Aalto University

I will discuss joint work with Eero Saksman (Helsinki) describing the statistical behavior of the Riemann zeta function on the critical line in terms of complex Gaussian multiplicative chaos. Time permitting, I will also discuss connections to random matrix theory as well as some recent joint work with Saksman and Adam Harper (Warwick) relating powers of the absolute value of the zeta function to real multiplicative chaos.

In this talk I will discuss some aspects of the potential theory, fine properties and boundary behaviour of the solutions to the Total Variation Flow. Instead of the classical Euclidean setting, we intend to work mostly in the general setting of metric measure spaces. During the past two decades, a theory of Sobolev functions and BV functions has been developed in this abstract setting.  A central motivation for developing such a theory has been the desire to unify the assumptions and methods employed in various specific spaces, such as weighted Euclidean spaces, Riemannian manifolds, Heisenberg groups, graphs, etc.

The total variation flow can be understood as a process diminishing the total variation using the gradient descent method.  This idea can be reformulated using parabolic minimizers, and it gives rise to a definition of variational solutions.  The advantages of the approach using a minimization formulation include much better convergence and stability properties.  This is a very essential advantage as the solutions naturally lie only in the space of BV functions. Our main goal is to give a necessary and sufficient condition for continuity at a given point for proper solutions to the total variation flow in metric spaces. This is joint work with Vito Buffa and Juha Kinnunen.

The classical Gehring lemma for elliptic equations with measurable coefficients states that an energy solution, which is initially assumed to be $H^1$ - Sobolev regular, is actually in a better Sobolev space space $W^{1,q}$ for some $q>2$. This a consequence of a self-improving property that so-called reverse Hölder inequality implies. In the case of nonlocal equations a self-improving effect appears: Energy solutions are also more differentiable. This is a new, purely nonlocal phenomenon, which is not present in the local case. The proof relies on a nonlocal version of the Gehring lemma involving new exit time and dyadic decomposition arguments. This is a joint work with G. Mingione and Y. Sire. 

Functions of bounded variation, abbreviated as BV functions, are defined in the Euclidean setting as very weakly differentiable functions that form a more general class than Sobolev functions. They have applications e.g. as solutions to minimization problems due to the good lower semicontinuity and compactness properties of the class. During the past decade, a theory of BV functions has been developed in general metric measure spaces, which are only assumed to be sets endowed with a metric and a measure. Usually a so-called doubling property of the measure and a Poincaré inequality are also assumed. The motivation for studying analysis in such a general setting is to gain an understanding of the essential features and assumptions used in various specific settings, such as Riemannian manifolds, Carnot-Carathéodory spaces, graphs, etc. In order to generalize BV functions to metric spaces, an equivalent definition of the class not involving partial derivatives is needed, and several other characterizations have been proved, while others remain key open problems of the theory.

Panu is visting Oxford until March 2015 and can be found in S2.48

We outline a path from polynomial numerical hulls to multicentric calculus for evaluating f(A). Consider
$$Vp(A) = {z ∈ C : |p(z)| ≤ kp(A)k}$$
where p is a polynomial and A a bounded linear operator (or matrix). Intersecting these sets over polynomials of degree 1 gives the closure of the numerical range, while intersecting over all polynomials gives the spectrum of A, with possible holes filled in.
Outside any set Vp(A) one can write the resolvent down explicitly and this leads to multicentric holomorphic functional calculus.
The spectrum, pseudospectrum or the polynomial numerical hulls can move rapidly in low rank perturbations. However, this happens in a very controlled way and when measured correctly one gets an identity which shows e.g. the following: if you have a low-rank homotopy between self-adjoint and quasinilpotent, then the identity forces the nonnormality to increase in exact compensation with the spectrum shrinking.
In this talk we shall mention how the multicentric calculus leads to a nontrivial extension of von Neumann theorem
$$kf(A)k ≤ sup |z|≤1
kf(z)k$$
where A is a contraction in a Hilbert space, and conclude with some new results on (nonholomorphic) functional calculus for operators for which p(A) is normal at a nontrivial polynomial p. Notice that this is always true for matrices.

We review potential theoretic aspects of degenerate parabolic PDEs of p-Laplacian type.

Solutions form a similar basis for a nonlinear parabolic potential theory as the solutions of the heat

equation do in the classical theory. In the parabolic potential theory, the so-called superparabolic

functions are essential. For the ordinary heat equation we have supercaloric functions. They are defined

as lower semicontinuous functions obeying the comparison principle. The superparabolic

functions are of actual interest also because they are viscosity supersolutions of the equation. We discuss

their existence, structural, convergence and Sobolev space properties. We also consider the

definition and properties of the nonlinear parabolic capacity and show that the infinity set of a superparabolic

function is of zero capacity.