring new ideas and methods. After reviewing how matrix integrals enable one to count planar “maps” and alternating links, I turn to the more difficult problem of counting and listing curves and knots, i.e. objects with a single component.

Lecture 2. Revisiting Horn’s problem
Wednesday April 3, 2:00-3.30 pm, Le Conte 402

Horn’s problem deals with the following question: what can be said about the spectrum of eigenvalues of the sum C=A+B of two Hermitian matrices of given spectrum?  The support of the spectrum of C is well understood, after a long series of works from Weyl (1912) to Horn (1952) to Klyachko (1998) and Knutson and Tao (1999). In this talk, after a short review of the problem, I show how to compute the probability distribution function of the eigenvalues of C, when A and B are independently and uniformly distributed on their orbit under the action of the unitary group. Comparison with the similar problem for real symmetric matrices and the action of the orthogonal group reveals unexpected differences…

Lecture 3. Horn’s problem and representation theory
Thursday April 4, 11:00-12.30 am, Le Conte 402

Horn’s problem has also amazing connections with group  theory and the decomposition of tensor product of representations. Recent progress in that direction will be discussed in that third lecture.



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