One of the new four-dimensional gauge theories is a set of equations discovered by Kapustin and Witten in 2005. This theory involves a boundary condition which in turn involves a knot K in a 3-manifold. A later conjecture by Gaiotto and Witten states that a sequence of numerical counts of elements of the moduli space of solutions determines the Jones polynomial of the knot. While this remains open, quite a lot has been discovered. I will briefly survey these developments, including the basic structural and regularity theory of these equations, which was joint work with Witten, and a fairly complete resolution of this problem in the dimensionally reduced case, on the product of a Riemann surface and a half-line, which was joint work with S. He.

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I will discuss some recent results concerning the existence and nature of asymptotic expansions for various classes of metrics with special geometry. The focus will be on explaining how these asymptotic expansions are useful for existence questions for these metrics, or for understanding certain global (spectral or geometric) invariants of these spaces.

I will discuss recent work with Akutagawa and Carron concerning the solvability of the Yamabe problem to compact stratified spaces. There are many new obstructions to solvability, some of which we identify quite explicitly. We also discuss regularity of the solutions along the singular strata, and identify some possible rigidity phenomena for cases where existence fails.

In previous joint work with Alexakis, we studied the renormalized area of properly embedded minimal surfaces in hyperbolic space, which turns out to be closely related to Willmore energy. In new work, we obtain some results about how control of this renormalized area of a minimal surface leads to some regularity of its asymptotic boundary.
Esta conferencia se enmarca dentro del convenio de colaboración entre las universidades de Granada y Stanford y forma parte de la acción integrada Granada-Stanford
Seminario de conferencias del doctorado interuniversitario Matemáticas