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Actions of non-abelian groups and measure rigidity

Abstract: Measure rigidity theorems (especially Ratner’s theorem) have played a big role in dynamics, with many applications outside the field. Traditionally problems of this type have
been considered in the context of homogeneous dynamics and other systems with a lot of structure. In this talk I will discuss a circle of ideas which suggests that, under some conditions, some measure rigidity theorems may hold for general smooth actions of non- abelian groups.

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Speaker: Alex Eskin (Chicago)

Reflecting Brownian motions and point-to-line last passage percolation

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Speaker: Jon Warren (Warwick)

Variants of RSK and polymers

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Speaker: Nikos Zygouras (Warwick)

Moments and tails of stochastic PDEs

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Speaker: Li-Cheng Tsai (Rutgers)

Limit shapes and local statistics for the stochastic six-vertex model

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Speaker: Amol Aggarwal (Columbia)

Symmetric functions from vertex models

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Speaker: Leonid Petrov (Virgina)

Random matrices as seen from infinite beta, part 1; part 2

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Speaker: Vadim Gorin (MIT, Wisconsin)

Algebraic combinatorics meets probability: statistical mechanics and asymptotics, part 2

Abstract: Algebraic Combinatorics sits at the intersection of Combinatorics and Representation Theory and Algebra, using algebraic methods to answer combinatorial questions and employs tools from discrete mathematics to study problems in group theory and representation theory. In this minicourse we will start with introducing basic tools and theorems from Algebraic Combinatorics which have seen wide applications in Statistical Mechanics: symmetric function theory, representation theory of the symmetric group, the RSK algorithm and nonintersecting lattice paths etc. We will then see how we can use them to derive probabilistic results like limit behavior in dimer models (mainly lozenge tilings). We will also discuss the application of probabilistic methods in enumerate asymptotically combinatorial objects.

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Speaker: Greta Panova (USC)

Limit shapes and variational principles, talk 3

Abstract: We discuss the notion of limit shape for random height models in 2+1 dimensions, giving basic examples and ideas of proofs. We discuss the analysis of the resulting variational problems, in particular the connection between the conformal structure and the surface tension. This is based on joint work with Istvan Prause.

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Speaker: Rick Kenyon (Yale)

On the free energy of the six-vertex model, talk 3

Abstract: In this series of lectures, we will explain how to compute the free energy of the six-vertex model with a given density of up arrows, and how one can relate the behaviour of this free energy to localization/delocalization of the corresponding height-function, and rotation invariance of the model in the delocalized regime.

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Speaker: Hugo Duminil-Copin (IHÉS)

An invitation of the q-Whittaker polynomials, talk 3

Abstract: The q-Whittaker polynomials are a family of symmetric functions that can be obtained as a degeneration of the famous Macdonald polynomials. They have played an important role in integrable probability, notably via the framework of Macdonald processes and their connection to the q-TASEP.

The aim of these lectures will be to give two combinatorial formulas for the q-Whittaker polynomials, using the theory of integrable vertex models. These formulas look completely different, but both of them exhibit the q-positivity of the q-Whittaker polynomials in an explicit way. In reaching this goal, we will pass through a number of important landmarks in the theory, including the coloured stochastic six-vertex model, fusion, and the Yang–Baxter equation.

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Speaker: Michael Wheeler (Melbourne)

Algebraic combinatorics meets probability: statistical mechanics and asymptotics, part 1

Abstract: Algebraic Combinatorics sits at the intersection of Combinatorics and Representation Theory and Algebra, using algebraic methods to answer combinatorial questions and employs tools from discrete mathematics to study problems in group theory and representation theory.
In this minicourse we will start with introducing basic tools and theorems from Algebraic Combinatorics which have seen wide applications in Statistical Mechanics: symmetric function theory, representation theory of the symmetric group, the RSK algorithm and nonintersecting lattice paths etc. We will then see how we can use them to derive probabilistic results like limit behavior in dimer models (mainly lozenge tilings). We will also discuss the application of probabilistic methods in enumerate asymptotically combinatorial objects.

Details

Speaker: Greta Panova (USC)