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Marginal polytopes are important geometric objects that arise in statistics as the polytopes underlying hierarchical log-linear models. These polytopes can be used to answer geometric questions about these models, such as determining the…
Given a scheme $Y$ equipped with a collection of globally generated vector bundles $E_1, \dots, E_n$, we study the universal morphism from $Y$ to a fine moduli space $\mathcal{M}(E)$ of cyclic modules over the endomorphism algebra of…
Graph neural networks (GNNs) are the most widely adopted model in graph-structured data oriented learning and representation. Despite their extraordinary success in real-world applications, understanding their working mechanism by theory is…
Complex geometry and symplectic geometry are mirrors in string theory. The recently developed generalised complex geometry interpolates between the two of them. On the other hand, the classical and quantum mechanics of a finite number of…
These are notes on the method of normalized graph cuts and its applications to graph clustering. I provide a fairly thorough treatment of this deeply original method due to Shi and Malik, including complete proofs. I include the necessary…
We generalize the scattering approach to quantum graphs to quantum graphs with with piecewise constant potentials and multiple excitation modes. The free single-mode case is well-known and leads to the trace formulas of Roth, Kottos and…
Let $G$ be a simple graph and let $\mathcal{L}(G)$ be the free partially commutative Lie algebra associated to $G$. In this paper, using heaps of pieces, we prove an expression for the generalized $\textbf k$-chromatic polynomial of $G$ in…
There is no known polynomial-time algorithm for graph isomorphism testing, but elementary combinatorial "refinement" algorithms seem to be very efficient in practice. Some philosophical justification is provided by a classical theorem of…
We study the generalized Lam\'e equation on an elliptic curve $E$ with multiple singularities. By restricting to the locus admitting solutions with quasi-periodic properties, we construct two curves: (i) The generalized Lam'e curve: with…
We introduce the notion of the space of parallel strings with partially summable labels, which can be viewed as a geometrically constructed group completion of the space of particles with labels. We utilize this to construct a machinery…
Recently, there has been interest in the question of whether a partial matrix in which many of the fully defined principal submatrices are PSD is approximately PSD completable. These questions are related to graph theory because we can…
Generalized planning is concerned with the characterization and computation of plans that solve many instances at once. In the standard formulation, a generalized plan is a mapping from feature or observation histories into actions,…
We review some definitions and basic notions relating to generalised spin structures and introduce the notion of reducibility. We discuss connections on these structures, define a covariant Lie derivative for associated bundles and develop…
This survey gives an overview of several fundamental algebraic constructions which arise in the study of splines. Splines play a key role in approximation theory, geometric modeling, and numerical analysis, their properties depend on…
We consider the whole-plane SLE conformal map f from the unit disk to the slit plane, and show that its mixed moments, involving a power p of the derivative modulus |f'| and a power q of the map |f| itself, have closed forms along some…
We define a general framework that includes objects such as tilings, Delone sets, functions and measures. We define local derivability and mutual local derivability (MLD) between any two of these objects in order to describe their…
This paper develops a new way to create smooth piecewise polynomial free-form spline surfaces from quad- meshes that include T-junctions, where surface strips start or terminate. All mesh nodes can be interpreted as control points of…
The aim of this article is to give a concise algebraic treatment of the modular symbols formalism, generalised from modular curves to Hecke triangle surfaces. A sketch is included of how the modular symbols formalism gives rise to the…
Extremal graph theory studies the maximum or minimum number of subgraphs isomorphic to a prescribed graph under given constraints. \textit{Localization} has recently emerged as a framework that refines such problems by assigning extremal…
The notion of symmetry in polynomial rings with several indeterminates is generalized to polynomial rings over finite fields. Families of extensions of the projective line over a finite field of constants possessing this property are…