Tag - Manifolds

Sergei Gukov: Surprising VOA Structures from Quantum Topology

6th June, 2022

In quantum topology, one usually constructs invariants of knots and 3-manifolds starting with an algebraic structure with suitable properties that can encode braiding and surgery operations in three dimensions. ln this talk, 1 review recent work on q-series invariants of 3-manifolds, associated with quantum groups at generic q, that provide a connection between quantum topology and algebra going in the opposite direction: starting with a 3-manifold and a choice of Spin-C structure, the q-series invariant turns out to be a character of a (logarithmic) vertex algebra that depends on the 3-manifold.

Jérôme Scherer: Floyd’s manifold is a conjugation space

7th December, 2021

This is joint work with Wolfgang Pitsch. We illustrate how equivariant stable homotopy methods can help us recognize the structure of a conjugation space, as introduced by Hausmann, Holm, and Puppe. We first explain their definition and present a characterization in terms of purity (obtained in previous joint work with Nicolas Ricka). We then perform equivariantly Floyd's construction from the 1970s of a pair of 5- and 10-dimensional manifolds with four cells, relying on Lück and Uribe’s work on equivariant bundles. The 10-dimensional one is a conjugation space.

William Meeks: Recent progress in the theory of CMC surfaces in 3-manifolds

1st May, 2017

In this talk I will report on some of the new results in the theory of constant mean curvature (CMC) surfaces M in Riemannian 3-manifolds N. I will mention just a few of the topics touched on in this talk. I first begin with the recent classification of CMC spheres in a homogeneous 3-manifold N and a sketch of its proof. The main result states that any two spheres in N with the same absolute mean curvature differ by an ambient isometry of N. Furthermore, the range of values of the mean curvature spheres are described in terms of the geometry of the universal cover X of N. In the case that X is diffeomorphic to ℝ3 then there exists a sphere of constant mean curvature H in N iff H is greater than half the Cheeger constant of X and otherwise there exists a sphere of constant mean curvature in N for every real number. These results generalize previous work of Hopf, of Abresch-Rosenberg and more recently of Danieil-Mira and of Meeks in the case of the Sol geometry.

Jointly with Tinaglia, we obtain curvature for embedded disks of fixed constant mean curvature H>0 in any fixed homogeneous 3-manifolds. In the ℝ3 setting this result implies that any complete embedded finite topology surface in ℝ3 of constant mean curvature is proper; this generalizes the previous work of Colding -Minicozzi in the case of minimal surfaces. Previous classification results then imply that the only complete embedded simply connected constant mean curvature surfaces in ℝ3 are the plane, the helicoid and round spheres. Another application of this work by Meeks-Tinaglia is to prove that complete embedded CMC surfaces of finite topology in a complete hyperbolic 3-manifold are proper if the mean curvature H is at least 1. On the other hand, Coskunuzer-Meeks-Tinaglia recently constructed for any H in [0,1) a non-proper, complete, stable embedded plane in hyperbolic 3-space having constant mean curvature H.

In 1982 Choi and Wang proved that an embedded closed minimal surface F in the the round 3-sphere S3 has a bound on its area that only depends on its genus; actually their result generalizes from the ambient space S3 to any closed 3-manifold M with positive Ricci curvature. This result was then used by Choi and Schoen to prove the compactness of the moduli space of such examples of fixed genus g in M. Tinaglia and I have been able to give the following related result in the case of connected closed surfaces M embedded in any Riemannian homology 3-sphere manifold N:

Theorem: For any H>0 and non-negative integer g, there exists a constant A(N,g,H) such that any closed surface embedded in M of genus g and constant mean curvature H has area at most A(N,g,H). This area estimate lead to a natural compactification of the moduli space of all such embedded constant mean curvature H examples in N with genus at most g, when H lies in a fixed compact interval [a,b] of positive numbers, and under a compact deformation of the Riemannian metric on N.

The recent classification of properly embedded minimal surfaces of genus 0 in ℝ3 given by Meeks-Perez-Ros, Lopez-Ros, Colin and of Meeks-Rosenberg play a role in the above area estimates, as do the curvature estimates of Meeks-Tinaglia for certain complete embedded CMC surfaces in a Riemannian 3-manifold.

At the end of my talk I will present a brief survey of some recent results on the existence and classification of CMC foliations of closed and non-closed 3-manifolds.

Ciprian Manolescu: Homology cobordism and triangulations

29th April, 2017

In the 1970s, Galewski-Stern and Matumoto studied the existence and the classification of triangulations on topological manifolds of dimension at least 5. They reduced these problems to questions about the three-dimensional homology cobordism group, ΘH3, and the Rokhlin homomorphism from this group to ℤ/2. The structure of the homology cobordism group is still unknown, but some information can be obtained using tools from gauge theory and symplectic geometry, such as the Seiberg-Witten Floer spectrum and involutive Heegaard Floer homology. I will describe the proof of the existence of non-triangulable high-dimensional manifolds (using gauge theory), and some open problems.

Dan Freed: Complex Chern-Simons invariants of 3-manifolds and abelianization

28th April, 2017

A hyperbolic 3-manifold M carries a flat PSL2(ℂ)-connection whose Chern-Simons invariant has been much studied since the early 1980s. For example, its real part is the volume of M. Explicit formulas in terms of a triangulation involve the dilogarithm. In joint work with Andy Neitzke we use 3-dimensional spectral networks to abelianize the computation of complex Chern-Simons invariants. The locality of the classical Chern-Simons invariant, expressed in the language of topological field theory, plays an important role.