# Mini-Geometry Gathering

**When: **Friday, March 6, 2020

**Where: **Sala 207,** **Bloco H, Campus do Gragoatá, UFF

**Speakers:**

Jason Lotay (Oxford)

Daniele Sepe (UFF)

Misha Verbitsky (IMPA)

Matheus Vieira (UFES)

**Schedule:**

10:00-11:00 : Jason Lotay

11:30-12:30: Daniele Sepe

Lunch Break

14:00-15:00: Misha Verbistky

15:30-16:30: Matheus Vieira

**Titles and Abstracts:**

Jason Lotay

Title: Bryant-Salamon G2 manifolds and coassociative fibrations

Abstract: The first examples of complete holonomy G2 manifolds were constructed by Bryant-Salamon and are thus of central importance in geometry, but also in physics. I will describe joint work with Spiro Karigiannis which realizes these Bryant-Salamon manifolds as fibrations by 4-dimensional calibrated (hence volume-minimizing) submanifolds known as coassociative 4-folds. In particular, I will discuss the relationship of this study to hyperkaehler geometry, conical singularities, (Kovalev)-Lefschetz fibrations and multi-moment maps.

Daniele Sepe

Title: Remarks on the local normal form of critical points in integrable systems with one degree of freedom.

Abstract: An integrable system on a symplectic manifold of dimension 2n is a collection of n (almost everywhere) independent functions in involution. Near regular points (where the functions are independent), there are no local invariants: this is the content of the Carathéodory-Jacobi-Lie theorem. For singular points (corresponding to equilibria) there are no general results. If a singular point is assumed to be non-degenerate (a symplectic Morse-Bott condition), a local normal form was established by Eliasson. In this talk we consider the simplest case: a non-degenerate singular point in an integrable system on a two-dimensional symplectic manifold. In this case, the non-degeneracy condition is precisely the Morse condition and the local normal form was first established by Colin de Verdiére and Vey. The aim of the talk is to illustrate how to go about proving such a normal form as well as proving a couple of (small!) remarks that seem to be folklore results in the literature.

Misha Verbitsky

Title: Sections of holomorphic Lagrangian fibrations

Abstract: Let M be a holomorphically symplectic manifold equipped with a holomorphic Lagrangian fibration $\pi$ and a section S. Then M admits another holomorphically symplectic structure such that S is holomorphic. This is a joint work with Fedor Bogomolov and Rodion Deev.

Matheus Vieira

Title: Gap theorems in Yang-Mills theory for four-dimensional manifolds

Abstract: The purpose of this lecture is to present some gap theorems in Yang-Mills theory over complete manifolds with a with weighted Poincaré inequality and complete manifolds with positive Yamabe invariant.