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Interactions and relations between objects may be pairwise or higher-order in nature, and so network-valued data are ubiquitous in the real world. The "space of networks", however, has a complex structure that cannot be adequately described…
In this paper, we define the notion of a virtually symmetric representation of representations of virtual braid groups and prove that many known representations are equivalent to virtually symmetric. Using one such representation, we define…
Trace diagrams are structured graphs with edges labeled by matrices. Each diagram has an interpretation as a particular multilinear function. We provide a rigorous combinatorial definition of these diagrams using a notion of signed graph…
We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general…
The Reidemeister theorem states that any link in $3$-space can be encoded by a diagram (a suitably decorated projection) on a plane, and provides a finite set of combinatorial moves relating two diagrams of the same link up to isotopy. In…
We describe an algorithm for compressing a partially ordered set, or \emph{poset}, so that it occupies space matching the information theory lower bound (to within lower order terms), in the worst case. Using this algorithm, we design a…
This chapter discusses the interplay between structure and dynamics in complex networks. Given a particular network with an endowed dynamics, our goal is to find partitions aligned with the dynamical process acting on top of the network. We…
Single-affiliation systems are observed across nature and society. Examples include collaboration, organisational affiliations, and trade-blocs. The study of such systems is commonly approached through network analysis. Multilayer networks…
Given a surface with boundary and some points on its boundary, a polygon diagram is a way to connect those points as vertices of non-overlapping polygons on the surface. Such polygon diagrams represent non-crossing permutations on a surface…
We propose a categorical framework to reason about scientific explanations: descriptions of a phenomenon meant to translate it into simpler terms, or into a context that has been already understood. Our motivating examples come from systems…
A permutation graph is a graph whose edges are given by inversions of a permutation. We study the Abelian sandpile model (ASM) on such graphs. We exhibit a bijection between recurrent configurations of the ASM on permutation graphs and the…
The partition algebras are algebras of diagrams (which contain the group algebra of the symmetric group and the Brauer algebra) such that the multiplication is given by a combinatorial rule and such that the structure constants of the…
In this paper we study the relation between two diagrammatic representations of links in lens spaces: the disk diagram and the grid diagram and we find how to pass from one to the other. We also investigate whether the HOMFLY-PT invariant…
Predicting missing links between entities in a knowledge graph is a fundamental task to deal with the incompleteness of data on the Web. Knowledge graph embeddings map nodes into a vector space to predict new links, scoring them according…
Square grids play a pivotal role in Robertson and Seymour's work on graph minors as planar obstructions to small treewidth. We introduce a three-sided bramble in a plane graph called a net, which generalizes the standard bramble of crosses…
Node representation learning in a network is an important machine learning technique for encoding relational information in a continuous vector space while preserving the inherent properties and structures of the network. Recently,…
Though the uniformization theorem guarantees an equivalence of Riemann surfaces and smooth algebraic curves, moving between analytic and algebraic representations is inherently transcendental. Our analytic curves identify pairs of circles…
A representation theorem relates different mathematical structures by providing an isomorphism between them: that is, a one-to-one correspondence preserving their original properties. Establishing that the two structures substantially…
This is an introduction to the group algebras of the symmetric groups, written for a quarter-long graduate course. After recalling the definition of group algebras (and monoid algebras) in general, as well as basic properties of…
New relations between algebraic geometry, information theory and Topological Field Theory are developed. One considers models of databases subject to noise i.e. probability distributions on finite sets, related to exponential families. We…