Georgia Institute of Technology College of Sciences

The mathematical theory of the recovery of quantum states stored in a quantum memory, is intimately related to the subadditivity property of the entropy function, and the class of states known as quantum Markov chains. In this talk we will introduce some of the basic ideas of this area of quantum information theory. We discuss a theorem regarding recovery of a widely studied class of quantum states, the matrix product states, and its implication for the mutual information stored over separated regions of a one dimensional quantum memory.

The Neumann-Poincaré (NP) operator arises in boundary value problems, and plays an important role in material design, signal amplification, particle detection, etc. The spectrum of the NP operator on domains with corners was studied by Carleman before tools for rigorous discussion were created, and received a lot of attention in the past ten years. In this talk, I will present our discovery and verification of eigenvalues embedded in the continuous spectrum of this operator.

A central question in ergodic theory is whether sequences obtained by sampling along the orbits of a given dynamical system behave similarly to sequences of i.i.d. random variables. Here we consider this question from a spectral-theoretic perspective. Specifically, we study large Hermitian matrices whose entries are defined by evaluating the exponential function along orbits of the skew-shift on the 2-torus with irrational frequency.

The Neumann-Poincaré (NP) operator arises in boundary value problems, and plays an important role in material design, signal amplification, particle detection, etc. The spectrum of the NP operator on domains with corners was studied by Carleman before tools for rigorous discussion were created, and received a lot of attention in the past ten years. In this talk, I will present our discovery and verification of eigenvalues embedded in the continuous spectrum of this operator.

Mathematical billiards naturally arise in mechanics, optics, acoustics, etc. They also form the most visual class of dynamical systems with evolution covering all the possible spectrum of behaviours from integrable (extremely regular) to strongly chaotic. Billiard is a (deterministic) dynamical system generated by an uniform (by inertia) motion of a point particle within a domain with piecewise smooth walls ("a billiard table"). I will introduce all needed notions on simple examples and outline some open problems.