Given a planar infinity harmonic function u, for each
$\alpha>0$ we show a quantitative $W^{1,\,2}_{\loc}$-estimate of
$|Du|^{\alpha}$, which is sharp when $\alpha\to 0$.  As a consequence we
obtain an $L^p$-Liouville property for infinity harmonic functions in
the whole plane
 

The theory of electromagnetic (EM) phenomena known as electrodynamics is one of the major theories in science. At macroscopic scales the interaction of the EM field with matter is described by the classical electrodynamics based on the Maxwell-Lorentz theory. Many of electromagnetic phenomena at microscopic scales are covered by the so-called semiclassical theory that treats the matter according to the quantum mechanics, whereas the EM field is treated classically. The subject of this presentation is a recently advanced by us neoclassical electromagnetic theory that describes EM phenomena at all spatial scales –microscopic and macroscopic. This theory modifies the classical electrodynamics into a theory that applies to all spatial scales including atomic and nanoscales. The neoclassical theory is conceived as one theory for all spatial scales in which the classical and quantum aspects are naturally unified and emerge as approximations. It is a classical Lagrangian field theory, and consequently it is a local and deterministic theory. Probabilistic aspects of the theory may arise in it effectively through complex nonlinear dynamical evolution. This presentation is to provide an introduction to our theory including a concise historical review.

(Joint work with Anatoli Babin)

Understanding joint behaviour of $(f(n),g(n+1))$ where f and g are given multiplicative functions play key role in analytic number theory with potentially profound consequences such as Riemann hypothesis, twin prime conjecture, Chowla's conjecture and many others.

In the the first part of this talk, I will discuss joint work with A. Mangerel, answering an old question of Katai about distribution of points $\{(f(n),g(n+1))\}_{n\ge 1}\in \mathbb{T}^2,$ where f and g are unimodular multiplicative functions.  

In the second part of the talk, which is based on a joint work with P. Kurlberg, answering a question of M. Lemanczyk, we construct deterministic example of multiplicative function $f:{\mathbb{N}\to \{+1,-1\}$ with various ergodic properties with respect to the Mirsky measure and discuss its relation to the interplay between Chowla conjecture and Riemann hypothesis. 

Free entropy theory is an analogue of information theory in a non-commutative setting, which has had great applications to the examination of structural properties of von Neumann algebras. I will discuss some ongoing joint work with Paul Skoufranis to extend this approach to the setting of bi-free probability which attempts to study simultaneously ``left'' and ``right'' non-commutative variables. I will speak in particular of an approach to a bi-free Fisher information and bi-free conjugate variables -- analogues of Fisher's information measure and the score function of information theory. The focus will be on constructing these tools in the non-commutative setting, and time permitting, I will also mention some results such as bi-free Cramer-Rao and Stam inequalities, and some quirks of the bi-free setting which are not present in the free setting.