Extremal combinatorics: Sperner systems, the Littlewood-Offord problem, intersecting hypergraphics, compression, Turan type problems, Ramsey theory, convexity, incidence problems, and algebraic methods. Contribute typo fixes here.
“The Haar measure allows integration over topological groups, which has many applications; for example, the use of the Haar measure in proving the Fully Quantum Slepian Wolf (or state transfer) protocol fundamental to quantum information theory is briefly discussed in section VI. But first we build the definitions necessary to define the Haar measure, and state some fundamental theorems. Next, we show a left Haar measure exists on locally compact Hausdorff topological groups, and is unique (up to scale) on such groups which are also $\sigma$-compact. While of course the Haar measure is unique up to scale on all locally compact Hausdorff groups, the simpler case admits an elegant proof. Lastly, the relationship between the right and left Haar measure is investigated.”
“The black hole information paradox concerns the intersection of quantum mechanics and general relativity, so its resolution could point the way towards a quantum theory of gravity. The paradox is a contradiction between our understanding of the limits of our physical theories and the consequences of them. With curvature on the scales of that in our solar system, current physical theories have been very successful experimentally; thus, there seems to exist a “solar system limit” in which the laws of physics reduce to those of quantum field theory. However, Hawking’s argument [Haw75], as formalized into a theorem by Mathur Mat11, shows that a natural interpretation of this limit, along with the existence of a “traditional” black hole(such as a Schwarzschild black hole), leads to a violation of the unitarity of quantum mechanics or physically-objectionable “remnants”. Two resolutions to this paradox are presented. The “firewalls” resolution formulated by [AMPS13] considers the observations of an external observer and an infalling observer, and concludes that high-energy quanta near the horizon of the black hole would prevent the paradox (and evade the theorem, as the black hole would no longer be “traditional”). A subsequent argument by Hayden and Harlow examines the computational complexity of the task presented to the infalling observer, and suggests that perhaps quantum field theory does not hold between “computationally-inaccessible” observables, allowing a resolution of the paradox without firewalls HH13.”
Introduction to metric spaces, topological spaces.
“Synthesizing some of the literature on non-local games and communication complexity scenarios provides a deeper understanding of the fundamental differences between classical mechanics and quantum mechanics, as exemplified by the quantum violation of Bell inequalities, which hold in classical mechanics. Quantum mechanical protocols are shown to provide significant advantages in certain tasks over classical protocols, and could create the first loophole-free demonstration of quantum non-locality. Non-local games are a scenario where two players attempt to perform a task without communication; communication complexity scenarios generalize non-local games by allowing but attempting to minimize communication. Completing the circle, communication complexity tasks naturally draw forth new non-locality scenarios.”