Combinatorial Group Theory: COMBINATORIAL GROUP THEORY

Let G be an inner form of a general linear group over a non-archimedean local field. We prove that the local Langlands correspondence for G preserves depths. We also show that the local Langlands correspondence for inner forms of special linear groups preserves the depths of essentially tame Langlands parameters.

Progress in Mathematics

Train tracks and automorphisms of free groups

Persistent homology (PH) is a method used in topological data analysis (TDA) to study qualitative features of data that persist across multiple scales. It is robust to perturbations of input data, independent of dimensions and coordinates, and provides a compact representation of the qualitative features of the input. The computation of PH is an open area with numerous important and fascinating challenges. The field of PH computation is evolving rapidly, and new algorithms and software implementations are being updated and released at a rapid pace. The purposes of our article are to (1) introduce theory and computational methods for PH to a broad range of computational scientists and (2) provide benchmarks of state-of-the-art implementations for the computation of PH. We give a friendly introduction to PH, navigate the pipeline for the computation of PH with an eye towards applications, and use a range of synthetic and real-world data sets to evaluate currently available open-source implementations for the computation of PH. Based on our benchmarking, we indicate which algorithms and implementations are best suited to different types of data sets. In an accompanying tutorial, we provide guidelines for the computation of PH. We make publicly available all scripts that we wrote for the tutorial, and we make available the processed version of the data sets used in the benchmarking.

/pdf/a-roadmap-for-the-computation-of-persistent-homology-4gjwkkj6rv.pdf

A roadmap for the computation of persistent homology

We summarize properties of 3-manifold groups, with a particular focus on the consequences of the recent results of Ian Agol, Jeremy Kahn, Vladimir Markovic and Dani Wise.

3-Manifold Groups

I. Introduction - II. A geometric approach to homotopy theory - III. Algebra - IV. Whitehead torsion in the category - Le sujet de ce cours est accessible a des etudiants ayant deja de bonnes notions de topologie algebrique : La theorie d'homotopie simple illustre l'effet reciproque entre geometrie et algebre en topologie et leur enrichissement mutuel

A Course in Simple-Homotopy Theory

Very small group actions on R-trees and dehn twist automorphisms

What Does a Basis of F(a, b) Look Like?

Simplicial Structures and Transverse Cellularity

Regular neighborhoods of polyhedra in manifolds have been discussed in many papers, most notably, in the present context, in [W], [H-Z], [H] and [S]. We study regular neighborhoods of X mod Y in Z where (Z, X, Y) is an arbitrary triad of finite dimensional, locally compact polyhedra. When applied to manifolds our theory generalizes previous results in that we make no compactness assumptions and place no restrictions on the polyhedra X and Y to be considered. The paper is organized as follows: 1. Definitions and notation 2. Simplicial preliminaries 3. The uniqueness theorem 4. The characterization of collared subpolyhedra 5. Cone-retracting and collaring regular neighborhoods 6. The stellar neighborhood theorem 7. Relationships between regular neighborhoods 8. Collapsibility and collapsible retractions 9. Regular neighborhoods via collapsibility. V is defined to be a regular neighborhood of X mod Y in Z if there exists a full triangulation (J, K, L; h) of (Z, X, Y) in which V underlies a relative first derived neighborhood, V=hN(K-L, J'). Thus, on the one hand, a given regular neighborhood can be presented in an explicit manner with a fixed triangulation. On the other hand, regular neighborhoods are piecewise linear rather than simplicial objects ("there exists a triangulation") and there are uncountably many regular neighborhoods of X mod Y in Z implicit in the choice of triangulating complex and triangulating homeomorphism. We develop the theory by first exploiting the seeming concreteness (??2-5) and then developing the implicit generality (??6-9). After presenting the basic simplicial data we turn immediately to the uniqueness theorem (3.1). (Existence is automatic.) This asserts that, given two regular neighborhoods of X mod Y in Z, there is an ambient piecewise linear isotopy taking one onto the other and keeping X u Y fixed. The proof is quite direct and carries a great deal of extra information. For example (3.4) if Z is an n-manifold the isotopy can be realized by a sequence of 2n "moves relative to X u Y."

https://www.ams.org/tran/1969-136-00/S0002-9947-1969-0248802-6/S0002-9947-1969-0248802-6.pdf

Marshall M. Cohen

Papers

A Course in Simple-Homotopy Theory

Very small group actions on R-trees and dehn twist automorphisms

What Does a Basis of F(a, b) Look Like?

Simplicial Structures and Transverse Cellularity

A general theory of relative regular neighborhoods