We will finish presenting Nyikos' counterexample to
Bozyakova's conjecture: If e(Y) = L(Y) for every subspace Y of X, must X
be hereditarily D?
Last week in the Kinderseminar I talked about a rough estimate on volumes of certain hyperbolic 3-manifolds. This time I will describe a different approach for similar estimates (you will not need to remember that talk, don't worry!), which is, in some sense, complementary to that one, as it regards mapping tori. A theorem of Jeffrey Brock provides bounds for their volume in terms of how the monodromy map acts on the pants graph (a relative of the better known curve complex) of the base surface. I will describe the setting and the relevance of this result (in particular the one it has for me); hopefully, I will also tell you part of its proof.
We show that string theories admit chiral infinite tension analogues in which only the massless parts of the spectrum survive. Geometrically they describe holomorphic maps to spaces of complex null geodesics, known as ambitwistor spaces. They have the standard critical space–time dimensions of string theory (26 in the bosonic case and 10 for the superstring). Quantization leads to the formulae for tree– level scattering amplitudes of massless particles found recently by Cachazo, He and Yuan. These representations localize the vertex operators to solutions of the same equations found by Gross and Mende to govern the behaviour of strings in the limit of high energy, fixed angle scattering. Here, localization to the scattering equations emerges naturally as a consequence of working on ambitwistor space. The worldsheet theory suggests a way to extend these amplitudes to spinor fields and to loop level. We argue that this family of string theories is a natural extension of the existing twistor string theories.
We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles. We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.
We discuss kinematic algebras associated to the scattering equations that arise in the description of the scattering of massless particles. We describe their role in the BCJ duality between colour and kinematics in gauge theory, and its relation to gravity. We find that the scattering equations are a consistency condition for a self-dual-type vertex and identify an extension of the anti-self-dual vertex, such that the two vertices are not conjugate in general. Both vertices correspond to the structure constants of Lie algebras. We give a prescription for the use of the generators of these Lie algebras in trivalent graphs that leads to a natural set of BCJ numerators. In particular, we write BCJ numerators for each contribution to the amplitude associated to a solution of the scattering equations. This leads to a decomposition of the determinant of a certain kinematic matrix, which appears naturally in the amplitudes, in terms of trivalent graphs. We also present the kinematic analogues of colour traces, according to these algebras, and the associated decomposition of that determinant.
A bit more than ten years ago, Peter Oszváth and Zoltán Szabó defined Heegaard-Floer homology, a gauge theory inspired invariant of three-manifolds that is designed to be more computable than its cousins, the Donaldson and Seiberg-Witten invariants for four-manifolds. This invariant is defined in terms of a Heegaard splitting of the three-manifold. In this talk I will show how Heegaard-Floer homology is defined (modulo the analysis that goes into it) and explain some of the directions in which people have taken this theory, such as knot theory and fitting Heegaard-Floer homology into the scheme of topological field theories.
Raushan Buzyakova asked if a space is hereditarily D provided
that the extent and Lindelöf numbers coincide for every subspace. We
will introduce interval topologies on trees and present Nyikos'
counterexample to this conjecture.