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Charlotte Ure, Michigan State University
The Brauer group of an elliptic curve $E$ is an important invariant with intimate connections to cohomology and rational points. Elements of this group can be described as Morita equivalence classes of central simple algebras over the function field. The Merkurjev-Suslin theorem implies that these classes can be written as tensor product of symbol (or cyclic) algebras. In this talk, I will describe an algorithm to calculate generators and relations of the $q$-torsion ($q$ a prime) of the Brauer group of $E$ in terms of these tensor products over any field of characteristic different from $2$,$3$, and $q$, containing a primitive $q$-th root of unity. This is work in progress.
Joseph Feneuil, Temple University
The Riesz transform $\nabla \Delta^{-1/2}$ on $\mathbb R^n$ is bounded on $L^p$ for all $p\in (1,+\infty)$. This well known fact can quickly be proved by using the Fourier transform. Strichartz asked then whether this property is transmitted to Riemannian manifold, more exactly, what are the geometric conditions needed on our manifold to get the boundedness of the Riesz transform.
We shall present (part of) the literature on the topic, including the results of the speaker (together with Li Chen, Thierry Coulhon, and Emmanuel Russ) on fractal-like spaces. We shall also talk about the case of graphs, that can be seen as discrete version of Riemannian manifolds, which will allow us to give concrete examples of application of our work.
If time permits, we will provide equivalent statements for an assumption frequently met when working on graphs (which implies $L^2$-analyticity of the Markov operator). In particular, we will see a way to weaken this assumption to $L^2$-analyticity.
There are no conferences next week.