We survey some new and old, positive and negative results on a priori estimates, regularity, and rigidity for special Lagrangian equations with or without certain convexity. The "gradient" graphs of solutions are minimal or maximal Lagrangian submanifolds, respectively in Euclidean or pseudo-Euclidean spaces. In the latter pseudo-Euclidean setting, these equations are just Monge-Ampere equations. Development on the parabolic side (Lagrangian mean curvature flows) will also be mentioned.

We complete the proof of the main theorem by showing that if X^3 is isomorphic to X, then X^{\omega} has a parity-reversing automorphism. By our previous results this implies X^2 is isomorphic to X as well. The proof generalizes to show that for any n > 1, if X^n is isomorphic to X, then X^2 is isomorphic to X. Time permitting we will discuss related results, including the existence of an A and X such that A^2X is isomorphic to X, while AX is not.

Public health workers are reaching out to mathematical scientists to use disease models to understand, and mitigate, the spread of emerging diseases. Mathematical and computational scientists are needed to create new tools that can anticipate the spread of new diseases and evaluate the effectiveness of different approaches for bringing epidemics under control. That is, these models can provide an opportunity for the mathematical scientists to collaborate with the public health community to improve the health of our world and save lives. The talk will provide an overview, for general audiences, of how these collaborations have evolved over the past decade. I will describe some recent advances in mathematical models that are having an impact in guiding pubic health policy, and describe what new advances are needed to create the next generation of models. Throughout the talk, I will share some of my personal experiences in used these models for controlling the spread of Ebola, HIV/AIDS, Zika, chikungunya, and the novel H1N1 (swine) flu. The talk is for a general audience.

There is a classical theorem of Iwasawa which concerns certain modules X for the formal power series ring Λ = Zp[[T]] in one variable.  Here p is a prime and Zp is the ring of p-adic integers.  Iwasawa's theorem asserts that X has no nonzero, finite Λ-submodules. We will begin by describing the modules X which occur in Iwasawa's theorem and explaining  how the theorem is connected with the title of my talk. Then we will describe generalizations of this theorem for  certain Λ-modules (the so-called "Selmer groups")  which arise naturally in Iwasawa theory.  The ring Λ can be a formal power series ring over Zp in any number of variables, or even a non-commutative analogue of such a ring.