The concentration of Lipschitz functions around their expectation is a classical topic and continues to be very active. In these talks, we will discuss some recent progress in detail, including: A tight log-Sobolev inequality for isotropic logconcave densities A unified and improved large deviation inequality for convex bodies An extension of the above to Lipschitz functions (generalizing the Euclidean squared distance)The main technique of proof is a simple iteration (equivalently, a Martingale process) that gradually transforms any density into one with a Gaussian factor, for which isoperimetric inequalities are considerably easier to establish. (Warning: the talk will involve elementary calculus on the board, sometimes at an excruciatingly slow pace). Joint work with Yin Tat Lee.
Quantum graphs are metric graphs with differential equations defined on the edges. Recent interest in control and inverse problems for quantum graphs
is motivated by applications to important problems of classical and quantum physics, chemistry, biology, and engineering.
In this talk we describe some new controllability and identifability results for partial differential equations on compact graphs. In particular, we consider graph-like networks of inhomogeneous strings with masses attached at the interior vertices. We show that the wave transmitted through a mass is more
regular than the incoming wave. Therefore, the regularity of the solution to the initial boundary value problem on an edge depends on the combinatorial distance of this edge from the source, that makes control and inverse problems
for such systems more diffcult.
We prove the exact controllability of the systems with the optimal number of controls and propose an algorithm recovering the unknown densities of thestrings, lengths of the edges, attached masses, and the topology of the graph. The proofs are based on the boundary control and leaf peeling methods developed in our previous papers. The boundary control method is a powerful
method in inverse theory which uses deep connections between controllability and identifability of distributed parameter systems and lends itself to straight-forward algorithmic implementations.
