In this talk we consider the moduli space of Yang-Mills instantons on the family of hyperkahler 4 manifolds known as multi-center TaubNUT spaces. We describe the Nahm transform for flat manifolds. Then we sketch its extension to the above hyperkahler family, where it defines an isometry between the moduli space of instantons on the multi-center TaubNUT and the moduli space of solutions of a rococo system of ordinary differential equations. This is joint work with Sergey Cherkis and Andres Larrain Hubach
In this talk, I will present some new topological obstructions for solving the Einstein equations (in Riemannian signature) on a large class of closed four-manifolds. Finally, I will show that complex-hyperbolic Einstein Dehn filling compactification cannot possibly performed in dimension four. This is in striking contrast with the real-hyperbolic case, and it answers (negatively) a fifteen years old question of Michael Anderson. If time permits, I will conclude with some tantalizing open problems both in dimension four and in higher dimensions. Part of this work is joint with M. Golla (Universit\’e de Nantes).
Though the study of isoperimetric problems for Laplacian eigenvalues dates back to the 19th century, the subject has undergone a renaissance in recent decades, due in part to the discovery of connections with harmonic maps and minimal surfaces. By the combined work of several authors, we now know that unit-area metrics maximizing the first nonzero Laplace eigenvalue exist on any closed surface, and are realized by minimal surfaces in spheres. At the same time, work of Fraser-Schoen, Matthiesen-Petrides and others yields analogous results for the first eigenvalue of the Dirichlet-to-Neumann map on surfaces with boundary, with maximizing metrics induced by free boundary minimal immersions into Euclidean balls. In this talk, I'll describe a series of recent results characterizing the (perhaps surprising) asymptotic behavior of these free boundary minimal immersions (and associated Steklov-maximizing metrics) as the number of boundary components becomes large. (Based on joint work with Mikhail Karpukhin.)
Over a field k, the zeroth homotopy group of the motivic sphere spectrum is given by the Grothendieck-Witt ring of symmetric bilinear forms GW(k). The Grothendieck-Witt ring GW(k) modulo the hyperbolic plane is isomorphic to the Witt ring of symmetric bilinear forms W(k) which further surjectively maps to Z/2. We may take motivic Eilenberg-Maclane spectra of Z/2, W(k) and GW(k). Voevodsky has computed the motivic Steenrod algebra of HZ/2 and solved the Bloch-Kato conjecture with its help. We move one step up in the above picture; we study the motivic Eilenberg-Maclane spectrum corresponding to the Witt ring and compute its dual Steenrod algebra.
To a cobordism between links, Khovanov homology assigns a linear map that is invariant under boundary-preserving isotopy of the cobordism. In this talk, we study those maps arising from surfaces in the 4-ball and apply our findings to existence and uniqueness questions regarding slice disks bounding a given knot. This reflects joint works with Jonah Swann and Kyle Hayden.