Speaker: Misha Verbitsky (IMPA)

Abstract: A hyperkahler manifold is a compact holomorphically symplectic manifold of Kahler type. We are interested in hyperkahler manifolds of maximal holonomy, that is, ones which are not flat and not decomposed as a product after passing to s finite covering. The group of automorphisms of such a manifold has a geometric interpretation: it is a fundamental group of a certain hyperbolic polyhedral space.

I will explain how to interpret the boundary of this hyperbolic group as the boundary of the ample cone of the hyperkahler manifold. This allows us to use the fractal geometry of the limit sets of a hyperbolic action to obtain results of hyperkahler geometry.

Speaker: Daniel Fadel (UFRJ)

Abstract: We consider the SU(2) Yang-Mills-Higgs energy functional with nonlinear potential term (the Higgs self-interaction), on compact Riemannian 3-manifolds. This is a gauge-invariant functional on pairs consisting of a connection and a section of the adjoint bundle (the Higgs field), whose critical points are solutions to a nonlinear second order elliptic PDE modulo gauge. These solutions were introduced by Higgs in 1964 as a generalization of Yang-Mills fields, and in the mathematics literature they were notably studied by Jaffe-Taubes (1980), whose work focused in the noncompact Euclidean space.

More recently, Pigati-Stern (2021), working in the compact setting, have shown a deep relation between critical points of the abelian version of the Yang-Mills-Higgs energy, and codimension 2 minimal submanifolds in dimensions 3 and higher, by means of an asymptotic analysis of sequences of critical points of the scaled functional as the scaling parameter goes to zero, while keeping the energies uniformly bounded. In this talk, we study the analogue asymptotic analysis problem for critical points of the scaled non-abelian SU(2) energy on compact 3-manifolds, showing in particular that these concentrate energy in codimension 3, in fact in finite sets, and if time permits we point out future work in higher dimensions. Th

is talk is based on ongoing joint work with Da Rong Cheng (Miami) and Luiz Lara (Unicamp)

Speaker: Carlos Andrés Toro Cardona (IMPA)

Abstract: We review the main constructions of free boundary minimal surfaces in the Euclidean unit ball $\mathbb{B}^3$ for compact orientable topologies. In the non-orientable setting we prove the non-existence of free boundary minimal Möbius bands in $\mathbb{B}^3$. This answers in the negative a question proposed by I. Fernández, L.Hauswirth and P. Mira.

Speaker: Raffaele Vitolo (Università del Salento)

Abstract: It is known (work by Ferapontov and Mokhov) that a system of N-dimensional WDVV equations can be written as a pair of N-2 commuting quasilinear systems (first-order WDVV systems). In recent years, particular examples of such systems were shown to possess two compatible Hamiltonian operators, of the first and third order. It was also shown that all $3$- dimensional first-order WDVV systems possess such bi-Hamiltonian formalism. We prove that, for arbitrary N, if one first-order WDVV system has the above bi-Hamiltonian formalism, than all other commuting systems do. The proof needs some interesting results on the geometric structure and properties of the WDVV equations that will be discussed as well. Joint work with S. Opanasenko.