Palestrante: Roberto Capistrano (UFPE)
Palestrante: Renato Vianna (UFRJ)
Título: Lifting Lagrangians from Donaldson Divisors
Resumo: A classical construction due to Paul Biran allows to lift a Lagrangian submanifold L from a Donaldson Y divisor to a Lagrangian L' in an ambient symplectic manifold X. In [BK], it is shown that if the minimal Chern number of Y is greater than 1, then the count of Maslov index 2 holomorphic disks with boundary on the lifted Lagrangian L' is equivalent to the similar count of disks with boundary on L plus one extra disk. We study this enumerative geometry problem in the case when the minimal Chern number of Y is 1. This reveals several new, previously unexplored connections it has with relative closed-string Gromov-Witten theory of the pair (X,Y). We explore applications, in particular, we use that to distinghish (up to action of Symp(X)) lifts of previously known Lagrangians.
Local: Sala C116
Palestrante: Gonçalo Oliveira (IMPA)
Título: Instantons on the Euclidean Schwarzschild manifold
Resumo: Instantons are "finite energy" solutions to a geometric PDE for a connection on a vector bundle. These have their origin in Physics but have also been extensively studied by Mathematicians. The first instanton on the Euclidean Schwarzschild manifold was found by Charap and Duff in 1977, only 2 years later than the famous BPST instantons on were discovered. While soon after, in 1978, the ADHM construction gave a complete description of the moduli spaces of instantons on , the case of the Euclidean Schwarzschild manifold resisted many efforts for the past 40
years. I shall explain, how using a duality between vortices and "spherically symmetric" instantons, Akos Nagy and I, recently gave a complete description of a connected component of the moduli space of unit energy instantons on the Euclidean Schwarzschild manifold. If time permits I will also explain how to use our techniques to:
1. Find new examples of instantons with non-integer energy;
2. Completely classify "spherically symmetric" instantons.
3. Give a counterexample to a conjectured possible non-Abelian extension of Birkhoff's Theorem.
(This is joint work with Akos Nagy)