Rationally and polynomially convex domains in ${\mathbb C}^n$ are fundamental objects of study in the theory of functions of several complex variables. After defining and illustrating these notions, I will explain joint work with Y.Eliashberg giving a complete characterization of the possible topologies of such domains in complex dimension at least three. The proofs are based on recent progress in symplectic topology, most notably the h-principles for loose Legendrian knots and Lagrangian caps.

 In a previous work, we introduced a refinement of Juhasz’s sutured Floer homology, and constructed a minus theory for sutured manifolds, called sutured Floer chain complex. In this talk, we introduce a new description of sutured manifolds as “tangles” and describe a notion of cobordism between them. Using this construction, we define a cobordism map between the corresponding sutured Floer chain complexes. We also discuss some possible applications. This is a joint work with Eaman Eftekhary.

This talk describes a method to compute supersymmetric tree amplitudes and loop integrands in ten-dimensional super Yang-Mills theory. It relies on the constructive interplay between their cubic graph organization and BRST invariance of the underlying pure spinor superspace description. After a general introduction to this kind of superspace, we discuss a canonical set of multiparticle building blocks which represent tree level subdiagrams and are guided by their BRST transformation. These building blocks are shown to yield a compact solution for tree level amplitudes, and the applicability of the BRST approach to loop integrands is exemplified through recent examples at one-loop.