In the talk we introduce the idea of an "integrable gas" as a collection of large random ensembles of special localized solutions (solitons, breathers) of a given integrable system. These special solutions can be treated as "particles". Known laws of pairwise elastic collisions allow one to write the heuristic "equation of state" for the gas of such particles.

In this talk we consider soliton and breather gases for the fNLS as special thermodynamic limits of finite gap (nonlinear multi phase wave) fNLS solutions. In this limit the rate of growth of the number of bands is linked with the rate of (simultaneous) shrinkage of the size of individual bands. This approach leads to the derivation of the equation of state for the gas and its certain limiting regimes (condensate, ideal gas limits), as well as construction of various interesting examples. We also discuss the recent progress and perspectives of future work, as well as some possible applications.

In this talk, we present a proof of local Holder regularity of solutions to generated Jacobian equations as a generalization of optimal transport case, which is proved by George Loeper. We compare structures of generated Jacobian equations with optimal transport, and point out differences with difficulties which the differences can cause. For local Holder regularity theory, we use (G3s) condition and solution in Alexandrov sense. (G3s) is a strict positiveness type condition on MTW tensor associated to the generating function G, and Alexandrov solution is a solution that satisfies pullback measure condition. (G3s) is used to show a quantitative version of (glp), which gives some room for G-subdifferentials of solutions. Then the inequality for Holder regularity is shown by comparing volumes of G-subdifferentials using the fact that our solutions is in Alexandrov sense.

The theory of periodic Jacobi matrices on the line is extremely rich and very well studied. Viewing the line as a regular tree of degree 2 leads to a natural generalization to periodic Jacobi matrices on general trees. This family of operators, which is at least as rich (by definition), but considerably less well understood, is at the center of this talk. We review some of the few known results, present some examples, and discuss open problems and directions for future research. The talk is based on joint work with Nir Avni and Barry Simon.

Properties of the parabolic equation associated with the fourth derivative operator (the bi-Laplacian) on $\mathbb R^d$, including a semi-explicit formula for the heat kernel, have been known since early investigations by Krylov, Hochberg, and Davies. On a domain or a metric graph conditions, the bi-Laplacian is generally not anymore acting as a squared operator, though: this strongly affects its analytic features. We are going to describe self-adjoint and eventually markovian extensions of the bi-Laplacian, focusing on how the properties of the semigroup generated by bi-Laplacians on metric graphs strongly depend on the boundary conditions. I will also discuss how classical extension theory can be applied to characterize realizations of higher order parabolic equations with dynamic boundary conditions. This is joint work with Federica Gregorio.