Magnitude 2023

Information

https://sites.google.com/view/magnitude2023/programme

Schon vor der Konferenz hielt Emily Roff eine Präsentation: Roff - Reachability homology and the magnitude-path spectral sequence

Presentations

There is a live blog here: https://golem.ph.utexas.edu/category/

Montag

1. Leinster - The many faces of magnitude

Dienstag

1. Roff - Iterated Magnitude homology
2. Meckes - A direct proof for the positive definiteness of four point metric spaces
3. Takatsu - Geometry of sliced-disintegrated Monge—Kantorovich metrics
4. Escolar - On interval covers and resolutions of persistence modules
5. Leinster - Magnitude homology equivalence of Euclidean sets (Joint work with Adrián Doña Mateo)

Mittwoch

1. Yoshinaga - Magnitude homotopy type
2. O’Hara - Residues of manifolds
3. Tsukamoto - Introduction to mean dimension
4. Caputi - On finite generation in magnitude (co)homology, and its torsion
5. Juan Pablo Vigneaux Ariztia - A combinatorial approach to Möbius inversion and pseudoinversion

Donnerstag

1. Giuliamaria Menara - Eulerian magnitude homology introduction and application to random graphs
2. Tsubasa Kamiyama - Metric fibrations over one-dimensional base spaces are trivial
3. Sho Shimoyama - Exploring the Uniqueness of Minimizing-Movement in Metric Spaces beyond p = 2
4. Yu Tajima - Magnitude homotopy type of graphs and Whitney twist via discrete Morse theory

Freitag

• Rayna Andreeva - Metric space magnitude and generalisation in neural networks
• Jun Yoshida - On a logical foundation for parametrized homologies
• Sergei O Ivanov - On the path homology of Cayley digraphs and covering digraphs
• Yasuhiko Asao - Classification of metric fibration
• Kiyonori Gomi - A direct proof for the positive definiteness of four point metric spaces
• Richard Hepworth - Bigraded path homology

Burning Questions!!!

Question

The topological entropy and the maximum diversity are connected by Epsilon-Covering-Numbers. Is there a connection? There should at least be an alternative bound to be made. (Meckes)

Question

Magnitude is connected to every kind of entropy except topological entropy. There needs to be some connection (Leinster)

Question

Gromov introduced the idea that geometric concepts can be dynamicalized. Can the whole concept of magnitude be dynamicalized? Or are dynamicalized systems a special case of magnitude? (Tsukamoto)

Question

Finite Metric spaces can be described by finite matrices. Can those matrices be studied as symbolic dynamics? Does the resulting topological entropy have any connection to magnitude/diversity concepts? (Vigneaux, Tsukamoto) Complex dynamics can be described by irreducible matrices, i.e. the graph is strongly connected. In the case of a graph what is the relation of magnitude to the spectral radius of the adjacency matrix? What happens if we generalize from finite to countable instead of compact metric spaces?

Question

The Agol Cycle Length is a kind of measure of length for a train track of a pseudo-anosov map. It ought to be describable by magnitude (homology). If so it is part of a measure of a metric space associated to the pseudo-anosov map. How does this metric space look like?

Question

Is there a nice way to switch between the adjacency matrix and the distance matrix of a graph?

Question

Can magnitude be used to determine the quality of Wikipedia-like graphs?

Question

Is there some kind of a fundamental group of a train track? Can I use HNN-to recreate the train track splittings? Maybe I should start by studying my ideas on train tracks of the torus.

Other Questions

• When searching for a Definition should you go to the original definition or a newer paper, where the definition might be more refined?
• Do you recommend to prepare for conference talks?