Video Lectures

Many of the major results of modern ergodic theory can be understood in terms of a sequence of finite metric measure spaces constructed from the marginal distributions of a shift-invariant process. Most simply, the growth rate of their covering...

We present the key ideas of a new proof of Landau damping for the Vlasov-Poisson equation obtained in a joint work with Bedrossian and Masmoudi. This nonlinear transport equation is a fundamental model for describing self-interacting plasmas or...

A planar set that contains a unit segment in every direction is called a Kakeya set. These sets have been studied intensively in geometric measure theory and harmonic analysis since the work of Besicovich (1919); we find a new connection to game...

We consider a typical situation in which probability model itself has non-negligible cumulated uncertainty. A new concept of nonlinear expectation and the corresponding non-linear distributions has been systematically investigated: cumulated...

I will survey recent results and open problems in several areas of quantum complexity theory, with emphasis on open problems which can be phrased in terms of classical complexity theory or mathematics but have implications for quantum computing: 1...

Let \[ f(t)=\sum_{N n 2N}a_nn^{-it} \] be a Dirichlet polynomial. We consider the weighted square mean value \[ I=\int_{-\infty}^{\infty}|f(t)|^2\exp\{-\Delta^{-2}(t-T)^2\}\,dt, \] where \(T\) is a large paremeter and \[ \Delta = \frac{T}{\log T}...

If \(M\) is a complex manifold, the Bott-Chern cohomology \(H_{\mathrm{BC}}^{(\cdot,\cdot)}\left(M,\mathbf{C}\right)\) of \(M\) is a refinement of de Rham cohomology, that takes into account the \((p,q)\) grading of smooth differential forms. By...

How can we produce randomness of almost perfect quality, in large quantities, and under minimal assumptions? This question is fundamental not only to modern day information processing but also to physics. Yet a satisfactory answer is still elusive...

We consider two classes of \(n \times n\) sample covariance matrices arising in quantum informatics. The first class consists of matrices whose data matrix has \(m\) independent columns each of which is the tensor product of \(k\) independent \(d\)...

The IP theorem, which asserts that IP = PSPACE (Lund et. al., and Shamir, in J. ACM 39(4)), is one of the major achievements of complexity theory. The known proofs of the theorem are based on the arithmetization technique, which transforms a...