Georgia Institute of Technology College of Sciences

We study the Legendre elliptic curve E: y^2=x(x+1)(x+t) over the field F_p(t) and its extensions K_d=F_p(mu_d*t^(1/d)). When d has the form p^f+1, in previous work we exhibited explicit points on E which generate a group V of large rank and finite index in the full Mordell-Weil group E(K_d), and we showed that the square of the index is the order of the Tate-Shafarevich group; moreover, the index is a power of p. In this talk we will explain how to use p-adic cohomology to compute the Tate-Shafarevich group and the quotient E(K_d)/V as modules over an appropriate group ring.

After briefly considering embeddings of the category of smooth manifolds into so called smooth toposes and arguing that we may ignore the details of the embedding and work from axioms if we agree to use intuitionistic logic, we consider axiomatic synthetic differential geometry. Key players are a space R playing the role of the "real line" and a space D consisting of null-square infinitesimals such that every function from D to R is "microlinear". We then define microlinear spaces and translate many definitions from differential geometry to this setting. As an illustration of the ideas, we prove Stokes' theorem. Time permitting, we show how synthetic differential geometry may be considered as an extension of differential geometry in that theorems proven in the synthetic setting may be "pulled back" to theorems about smooth manifolds.

We consider a class of dynamical systems with random switching with the following specifics: Given a finite collection of smooth vector fields on a finite-dimensional smooth manifold, we fix an initial vector field and a starting point on the manifold. We follow the solution trajectory to the corresponding initial-value problem for a random, exponentially distributed time until we switch to a new vector field chosen at random from the given collection. Again, we follow the trajectory induced by the new vector field for an exponential time until we make another switch. This procedure is iterated. The resulting two-component process whose first component records the position on the manifold, and whose second component records the driving vector field at any given time, is a Markov process. We identify sufficient conditions for its invariant measure to be unique and absolutely continuous. In the one-dimensional case, we show that the invariant densities are smooth away from critical points of the vector fields and derive asymptotics for the invariant densities at critical points.

Given a Riemannian manifold $(M,g)$, does there exist a metric $g'$ on $M$ conformal to $g$ such that $g'$ has constant scalar curvature? This question is known as the Yamabe problem. Aim of this talk is to give an overview of the problem and discuss and develop methods that go into solving a few of intermediate results in the solution to the problem in full generality.