135 research outputs found
Simplicial decompositions of graphs: a survey of applications
AbstractWe survey applications of simplicial decompositions (decompositions by separating complete subgraphs) to problems in graph theory. Among the areas of application are excluded minor theorems, extremal graph theorems, chordal and interval graphs, infinite graph theory and algorithmic aspects
Algebraic Topology for Data Scientists
This book gives a thorough introduction to topological data analysis (TDA),
the application of algebraic topology to data science. Algebraic topology is
traditionally a very specialized field of math, and most mathematicians have
never been exposed to it, let alone data scientists, computer scientists, and
analysts. I have three goals in writing this book. The first is to bring people
up to speed who are missing a lot of the necessary background. I will describe
the topics in point-set topology, abstract algebra, and homology theory needed
for a good understanding of TDA. The second is to explain TDA and some current
applications and techniques. Finally, I would like to answer some questions
about more advanced topics such as cohomology, homotopy, obstruction theory,
and Steenrod squares, and what they can tell us about data. It is hoped that
readers will acquire the tools to start to think about these topics and where
they might fit in.Comment: 322 pages, 69 figures, 5 table
The logarithmic Picard group and its tropicalization
We construct the logarithmic and tropical Picard groups of a family of
logarithmic curves and realize the latter as the quotient of the former by the
algebraic Jacobian. We show that the logarithmic Jacobian is a proper family of
logarithmic abelian varieties over the moduli space of Deligne-Mumford stable
curves, but does not possess an underlying algebraic stack. However, the
logarithmic Picard group does have logarithmic modifications that are
representable by logarithmic schemes, all of which are obtained by pullback
from subdivisions of the tropical Picard group.Comment: 3 figures (2 added), 66 pages; comments welcome
Multicoloured Random Graphs: Constructions and Symmetry
This is a research monograph on constructions of and group actions on
countable homogeneous graphs, concentrating particularly on the simple random
graph and its edge-coloured variants. We study various aspects of the graphs,
but the emphasis is on understanding those groups that are supported by these
graphs together with links with other structures such as lattices, topologies
and filters, rings and algebras, metric spaces, sets and models, Moufang loops
and monoids. The large amount of background material included serves as an
introduction to the theories that are used to produce the new results. The
large number of references should help in making this a resource for anyone
interested in beginning research in this or allied fields.Comment: Index added in v2. This is the first of 3 documents; the other 2 will
appear in physic
Graph Theory
Highlights of this workshop on structural graph theory included new developments on graph and matroid minors, continuous structures arising as limits of finite graphs, and new approaches to higher graph connectivity via tree structures
Noetherian modules over combinatorial categories
In this project, we analyse subcategories C of the category of finite sets and functions. A C-module over a ring k is a functor from C to the category of left k-modules. We investigate whether the category of C-modules is Noetherian whenever the ring k is left-Noetheria
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