- Geometry Topology Seminar
- Monday, December 12, 2022 - 14:00 for 1 hour (actually 50 minutes)
- Skiles 006
- Giuseppe Martone – Yale
This talk is based on joint work with Sara Maloni, Filippo Mazzoli and Tengren Zhang.
We introduce a decomposition of a 4-manifold called a multisection, which is a mild generalization of a trisection. We show that these correspond to loops in the pants complex and provide an equivalence between closed smooth 4-manifolds and loops in the pants complex up to certain moves. In another direction, we will consider multisections with boundary and show that these can be made compatible with a Weinstein structure, so that any Weinstein 4-manifold can be presented as a collection of curves on a surface.
In this talk, I will talk about the (geometric) intersection number between closed geodesics on finite volume hyperbolic surfaces. Specifically, I will discuss the optimum upper bound on the intersection number in terms of the product of hyperbolic lengths. I also talk about the equidistribution of the intersection points between closed geodesics.
Given a Legendrian knot L in a contact 3 manifold, one can associate a so-called LOSS invariant to L which lives in the knot Floer homology group. We proved that the LOSS invariant is natural under the positive contact surgery. In this talk I will review some background and definition, try to get the ideal of the proof and talk about the application which is about distinguishing Legendrian and Transverse knot.
Let M be the 3-manifold obtained by r-surgery on the right handed trefoil knot. Classification of contact structures on such manifolds have been mostly understood for r \geq 1 and r=0. Etnyre-Min-Tosun has an upcoming work on the classification of the tight contact structures for all r. The fillability of contact structures on M is mostly understood if r is not between 0 and 1/2. In this talk, we will discuss the fillability of the contact structures M for 0
We study reducible surgeries on knots in S^3, developing thickness bounds for L-space knots that admit reducible surgeries and lower bounds on the slice genus of general knots that admit reducible surgeries. The L-space knot thickness bounds allow us to finish off the verification of the Cabling Conjecture for thin knots. Our techniques involve the d-invariants and mapping cone formula from Heegaard Floer homology. This is joint work with Holt Bodish.
We will introduce an analogue of big mapping class groups as defined by Algom-Kfir and Bestvina which hopes to answer the question: What is “Big Out(Fn)”? This group will consist of proper homotopy classes of proper homotopy equivalences of locally finite, infinite graphs. We will then discuss some classification theorems related to the coarse geometry of these groups. This is joint work with Hannah Hoganson and Sanghoon Kwak.
How good of an invariant is the Jones polynomial? The question is closely tied to studying braid group representations since the Jones polynomial can be defined as a (normalized) trace of a braid group representation.
In this talk, I will present my work developing a new theory to precisely characterize the entries of classical braid group representations, which leads to a generic faithfulness result for the Burau representation of B_4 (the faithfulness is a longstanding question since the 1930s). In forthcoming work, I use this theory to furthermore explicitly characterize the Jones polynomial of all 3-braid closures and generic 4-braid closures. I will also describe my work which uses the class numbers of quadratic number fields to show that the Jones polynomial detects the unknot for 3-braid links - this work also answers (in a strong form) a question of Vaughan Jones.
I will discuss all of the relevant background from scratch and illustrate my techniques through simple examples.
I'll talk about some 2D billiards, the most visual class of dynamical systems, where orbits (rays) move along straight lines within a billiard table with elastic reflections off the boundary. Elliptic flowers are built “around" convex polygons, and the boundary of corresponding billiard tables consists of the arcs of ellipses. It will be explained why some classes of such elliptic flowers demonstrate a never expected before dynamics, and why it raises a variety of (seemingly new) questions in geometry (particularly in 3D), in bifurcation theory (particularly about singularities of wave fronts and creation of wave trains), in statistical mechanics, quantum chaos, and perhaps some more. The talk will be concluded by showing a free movie. Everything (including various definitions of ellipses) will be explained/reminded.
Please Note: Joint Topology Seminar @ GaTech
There exist many different diagrammatic descriptions of 4-manifolds, with the usual claim that "such and such a diagram uniquely determines a smooth 4-manifold up to diffeomorphism". This raises higher order questions: Up to what diffeomorphism? If the same diagram is used to produce two different 4-manifolds, is there a diffeomorphism between them uniquely determined up to isotopy? Are such isotopies uniquely determined up to isotopies of isotopies? Such questions become important if one hopes to use "diagrams" to study spaces of diffeomorphisms between manifolds. One way to achieve these higher order versions of uniqueness is to ask that a diagram uniquely determine a contractible space of 4-manifolds (i.e. a 4-manifold bundle over a contractible space). I will explain why some standard types of diagrams do not do this and give at least one type of diagram that does do this.