Abstract. It is a well-known fact that the Poincare inequality does not hold on a connected sum Rⁿ#Rⁿ for n>2. In this talk, we discuss the estimates of the Poincare constant on manifolds with ends, where the Poincare constant is given by the supremum of variance over the Dirichlet energy form of functions on balls. This talk is based on a joint work with A. Grigor'yan and L. Saloff-Coste.

Abstract. We study removable sets (in the sense of Mazja) and L^p-uniqueness problems on manifolds and metric measure spaces. The Laplacian, considered on suitable spaces of test functions vanishing in a neighborhood of a 'removed' null set, has a unique extension if and only if this set has vanishing second order capacity. If the semigroup is strong Feller and satisfies a specific gradient inequality then two types of capacities are comparable and the critical size of the removed set can be expressed in terms of Hausdorff measures. As an application we show that from the point of view of essential self-adjointness the Laplacian on certain Gromov-Hausdorff limits of non-collapsing spaces 'ignores' the singular part of the limit space. This is joint work with Jun Masamune (Sapporo) and Kohei Suzuki (Pisa).

Abstract. By classical results of G.C.Evans and G.Choquet on ''good kernels G in potential theory'', for every polar F_s-set P, there exists finite measure μ on P such that Gμ=∞ on P, and a set P admits a finite measure μ on P such that 
{Gμ=∞}=P if and only if P is a polar G_d-set. A known application of Evans' theorem yields the solutions of the generalized Dirichlet problem for open sets by the Perron-Wiener-Brelot method using only harmonic  upper and lower functions.

In this talk it is shown that, by elementary ''metric'' considerations and without using any potential theory, such results can be obtained for general kernels G satisfying a local triangle property. The particular case, G(x,y)=|x−y|^a−d on R^d  where  2<a<d,  solves an open problem from the book of N.S.Landkoff on ''Foundations of Modern Potential Theory''.

Abstract. The notion of conformal walk dimension serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that the finiteness of the conformal walk dimension characterizes the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension is the infimum of all possible values of the walk dimension that can be attained by a time-change of the process and by a quasisymmetric change of the metric. Two natural questions arise 
(a) What are the possible values of the conformal walk dimension? 
(b) When is the infimum attained? 
In this talk, I will explain the answer to (a), and mention partial progress towards (b). 
This talk is based on joint work with Naotaka Kajino.

Abstract. In this talk we introduce (in some sense natural) diffusion processes on a parametric family of fractals called generalized diamond fractals. These spaces arise as scaling limits of diamond hierarchical lattices, which are studied in the physics literature in relation to random polymers, Ising and Potts models among others. In the case of constant parameters, diamond fractals are self-similar sets. This property was exploited in earlier investigations by Hambly and Kumagai to study the properties of the corresponding diffusion process and its associated heat kernel. These questions are of interest in particular because in this setting some usual assumptions like volume doubling are not satisfied. Alternatively, a diamond fractal can also be regarded as an inverse limit of metric measure graphs. Through a procedure proposed by Barlow and Evans, one can construct a canonical diffusion process for more general parameters, also in the absence of self-similarity. It turns out that it is possible to give a rather explicit expression of the associated heat kernel, which is in particular uniformly continuous and admits an analytic continuation.

Abstract.The Zakharov system is a system of coupled Schrödinger-wave equations which was originally derived as a model in plasma physics. We show that in dimensions d>3 for large wave data, and small Schrödinger data, solutions to the Zakharov system exist globally in time and scatter. Moreover, we extend the regularity region for well-posedness to the sharp region. The key step is to prove a Strichartz estimate for the Schrödinger equation with a potentially large free wave potential. In contrast to previous work on the Zakharov system, we avoid the use of normal forms, and instead work with spaces which are carefully adapted to control bilinear interactions between solutions to the Schrödinger and wave equations. Avoiding the use of normal forms allows us to consider data with regularity in the extended full region of local well-posedness. This is joint work with Sebastian Herr and Kenji Nakanishi.

Abstract. We consider symmetric Dirichlet forms that consist of strongly local (diffusion) part and non-local (jump) part on a metric measure space. Under general volume doubling condition and some mild assumptions on scaling functions, we establish stability of two-sided heat kernel estimates in terms of Poincare inequalities, jumping kernels and generalized capacity inequalities. We also discuss characterizations of the associated parabolic Harnack inequalities. Our results apply to symmetric diffusions with jumps even when the underlying spaces have walk dimensions larger than 2. This is a joint work with Z.Q. Chen (Seattle) and J. Wang (Fuzhou). 

Abstract. If a random variable X has an absolutely continuous density f(x), its score is defined to be the random variable 
\rho(X) = f'(X)/f(X), where f'(x) is the derivative of f. We will discuss upper bounds on the moments of the scores, especially in the case when X represents the sum of independent random variables.

Abstract. Within the framework of balayage spaces (the analytical equivalent of nice Hunt processes), we prove  equicontinuity of bounded families of harmonic functions and apply it to obtain criteria for compactness of potential kernels.

Abstract. An important problem in the theory of random processes and random fields (Dynkin, Dobrushin etc.) is to describe the probability measures with given co-transition distributions, e.g. to find all Markov processes with the same co-transition distributions as a given Markov process. The set of all such measures is referred to as the Absolute. This problem is solved on certain groups, including commutative groups, nilpotent groups, trees. Connections with the theory of harmonic functions, Poisson-Furstenberg boundaries and others will be explained.