HColl_2014

Analysis is replete with inequalities stating that some linear operator is bounded from one Banach space to another, commonly with an unspecified operator norm. However, for a few fundamental inequalities, optimal constants and extremizing functions are known. This arises typically for inequalities with a high degree of symmetry. The character of extremizers reflects and sheds light on underlying algebraic or geometric structure.
This talk is concerned with characterization of functions (or sets) that nearly, but not exactly, extremize certain classical inqualities set in Euclidean space: Young's inequality concerning convolutions of functions, the Brunn-Minkowski inequality concerning sums of sets, the Riesz-Sobolev inequality concerning convolutions of indicator functions of sets, and the Hausdorff-Young inequality concerning the Fourier transform.
Analyses that identify extremizers are often unstable, and may reveal nothing about near extremizers. Recent progress on near extremizers relies on inverse theorems from additive combinatorics. Arithmetic progressions play a central role.
The four inequalities, their extremizers, and two combinatorial inverse theorems will be reviewed. Interconnections will be sketched.

Nicola Gigli: Spaces with Ricci curvature bounded from below

We will discuss the definition of metric measure spaces with a lower bound on the Ricci curvature and some of their analytic and geometric properties.

Valentin Blomer: Trace formulae in number theory

Starting from the Poisson summation formula, I discuss spectral summation formulae on locally symmetric spaces and present a variety of applications to automorphic forms, analytic number theory, and arithmetic.

Aaron Naber: Characterizations of Bounded Ricci Curvature on Smooth and Nonsmooth Spaces

In this talk we discuss several new estimates on manifold with bounded Ricci curvature, and in particular Einstein manifolds. In fact, the estimates are not only implied by bounded Ricci curvature, but turn out to be equivalent to bounded Ricci curvature. We will see that bounded Ricci curvature controls analysis on the path space P(M) of a manifold in much the same way that lower Ricci curvature controls analysis on M. There are three distinct such characterizations given. The first is a gradient estimate that acts as an infinite dimensional analogue of the Bakry-Emery gradient estimate on path space. The second is a C^{1/2}-Holder estimate on the time regularity of the martingale decomposition of functions on path space.
For the third we consider the Ornstein-Uhlenbeck operator, a form of infinite dimensional laplace operator, and show that bounded Ricci curvature is equivalent to an appropriate spectral gap. One can use these notions to make sense of bounded Ricci curvature on abstract metric-measure spaces.

Clément Mouhot: Factorization of semigroups and applications to PDEs

We will present a general factorization method for estimating growth of semigroups in Banach spaces for a class of operators, and discuss various applications for linear and nonlinear PDEs of Boltzmann and Fokker-Planck types.

Zhan Shi: Biased random walks on trees

I am going to make some elementary discussions on the asymptotic properties of recurrent biased random walks on rooted trees.