Reading seminar on integrable probability in Spring 2019
Fridays, 11:00am - 12:30pm, Kerchof 128
We will discuss the classical problem of describing analogues of irreducible representations of the infinite-dimensional unitary group (Edrei-Voiculescu theorem), and its connections with
- representations of finite-dimensional unitary groups
- symmetric functions
- random lozenge tilings and asymptotic combinatorics
- total positivity
- classical analysis (e.g. B-splines)
- von Neumann factor and spherical representations of the infinite-dimensional unitary group
- random matrices
Schedule of meetings
The schedule is subject to change as we progress with the topics.
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January 18, 2019, special location - Ruffner Hall 127.
Leo Petrov - Introduction “The many facets of the Edrei-Voiculescu theorem”
-
January 25, 2019, special location - Ruffner Hall 123.
Liron Speyer - $U(N)$: weights, irreducible characters, Schur polynomials, branching
[standard textbooks like the H. Weyl’s], a summary is in [section 2.2 in arXiv:1310.8007] -
February 1, 2019
George Seelinger - Extreme characters of $U(\infty)$: definition and two elementary examples [PDF notes]
[for example, section 4.1.10 in these draft lecture notes] -
February 8, 2019
Axel Saenz - Functional equation for irreducible characters and the ring theorem for extreme characters of $U(\infty)$
[G. Olshanski, On semigroups related to infinite-dimensional groups. In: Topics in Representation Theory (A. A. Kirillov, ed.). Advances in Soviet Math.] or [A. Borodin, G. Olsahnski “Representations of the Infinite Symmetric Group”, exercises 1.5 and 4.3] -
February 15, 2019
Ethan Zell - Projection formula for Gibbs measures on tilings (formula for the number of trapezoidal Gelfand-Tsetlin schemes)
[arXiv:1109.1412 and arXiv:1208.3443] -
February 22, 2019
Leo Petrov - Idea of proof of Edrei-Voiculescu theorem
[arXiv:1109.1412] -
March 1, 2019
No meeting
-
March 8, 2019
Joseph Eisner - Characters and von Neumman factor representations. GNS construction
-
March 22, 2019
Axel Saenz - Pascal triangle as a branching graph [PDF notes]
[Kerov, “Combinatorial examples in the theory of AF-algebras”], [A. Borodin, G. Olsahnski “Representations of the Infinite Symmetric Group”], these draft lecture notes -
March 29, 2019
Arun Kannan - Diffusions from dynamics on the Pascal triangle
[arXiv:0706.1034, section 2] -
April 5, 2019
No meeting
-
April 12, 2019
George Seelinger - A q-analogue of de Finetti’s theorem
[arXiv:0905.0367] -
April 19, 2019
Liron Speyer - Young graph and its boundary
[A. Borodin, G. Olsahnski “Representations of the Infinite Symmetric Group”] -
April 26, 2019
Joseph Eisner - Construction of extreme representations of the infinite symmetric group
Chapter 10, [A. Borodin, G. Olsahnski “Representations of the Infinite Symmetric Group”]
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