Geometric flows from unified string theories

Unified string theories have led to new partial differential equations, many requiring a novel and unfamiliar geometry. Calabi-Yau manifolds were the earliest example, but many other equations have emerged since, each of which can be interpreted as defining a new notion of canonical metric in a non-Kaehler setting. Because of cohomological constraints, geometric flows seem particularly appropriate for the study of these equations. We provide a survey of them, with emphasis on the many open questions.

Multilinear Kakeya and Michael-Simon inequality for anisotropic stationary varifolds

  • Speaker: Guido De Philippis
  • Time: 10:10-11:10
  • Abstract: Michael Simon inequality is a fundamental tool in geometric analysis and geometric measure theory. Its extension to anisotropic integrands will allow to extend to anisotropic integrands a series of results which are currently known only for the area functional. In this talk I will present an anistropic version of the Michael-Simon inequality, for two-dimensional varifolds in R3, provided that the integrand is close to the area in the C1-topology. The proof is deeply inspired by posthumous notes by Almgren, devoted to the same result. Although our arguments overlap with Almgren’s, some parts are greatly simplified and rely on a nonlinear version of the planar multilinear Kakaeya inequality.

Computations of instanton knot Floer homology with local coefficients

  • Speaker: Tom Mrowka
  • Time: 11:20-12:20
  • Abstract: This talk will discuss some motivating problems for developing a version of instanton knot Floer homology with local coefficients. These Floer homology groups are modules over Laurent polynomial rings. Though more complicated than the vanilla version they turn out to be easier to compute. We’ll describe some of computations of these modules. One emphasis will be on the product case of a trivial n-strand braid in S1xS2, generalizing work of Ethan Street. This is joint work with Peter Kronheimer.

Infinite Staircases in Hirzebruch surfaces

  • Speaker: Dusa McDuff
  • Time: 2:00-3:00
  • Abstract: The ellipsoidal capacity function \(c_X(z)\) of a symplectic four-manifold \(X\) measures how much the form on \(X\) must be dilated in order for it to admit an embedded ellipsoid of eccentricity \(z\). In most cases there are just finitely many obstructions to such an embedding besides the volume. If there are infinitely many obstructions, \(X\) is said to have a staircase. This talk will give an almost complete description of these staircases when \(X\) is a Hirzebruch surface \(H_b\) formed by blowing up the projective plane with weight \(b\). There is an interweaving, recursively defined, family of obstructions that show there is an open dense set of shape parameters \(b\) that are blocked, i.e., have no staircase, and an uncountable number of other values of \(b\) that do admit staircases. Moreover, there are interesting symmetries that act on the set of staircases. This is joint work with Nicki Magill and Morgan Weiler.

Knotted handlebodies

Often, interesting knotting vanishes when allowed one extra dimension, e.g. knotted circles in 3-space all become isotopic when included into 4-space. Hughes, Kim and I recently found a new counterexample to this principle: for g>1, there exists a pair of 3-dimensional genus-g solids in the 4-sphere with the same boundary, and that are homeomorphic relative to their boundary, but do not become isotopic rel boundary even when their interiors are pushed into the 5-dimensional ball. This proves a conjecture of Budney and Gabai (who previously constructed 3-balls in the 4-sphere with the same boundary that are not isotopic rel boundary) for g>1 in a very strong sense. In this talk, I’ll describe some interesting background theorems on codimension-2 knotting in higher dimensions and talk about related open problems in dimensions 3 and 4. This is joint work with Mark Hughes and Seungwon Kim.

Hidden symmetries of almost Kähler manifolds

  • Speaker: Joana Cirici
  • Time: 4:20-5:20
  • Abstract: I will explain how local identities for almost Kähler manifolds lead to various unexpected symmetries on certain subspaces of their cohomology. This allows to deduce some geometric and topological consequences. In particular, we obtain new obstructions to the existence of a symplectic form compatible with a given almost complex structure. This is joint work with Scott Wilson.