We revisit some properties of AdS2 Einstein-Maxwell gravity with the aim of reconciling apparently conflicting results in prior literature. We show that the two dimensional theory can be obtained as a dimensional reduction of the three dimensional Einstein gravity with the CSS boundary condition. In general, this theory on AdS2 can be viewed as an effective theory of the near horizon geometry of a (near-)extremal black hole. We provide an interpretation of our results in terms of the microscopic description of an extremal black hole.

In this expository note we discuss some arithmetic aspects of the mirror symmetry for plane cubic curves. We also explain how the Picard-Fuchs equation can be used to reveal part of these arithmetic properties. The application of Picard-Fuchs equations in studying the genus zero Gromov-Witten invariants of more general Calabi-Yau varieties and the Weil-Petersson geometry on their moduli spaces will also be discussed.

We give a new type of Schur-Weyl duality for the representations of a family of quantum subgroups and their centralizer algebra. We define and classify singly-generated, Yang-Baxter relation planar algebras. We present the skein theoretic construction of a new parameterized planar algebra. We construct infinitely many new subfactors and unitary fusion categories, and compute their trace formula as a closed-form expression, in terms of Young diagrams.

There are two important statements regarding the Trautman-Bondi mass at null infinity: one is the positivity, and the other is the Bondi mass loss formula, which are both global in nature. The positivity of the quasi-local mass can potentially lead to a local description at null infinity. This is confirmed for the Vaidya spacetime in this note. We study the Wang-Yau quasi-local mass on surfaces of fixed size at the null infinity of the Vaidya spacetime. The optimal embedding equation is solved explicitly and the quasi-local mass is evaluated in terms of the mass aspect function of the Vaidya spacetime.

We determine exact exponential asymptotics of eigenfunctions and of corresponding transfer matrices of the almost Mathieu operators for all frequencies in the localization regime. This uncovers a universal structure in their behavior, governed by the continued fraction expansion of the frequency, explaining some predictions in physics literature. In addition it proves the arithmetic version of the frequency transition conjecture. Finally, it leads to an explicit description of several non-regularity phenomena in the corresponding non-uniformly hyperbolic cocycles, which is also of interest as both the first natural example of some of those phenomena and, more generally, the first non-artificial model where non-regularity can be explicitly studied.