The seminar is devoted to the study of Gibbs (or central) measures on graded graphs. The general theory has appeared in works of Vershik, Kerov, and Olshanski and continues to develop nowadays. On top of its intrinsic beauty, it has connections with classical probability questions, combinatorics, dynamical systems, representation theory, and even biology.
We will start with earlier (and more elementary) examples related to de Finetti's theorem and Kingman's partition structures, and their variations.
If this will go well, we will discuss more complicated settings: the Gibbs measures on the Young graph and the Gelfand-Tsetlin graph.
Friday 14:00 -- 16:00, room 0.007 , Endenicher Allee 60
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Schedule
5 April -- Introduction ( by Alexey)
12 April -- no seminar
19 April --Holiday
26 April -- no seminar
3 May -- Pascal's graph and de Finetti's theorem ( Alexey)
10 May -- Kingman partition structures: motivation, definition, examples
17 May -- Poisson-Dirichlet process
24 May -- Kingman partition structures: classification
31 May -- Symmetric functions and branching graphs.
7 June -- general de Finetti's theorem
14 June -- Holiday
21 June -- Coherent measures on Young's graph: motivation, examples.
28 June -- no seminar
5 July --- TBA
12 July -- no seminar
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Possible topics
0) Introduction (by Alexey)
1) de Finetti's theorem and Pascal graph. ( https://projecteuclid.org/euclid.aop/1176995566 , .... )
2) Kingman's partition structures: motivation, definition, examples. ( Sections 2 and 3 of https://royalsocietypublishing.org/doi/abs/10.1098/rspa.1978.0089 )
3) Kingman's partition structures: classification ( https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/jlms/s2-18.2.374 )
4) Poisson-Dirichlet process.
5) Symmetric functions and branching. Kingman's branching graph.
6) Young's graph. Classification of Gibbs measures.
7) RSK algorithm. Sampling of Gibbs measures on Young's graph.
8) Gelfand-Tsetlin graph. Classification of Gibbs measures.
9) Gibbs measures and total positivity.