Seminar
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| Location: | SLMath: Eisenbud Auditorium, Online/Virtual |
Morse Index Stability for Conformally Invariant Lagrangians in Dimension Two
Isabel Beach, Title: The Quantitative Geometry of Geodesics
Abstract: The goal of quantitative geometry is to provide effective versions of known existence theorems for geometric objects. For example, following Serre's proof of the existence of infinitely many geodesics connecting any two points on a closed Riemannian manifold, one may attempt to prove a length bound for these geodesics. In this talk, we will
provide a survey of current quantitative theorems concerning geodesics and explore how such results can be proven. In particular, we will prove that any disk with a convex boundary admits two “short” simple geodesics that meet the boundary orthogonally.
Matilde Gianocca, Title: Morse Index Stability for Conformally Invariant Lagrangians in Dimension Two
Abstract: In this talk, I will discuss compactness properties of sequences of (approximate) harmonic maps in two dimensions, with a focus on their energy distribution and stability. A well-known result in this context is the energy identity, which asserts that the total energy of a sequence of harmonic maps converges to the sum of the energy of the
weak limit and the energies of finitely many harmonic "bubbles". The limiting objects (unions of harmonic maps with some additional properties) are called "bubble trees". Building on this, I will present a joint result with T. Rivière and F. Da Lio, showing that the extended Morse index, a measure of instability for the given harmonic map, is upper semi-continuous in the bubble tree convergence.The main ideas and techniques involved in the proof will be outlined. If time
permits, I will also discuss how these methods extend to Ginzburg-Landau sequences, which approximate harmonic maps into spheres, as developed in collaboration with F. Da Lio.