Friday, November 17, 2023
All talks will take place at the FU Berlin in the 'Seminarraum der Villa', Arnimallee 2, 14195 Berlin.
10:00 – 10:30
Arrival / Refreshments
10:30 – 11:30
Abstract: Using periodic surfaces as a scaffold is a convenient route to making periodic entanglements. I will present a systematic way of enumerating new tangled periodic structures, using low-dimensional topology and combinatorics, posing the question of how to best characterise these new patterns. I will also give an insight into applications of these structures.
11:45 – 12:30
Abstract: The class of two-bridge links are known to possess a nice geometry in the sense that an ideal triangulation of their complements can be read directly from the link diagram. A certain infinite family of these triangulations, first studied by Sakuma and Weeks, have been shown to be minimal through volume bound arguments by Ishikawa and Nemoto. The complexity for the vast majority of these link complements, however, remains unknown.
In this talk we will define the Sakuma-Weeks triangulations of two-bridge link complements and discuss recent progress on determining their complexity. This is joint work with Jonathan Spreer (University of Sydney, Australia).
12:30 – 14:30
Lunch break / Discussions
14:30 – 15:15
Abstract: We discuss the structure of the Khovanov homology groups in homological grading 1 of positive links. More concretely, the first Khovanov homology is supported in a single quantum grading determined by the Seifert genus of the link, where the group is free abelian and of rank determined by the Seifert graph of any of its positive link diagrams. In particular, for a positive link, the first Khovanov homology vanishes if and only if the link is fibered.
This is based on joint work with M. Kegel, L. Mousseau, and M. Silvero.
15:30 – 16:15
Lucy Tobin
Title: Many-Vertex 4-Manifolds, Complexity Bounds, and Building Handlebodies from Chained Triangulations
Abstract: In this talk, we will ask a simple question about 4-manifold triangulations - what is the largest number of vertices possible for a triangulation of a given manifold with n pentachora? An accurate answer would immediately give us a lower bound on the complexity of the manifold, using some simple combinatorial observations and Dehn-Sommerville type relations. I will give a conjecture on the answer to this question, relate it to conjectures on complexity, and discuss how to build many-vertex triangulations for many simply connected 4-manifolds in a way that mimics a handlebody decomposition.
This is joint work with Jonathan Spreer.
16:15 – 17:15
Refreshments / Discussions
17:15 – 18:00
Justin Lanier
Title: Braids, areas, and the Gauss–Lucas theorem
Abstract: A polynomial and its derivative each produce a finite set of roots sitting in the complex plane. Varying the polynomial varies the sets of roots. What shapes can be formed by the two sets? How do the areas of these shapes compare? And what kinds of braids can these points trace out? All of these questions are impacted by the Gauss–Lucas theorem, which says that the derivative’s roots lie in the convex hull of the polynomial’s roots. After making these questions more precise, I’ll share some results of Anderson and Salter and point to some open directions.
19:00 – ...