Speakers June 8th

(Joint work with Claudia Landi)

We present a pipeline that takes as an input a gradient-like Morse-Smale vector field on a surface, produces a parametrized epimorphic chain complex and encodes it as a barcode. The chain maps in the parametrized chain complex are quotient maps corresponding to topological simplifications of the vector field, and the times of taking the quotients depend on the value of a parameter measuring the local robustness of the vector field. In the end we are left with a vector field that has a very simple topological structure. Remembering the times when the simplifications were applied yields a barcode. Similarly to the usual persistent homology construction for real valued functions, this pipeline paves the way for the development of a theory of persistence for vector fields.

In 1982, Gromov introduced a homotopy invariant of manifolds called simplicial volume. The vanishing of this invariant in the case of closed manifolds is implied by the amenability of the fundamental group. For open manifolds the situation is different, since an open manifold with amenable fundamental group can have either vanishing or infinite simplicial volume; things get clearer if we look at the fundamental group at infinity. In particular, generalizing a result of Loeh, we prove that an open manifold with amenable fundamental group at infinity has finite simplicial volume. We also prove that if a finitely-many-ended open manifold is simply connected at infinity then it has finite simplicial volume.

The triangulation complexity of a closed oriented manifold M is the minimum number of simplices that a triangulation of M requires, while the integral simplicial volume is the “minimal number” of simplices that an integral cycle of M must be made of.
At first sight these two invariants may appear really similar: they both measure the complexity of a manifold in terms of simplices, they coincide in dimensions 1 and 2, and both are known to be difficult to compute. It is then a natural question whether one can use one of these two quantities to approximate the other, even in dimension greater than 2. The poster will deal with three different approaches that one can use to face this question.

This article is motivated by the equivalence of links in 3-manifolds of Heegaard genus two.
We construct an algorithm (implemented in C++) which, starting from a description of such a manifold introduced by Casali and Grasselli that uses 6-tuples of integers and determines a Heegaard decomposition of the manifold, allows to find the words in B2,2n, the braid group on 2n strands of a surface of genus two, that realize the plat-equivalence for links in that manifold. In this way we extend the result obtained by Cattabriga and Gabrovšek for 3-manifolds of Heegaard genus one to the case of genus two. We describe explicitly the words for a notable family of 3-manifolds.

Specialising the Burau representation of braid groups at t = −1 we obtain a symplectic represen- tation of braid groups, called the integral Burau representation. The level m congruence subgroups of the braid groups are the kernels of the (mod m) reductions of the integral Burau representation. In a recent paper, Dan Margalit asked some questions about natural generating sets and properties of these groups. We discuss some of these questions, and then we move on to consider crystallographic structures on quotients of braid groups by certain congruence subgroups, building on the work of Gon ̧calves, Guaschi and Ocampo. This is a joint work with Paolo Bellingeri, Alan McLeay, Oscar Ocampo, and Charalampos Stylianakis.

A famous theorem of Alexander (1923) states that every link can be obtained as the closure of a braid. Later, in 1936, Markov proved an equally celebrated theorem which specifies necessary and sufficient conditions for two braids to have the same closure. We will see a generalization of these theorems for strongly invertible knots. These are knots with a particular kind of symmetry that are being widely studied for both their 3-dimensional and 4-dimensional properties.

Based on previous results in crystallization theory, we give a procedure to construct trisections for closed manifolds generated by colored tensor models without restrictions on the number of simplices in the triangulation. We give a description of how trisection diagrams can arise from colored tensor model graphs for closed 4-manifold, although a large amount of redundant information is generated in the process.