Magnitude 2023

Schedule

Titles and abstracts

Rayna Andreeva

Title : Metric space magnitude and generalisation in neural networks

Abstract : Deep learning models have seen significant successes in numerous applications, but their inner workings remain elusive. The purpose of this work is to quantify the learning process of deep neural networks through the lens of a novel topological invariant called magnitude. Magnitude is an isometry invariant; its properties are an active area of research as it encodes many known invariants of a metric space. We use magnitude to study the internal representations of neural networks and propose a new method for determining their generalisation capabilities. Moreover, we theoretically connect magnitude dimension and the generalisation error, and demonstrate experimentally that the proposed framework can be a good indicator of the latter.

Yasuhiko Asao

Title : Classification of metric fibration

Abstract : The notion of metric fibration is a metric space analogue of topological fiber bundle, which is  introduced by Leinster. In this talk, we classify metric fibrations in a way parallel to topological case. We give two equivalent definitions of metric fibration, which are by `the lifting property ’ and  by `transition functions’. The latter helps us to consider `principal fibrations’ and classify them by `the metric fundamental group’. Here the notion of `group’ is not mere a usual group but is a group object in the category of metric spaces. Moreover, we can define `the 1-Cech cohomology of metric spaces with the coefficient in metric groups’ that also classifies metric fibration.

Luigi Caputi

Title : On finite generation in magnitude (co)homology, and its torsion

Abstract : The aim of the talk is to show that magnitude cohomology yields finitely generated modules over categories of (directed) graphs with bounded genus. We will use the theory of quasi-Groebner categories, as recently developed by Sam and Snowden, and present the case of directed graphs. Then, we will discuss two main consequences. First, the ranks of magnitude homology of graphs with bounded genus, in a fixed degree, grow at most polynomially in the number of vertices. Second, the order of its torsion, in each fixed degree, is bounded. 

Emerson Gaw Escolar

Title : On interval covers and resolutions of persistence modules

Abstract : In topological data analysis, one-parameter persistent homology can be used to describe the topological features of data in a multiscale way via the persistence barcode, which can be formalized as a decomposition of a persistence module into interval representations. Multi-parameter persistence modules, however, do not necessarily decompose into intervals and thus (together with other reasons) are complicated to describe. In this talk I will discuss some recent results concerning the use of relative homological algebra with respect to intervals in the study of multi-parameter persistence modules and more generally persistence modules over posets. In particular, I will discuss a nice property of "interval covers" and its potential interpretation as an invariant, a monotonicity property of the "interval global dimension" of posets, and a result classifying all posets of interval global dimension zero. This talk is mainly based on joint work with Toshitaka Aoki and Shunsuke Tada (https://arxiv.org/abs/2308.14979).

Kiyonori Gomi

Title : A direct proof for the positive definiteness of four point metric spaces

Abstract : The positive definiteness is a property of a metric space which ensures nice behaviour of its magnitude. For any four-point metric space, the positive definiteness was first established by Meckes, where an embedding theorem is invoked. The aim of my talk is to explain a direct proof for this fact without invoking an embedding theorem. I also discuss a possible condition for the magnitude of a finite metric space to obey the inclusion-exclusion principle with respect to a specific choice of subspaces. This condition is suggested by the direct proof, and its validity is verified when the number of points is small.

Richard Hepworth

Title: Bigraded path homology

Abstract : Important recent work of Asao conclusively demonstrated that magnitude homology and path homology of graphs, both previously unrelated, are in fact just two aspects of a much larger object, the magnitude-path spectral sequence or MPSS.  Magnitude homology is precisely the E1 page of this sequence, while path homology is an axis of the E2 page. In this talk I will present joint work in progress with Emily Roff.  We show that the E2 page of the MPSS satisfies Kunneth, excision and Mayer-Vietoris theorems.  These, together with the homotopy-invariance property proved by Asao, show that the entire E2 term should be regarded as a homology theory, which we call "bigraded path homology".  Second, we show that bigraded path homology is a strictly stronger invariant than path homology, by demonstrating that it can completely distinguish the directed cycles, none of which can be told apart under the original path homology.

Sergei O Ivanov

Title : On the path homology of Cayley digraphs and covering digraphs

Abstract : We develop a theory of covering digraphs, similar to the theory of covering spaces. By applying this theory to Cayley digraphs, we build a "bridge" between GLMY-theory and group homology theory, which helps to reduce path homology calculations to group homology computations. We show some cases where this approach allows us to fully express path homology in terms of group homology. To illustrate this method, we provide a path homology computation for the Cayley digraph of the additive group of rational numbers with a generating set consisting of inverses to factorials. The main tool in our work is a filtered simplicial set associated with a digraph, which we call the filtered nerve of a digraph, and whose quotients have homology isomorphic to the magnitude homology.

Tsubasa Kamiyama

Title : Metric fibrations over one-dimensional base spaces are trivial

Abstract : The notion of a metric fibration was first introduced by Leinster. A metric fibration is said to be trivial if the total space is obtained by the \ell_1 product of the base space and the fiber. A notable property of this is that its magnitude coincides with the product of the magnitudes of the base space and the fiber. It is known that for any odd-cycle graph, non-trivial fibrations do exist over the graph. On the other hand, any metric fibration over any even-cycle graph is trivial.  In this talk, we focus on the existence of non-trivial metric fibrations. As a main result, we provide elementary proof that if the base space is a one-dimensional metric space, then only a trivial fibration exists.

Tom Leinster

Title : Magnitude homology equivalence of Euclidean sets (Joint work with Adrián Doña Mateo)

Abstract : I will present a theorem describing when two closed subsets X and Y of Euclidean space have the same magnitude homology in the following strong sense: there are distance-decreasing maps X \to Y and Y \to X inducing mutually inverse isomorphisms in positive-degree magnitude homology. The theorem states that this is equivalent to a certain concrete geometric condition. This condition involves the notion of the "inner boundary" of a metric space, which consists of those points adjacent to some other point. The proof of the theorem builds on Kaneta and Yoshinaga's structure theorem for magnitude homology.

Tom Leinster (Colloquium)

Title : The many faces of magnitude

Abstract : The magnitude of a square matrix is the sum of all the entries of its inverse. This strange definition, suitably used, enables us to define the "magnitude" of many objects in different contexts across mathematics. All of them can be understood as measures of size. For example, the magnitude of a metric space combines classical quantities such volume, surface area, and dimension. The magnitude of a category is closely related to Euler characteristic. The magnitude of a graph is an invariant sharing features with the Tutte polynomial (but not a specialization of it). Magnitude also appears in the difficult problem of quantifying biological diversity: under certain circumstances, the greatest possible diversity of an ecosystem is exactly its magnitude. And there is now a theory of magnitude homology, which has the same relationship to magnitude as ordinary homology does to Euler characteristic. I will give an aerial view of this landscape.

Katharina Limbeck

Title : Metric space magnitude for evaluating unsupervised representation learning

Abstract : The magnitude of a metric space was recently established as a novel invariant, providing a measure of the `effective size' of a space across multiple scales. By capturing both geometrical and topological properties of data, magnitude is poised to address challenges in unsupervised representation learning tasks. We formalise a novel notion of a dissimilarity between magnitude functions of finite metric spaces and use them to derive a quality measure for dimensionality reduction tasks. Our measure is provably stable under perturbations of the data, can be efficiently calculated, and enables a rigorous multi-scale comparison of embeddings. We show the utility of our measure in an experimental suite that comprises different domains and tasks, including the comparison of data visualisations.

Mark Meckes --- Slides

Title : Inequalities for magnitude and maximum diversity

Abstract : Inequalities lie at the heart of the study and applications of classical numerical geometric invariants like volume, surface area, and metric entropy.  (Think for example of the isoperimetric inequality, or the Brunn-Minkowski inequality which implies it).  I will survey the inequalities that are known for magnitude and maximum diversity, some applications, and open questions.

Giuliamaria Menara

Title : Eulerian magnitude homology: introduction and application to random graphs

Abstract : Hepworth, Willerton, Leinster and Shulman introduced magnitude homology groups for enriched categories, and in particular for graphs. Although the construction of the boundary map suggests magnitude homology groups are strongly influenced by graph substructures it is not straightforward to detect such subgraphs. In this talk, we introduce eulerian magnitude homology and highlight its relation with magnitude homology. We then show how eulerian magnitude homology enables a more accurate analysis of graph substructures and apply these results to Erdos-Renyi random graphs and random geometric graphs, obtaining an asymptotic estimate for the Betti numbers of the eulerian magnitude homology groups on the first diagonal.

Jun O’Hara

Title : Residues of manifolds

Abstract : Brylinski's beta function of a manifold $M$ is a meromorphic function of a complex variable $z$ which is given by the double integral on the product space $M\times M$ of the distance between a pair of points to the power $z$. It has only simple poles, and its residues behave somewhat similarly to magnitudes. I will introduce the relation shown by Gimperlein et.al. and show some properties of residues. 

Emily Roff

Title : Iterated magnitude homology

Abstract : In 2017, extending Hepworth and Willerton’s construction for graphs, Leinster and Shulman introduced the magnitude homology of an enriched category. Their construction generalizes ordinary categorical homology: the magnitude homology of an ordinary small category is the homology of its classifying space. But categorical homology can also be generalized in a different direction. Duskin and Street established back in the 1980s that the classifying space extends naturally from categories to bicategories, lifting the theory to a second categorical dimension. This talk will explore a hybrid of the two ideas: an iterated magnitude homology theory for categories with a second-order enrichment. Such objects occur in nature more often than one might imagine---in particular, various commonly-encountered structures on groups can be described as second-order enrichments in familiar base categories. To keep things down-to-earth we will focus on such examples, investigating the content and behaviour of iterated magnitude homology when interpreted for groups equipped with extra structure such as a congruence, a partial ordering, or a bi-invariant metric. This talk will be based on the preprint https://arxiv.org/abs/2309.00577.

Sho Shimoyama

Title : Exploring the Uniqueness of Minimizing-Movement in Metric Spaces: beyond p = 2.

Abstract : The p-curve of maximal slopes (p-CMS) for a function defined on a metric space plays a similar role to the gradient flow in the Euclidean setting, decreasing the function along it in the steepest descent direction. Minimizing-movement (MM) schemes is one of the methods for constructing p-CMS. In the case p=2, Mayer proved that MM is (i) unique and (ii) p-CMS on nonpositively curved metric spaces. Ambrosio-Gigli-Savar\'e studied the case p=2 and showed that the above result holds for a wider class of metric spaces including positively curved spaces. However, the case of p \neq 2 is still widely open. In this talk, after an introduction to related notions, we provide the first example satisfying (i) for a fixed p \gt 2 and then discuss (ii) for the example.

Yu Tajima

Title : Magnitude homotopy type of graphs and Whitney twist via discrete Morse theory

Abstract : The invariance of magnitudes for graphs differ by a Whitney twist (with a certain condition) was proved by Leinster, and the problem that "Are their magnitude homology groups isomorphic" is still open. We approach the problem using discrete Morse theory on magnitude homotopy types. In the course of this approach, we had another proof of the invariance of magnitudes under a Whitney twist. This is joint work with Masahiko Yoshinaga (Osaka University).

Asuka Takatsu

Title : Geometry of sliced/disintegrated Monge--Kantorovich metrics

Abstract : The Monge--Kantorovich transport problem is a variational problem on the space of probability measures over a complete separable metric space. This provides the so-called Monge--Kantorovich metric on the space of probability measures and the metric is applied in the fields of geometric analysism, PDE's and applied mathematics. Since the Monge--Kantorovich metric is expensive to compute, there is great interest in alternative metrics on spaces of probability measures. In this talk, I explain two different two-parameter families of metrics derived from a slice-wise/disintegrated optimal transport problem. One family contains sliced and max-sliced Wasserstein metrics, which are used in applied mathematics. This talk is based on joint work with Jun KITAGAWA (Michigan State University).

Masaki Tsukamoto

Title : Introduction to mean dimension.

Abstract : Mean dimension is a topological invariant of dynamical systems introduced by Gromov in 1999. It is a dynamical version of topological dimension. It evaluates the number of parameters per unit time for describing a given dynamical system. Gromov introduced this notion for the purpose of exploring a new direction of geometric analysis. Independently of this original motivation, Elon Lindenstrauss and Benjamin Weiss found deep applications of mean dimension in topological dynamics. I plan to survey some highlights of the mean dimension theory.

Juan Pablo Vigneaux Ariztia

Title : A combinatorial approach to Möbius inversion and pseudoinversion

Abstract : Firstly,  I’ll explain how Cramer's formula for the inverse of a matrix and a combinatorial expression for determinants give a novel combinatorial interpretation of the Möbius inverse whenever it exists. The sums thus obtained are indexed by linear connections on associated digraphs (each given by a path without cycles and a cycle decomposition of its complement). My result contains, as particular cases, previous theorems by P. Hall and T. Leinster.

Secondly, I’ll present a new definition of magnitude (valid for any finite category) in terms of the pseudo-Möbius function (Moore-Penrose pseudoinverse of the Dirichlet zeta matrix). This definition has been recently introduced by Akkaya and Ünlü and, independently, by Chen and me.  I’ll show how Berg's formula for the computation of the Moore-Penrose pseudoinverse, the natural generalization of Cramer's rule, yields a combinatorial interpretation of the pseudo-Möbius function, and therefore also of magnitude in all cases. This combinatorial interpretation a priori differs from the one given by magnitude homology.

Jun Yoshida

Title : On a logical foundation for parametrized homologies

Abstract : The magnitude homology has a variant called the blurred magnitude homology. It is constructed from a filtration by a distance function on a simplex spanned by the points on a metric space.

From this point of view, it is thought of as a sibling of the persistent homology. The goal of this talk is to introduce a common foundation for this type of parameterized homologies. More precisely, I will explain how they are understood as semantics of ordinary simplicial homology in the intuitionistic logic. As an application, one can make a provably correct implementation of these homologies.

Masahiko Yoshinaga

Title : Magnitude homotopy type

Abstract : Magnitude homology group was introduced by Hepworth-Willerton and Leinster-Shulman. It was mentioned that the magnitude homology group can be considered as the reduced homology of certain simplicial set (also by Bottinelli-Kaiser for general metric spaces). Inspired by Asao-Izumihara's work for finite graphs, we introduce a pair of simplicial complexes whose quotient is isomorphic to the above mentioned simplicial set. The construction is functorial and enables us to apply discrete Morse theory on magnitude homology groups. This is joint work with Y. Tajima.