Related papers: Cellular Sheaves on Higher-Dimensional Structures
We employ techniques from topological data analysis to model sensor networks. Our approach to sensor integration uses the topological method of sheaves over cell complexes. The internal consistency of data from individual sensors is…
The discovery of neural architectures from simple building blocks is a long-standing goal of Neural Architecture Search (NAS). Hierarchical search spaces are a promising step towards this goal but lack a unifying search space design…
We survey recent advances in the theory of moduli spaces of stable sheaves on hyperk\"ahler manifolds of dimension greater than $2$. We start by recalling the well-known theory in dimension $2$, i.e.~for $K3$ surfaces, emphasizing the…
Volumetric brain reconstructions provide an unprecedented opportunity to gain insights into the complex connectivity patterns of neurons in an increasing number of organisms. Here, we model and quantify the complexity of the resulting…
As data structures and mathematical objects used for complex systems modeling, hypergraphs sit nicely poised between on the one hand the world of network models, and on the other that of higher-order mathematical abstractions from algebra,…
In this paper, we investigate the properties of $A$-coherent and $A$-quasi-coherent sheaves within the framework of algebraic geometry over non-algebraically closed fields. We define an $\mathcal{O}_X$-module to be $A$-coherent (resp.…
Off-shell $(4,0)$ supermultiplets in 2-dimensions are formulated. These are used to construct sigma models whose target spaces are vector bundles over manifolds that are hyperk\"ahler with torsion. The off-shell supersymmetry implies that…
In Schiebinger et al. (2017), the authors use optimal transport of measures on empirical distributions arising from biological experiments to relate the single cell RNA sequencing profiles for induced pluripotent stem cells differentiating.…
We extend Cellular Automata to time-varying discrete geometries. In other words we formalize, and prove theorems about, the intuitive idea of a discrete manifold which evolves in time, subject to two natural constraints: the evolution does…
Modern deep learning architectures increasingly contend with sophisticated signals that are natively infinite-dimensional, such as time series, probability distributions, or operators, and are defined over irregular domains. Yet, a unified…
Single-cell data analysis seeks to characterize cellular heterogeneity based on high-dimensional gene expression profiles. Conventional approaches represent each cell as a vector in Euclidean space, which limits their ability to capture…
The purpose of this contribution is to give a coherent account of a particular narrative which links locales, geometric theories, sheaf semantics and constructive commutative algebra. We are hoping to convey a firm grasp of three ideas: (1)…
Over a smooth complex projective curve, we study an algebraic versal deformation space with fixed determinant of a coherent sheaf. The algebraic versal deformation space decomposes into a disjoint union of Shatz strata, namely locally…
The mechanical response of cellular materials with spinodal topologies is numerically and experimentally investigated. Spinodal microstructures are generated by the numerical solution of the Cahn-Hilliard equation. Two different topologies…
We develop a statistical mechanical approach based on the replica method to study the design space of deep and wide neural networks constrained to meet a large number of training data. Specifically, we analyze the configuration space of the…
This article reviews recent progress in the development of the computing framework vector symbolic architectures (VSA) (also known as hyperdimensional computing). This framework is well suited for implementation in stochastic, emerging…
We present a new construction of high dimensional expanders based on covering spaces of simplicial complexes. High dimensional expanders (HDXs) are hypergraph analogues of expander graphs. They have many uses in theoretical computer…
Multilayer networks allow one to represent diverse and coupled connectivity patterns --- e.g., time-dependence, multiple subsystems, or both --- that arise in many applications and which are difficult or awkward to incorporate into standard…
A derivative structure is a nonequivalent substitutional atomic configuration derived from a given primitive cell. The enumeration of derivative structures plays an essential role in searching for the ground states in multicomponent…
Biomembranes adopt varying morphologies that are vital to cellular functions. Many studies use computational modeling to understand how various mechanochemical factors contribute to membrane shape transformations. Compared to…