Georgia Institute of Technology College of Sciences

Real-valued smooth differential forms on Berkovich analytic spaces were introduced by Chambert-Loir and Ducros. They show many fundamental properties analogous to smooth real differential forms on complex manifolds, which are used for example in Arakelov geometry. In particular, these forms define a real valued bigraded cohomology theory for Berkovich analytic space, called tropical Dolbeault cohomology. I will explain the definition and properties of these forms and their link to tropical geometry. I will then talk about results regarding the tropical Dolbeault cohomology of varietes and in particular curves. In particular, I will look at finite dimensionality and Poincar\'e duality.

In this talk, we will introduce a family of stochastic processes on the Wasserstein space, together with their infinitesimal generators. One of these processes is modeled after Brownian motion and plays a central role in our work. Its infinitesimal generator defines a partial Laplacian on the space of Borel probability measures, taken as a partial trace of a Hessian. We study the eigenfunction of this partial Laplacian and develop a theory of Fourier analysis. We also consider the heat flow generated by this partial Laplacian on the Wasserstein space, and discuss smoothing effect of this flow for a particular class of initial conditions. Integration by parts formula, Ito formula and an analogous Feynman-Kac formula will be discussed. We note the use of the infinitesimal generators in the theory of Mean Field Games, and we expect they will play an important role in future studies of viscosity solutions of PDEs in the Wasserstein space.

Sage is widely considered to be the defacto open-source alternative to Mathematica that is freely available for download to users on most standard platforms at sagemath.org. New users to Sage are also able to use its capabilities from any webbrowser and other useful Linux-only software by registering for a free account on the Sage Math Cloud platform (SMC). In addition to providing users with excellent documentation, Sage allows its users to develop spohisticated mathematics applications using Python and other excellent open-source developer tools that are well tested under both Unix / Linux and Windows environments. In this two-week workshop we provide a user-friendly introduction to Sage for beginners starting from first principles in Python, though some coding experience in other languages will of course be helpful to participants. The main project we will be focusing on over the course of the workshop is an extension of the open-source library provided by the Tilings Gap Distributions and Pair Correlation Project developed by the workshop guide at the University of Washington this and last year. This application will allow participants in the workshop to hone their coding skills in Sage by working on an extension of a real-world computational mathematics application in statistics and geometry. Prospective participants can gain a heads-up on the workshop by visiting the syllabus webpage freely available for modification online at https://github.com/maxieds/WXMLTilingsHOWTO/wiki. The workshop guide will also offer continued free technical support on Sage, Python programming, and Linux to participants in the workshop after the two-week session is complete. Future AMS workshop sessions focusing on other Sage programming topics may be run later based on feedback from this proto-session. Faculty and postdocs are welcome to attend. See you all there on Friday!

Starting from the classical Berezin- and Li-Yau-bounds onthe eigenvalues of the Laplace operator with Dirichlet boundaryconditions I give a survey on various improvements of theseinequalities by remainder terms. Beside the Melas inequalitywe deal with modifications thereof for operators with and withoutmagnetic field and give bounds with (almost) classical remainders.Finally we extend these results to the Heisenberg sub-Laplacianand the Stark operator in domains.