Used by the work of Koch-Tataru on Navier-Stokes equations, the theory of tent spaces turns out to be useful to deal with evolution equations with very rough initial data. In this talk, we shall discuss the recent progress on studying linear parabolic equations with time-independent, uniformly elliptic, bounded measurable complex coefficients via tent spaces. The talk is based on a joint work with Pascal Auscher.

We'll define stabilization for codim=2 contact manifolds of dim>3 contact manifolds so that the following holds: A contact manifold is overtwisted iff its "standard contact unknot" is a stabilization. This means that many dim=2n+1>3 contact manifolds contain dim=2n-1 spheres which are unknotted smoothly but "knotted" from a contact-topological point of view.

(Joint with Erwan Brugallé and Lucía López de Medrano.) Combinatorial patchworking is a technique introduced by Viro to study the topology of real algebraic hypersurfaces. It's base ingredients are a triangulation of a lattice polytope and a sign distribution on its vertices. If the triangulation is "convex", the construction can be translated to the dual setting of tropical varieties. In recent years, this dual viewpoint has promoted new results using tropical homology theories. For example, Renaudineau and Shaw proved a conjecture by Itenberg bounding the Betti numbers of the real part of a (unimodular) patchwork hypersurface in terms of the Hodge numbers of the complex part.

The subject jokingly called "3.5-dimensional topology" concerns itself with the interactions between 3-manifolds and 4-manifolds, asking, given a 3-manifold, what 4-manifolds it is the boundary of. One big question in 3.5-dimensional topology is when a rational homology 3-sphere is the boundary of a rational homology 4-ball. (Guess what? Almost never.) We will discuss this question for a particular class of rational homology 3-spheres described by weighted graphs, presenting some results, conjectures, and ways forward.

In this talk, we will review the recent works by Petersen and Wink regarding new curvature conditions for Bochner techniques on closed manifolds and its applications. Then, we continue there techniques to study non-compact complete manifolds and show several rigidity results of harmonic tensors in terms of Lichnerowicz Laplacians. Several applications to study geometry of curved manifolds and immersed submanifolds are also given .