The one dimensional quantum Ising model is used in quantum statistical physics to model interracting particles on a discrete lattice. While the classical model (in one and two dimensions) has long been solved (its origin dates back to 1930's), its quasiperiodic analog (dating back about 25 years) is still a source of interesting problems. We shall discuss our solution to one such problem: we'll rigorously prove that the energy spectrum of the one dimensional quantum quasiperiodic Ising model is a Cantor set, as has been long believed, and discuss some of its properties.
This is the first in a series of two seminars dedicated to this topic. In this seminar we'll present the problem and set up the main ideas.

A survey of automatic continuity in Banach algebras with a focus on the case of a derivation on a Banach Jordan triple. (Joint work with Antonio Peralta)

The Weil Conjectures are one of the most beautiful theorems in mathematics. In the number field context zeta and L-functions are transcendental. It is well known, for example, that zeta(2)=pi^2/6. The values of these functions, even at integers, are not well understood. The Weil conjectures state the perhaps shocking result that the function field analogues of these functions are almost as simple as possible: they are rational functions. Further, they include the analogue of the Riemann Hypothesis for function fields. In this talk we will explore what the Weil conjectures say, as well as how they are proven.

We obtain explicit formulas for the number of non-isomorphic
elliptic curves with a given group structure (considered as an abstract abelian group).
Moreover, we give explicit formulas for the number of distinct group structures of all
elliptic curves over a finite field. We use these formulas to derive
some asymptotic estimates and tight upper and lower bounds for
various counting functions related to classification of elliptic
curves accordingly to their group structure. Finally, we present
results of some numerical tests which exhibit several interesting
phenomena in the distribution of group structures.

In this talk, we present the family of generalized Hessian curves.

The family of generalized Hessian curves covers more isomorphism classes of elliptic curves than Hessian curves.

We provide efficient unified addition formulas for generalized Hessian curves. The formulas even feature completeness for suitably chosen curve parameters.

We also also present extremely fast addition formulas for generalized binary Hessian curves. The fastest projective addition formulas require $9\M+3\s$, where $\M$ is the cost of a field multiplication and $\s$ is the cost of a field squaring. Moreover, very fast differential addition and doubling formulas are provided that need only $5\M+4\s$ when the curve is chosen with small parameters.