Related papers: Lecture Notes on Differential Forms
The content of this paper can be roughly organized into a three-level hierarchy of generality. At the first, most general level, we introduce a new language which allows us to express various categorical structures in a systematic and…
In order to solve two problems in deformation theory, we establish natural structures of homotopy Lie algebras and of homotopy associative algebras on tensor products of algebras of different types and on mapping spaces between coalgebras…
This is a long summary of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html . A shorter survey paper on the book, focussing on…
We study sheaves of differential forms and their cohomology in the h-topology. This allows to extend standard results from the case of smooth varieties to the general case. As a first application we explain the case of singularities arising…
A general theory of topological classification of defects is introduced. We illustrate the application of tools from algebraic topology, including homotopy and cohomology groups, to classify defects including several explicit calculations…
We propose TopDis (Topological Disentanglement), a method for learning disentangled representations via adding a multi-scale topological loss term. Disentanglement is a crucial property of data representations substantial for the…
Minkowski tensors are comprehensive shape descriptors that robustly capture n-point information in complex random geometries and that have already been extensively applied in the Euclidean plane. Here, we devise a novel framework for…
Category theory provides a means through which many far-ranging fields of mathematics can be related by their similar structure. In a paper by Robinson [2], this interconnectivity afforded by categorical perspectives allowed for the…
This is a introductory course focusing some basic notions in pseudodifferential operators ($\Psi$DOs) and microlocal analysis. We start this lecture notes with some notations and necessary preliminaries. Then the notion of symbols and…
We introduce a topology, which we call the regional topology, on the space of all real functions on a given locally compact metric space. Next we obtain a new versions of Schauder's fixed point theorem and Ascoli's theorem. We use these…
In this paper, we aim to provide a notion of "relative objects", i.e. objects equipped with some sort of subobjects, in differential topology. In spite of active researches relating them, e.g. knot theory or the theory of manifolds with…
These course notes are about computing modular forms and some of their arithmetic properties. Their aim is to explain and prove the modular symbols algorithm in as elementary and as explicit terms as possible, and to enable the devoted…
We construct and study a morphism of spectra implementing the Anderson duality of topological modular forms ($\mathrm{TMF}$). Its differential version will then be introduced, allowing us to pair elements of $\pi_d\mathrm{TMF}$ with spin…
Low rank tensor decompositions are a powerful tool for learning generative models, and uniqueness results give them a significant advantage over matrix decomposition methods. However, tensors pose significant algorithmic challenges and…
In many scenarios, especially biomedical applications, the correct delineation of complex fine-scaled structures such as neurons, tissues, and vessels is critical for downstream analysis. Despite the strong predictive power of deep learning…
Higher-order tensors appear in various areas of mechanics as well as physics, medicine or earth sciences. As these tensors are highly complex, most are not well understood. Thus, the analysis and the visualization process form a highly…
We study "higher-dimensional" generalizations of differential forms. Just as differential forms can be defined as the universal commutative differential algebra containing C^\infty(M), we can define differential gorms as the universal…
The invariant is one of central topics in science, technology and engineering. The differential invariant is essential in understanding or describing some important phenomena or procedures in mathematics, physics, chemistry, biology or…
This is a survey of the author's book "D-manifolds and d-orbifolds: a theory of derived differential geometry", available at http://people.maths.ox.ac.uk/~joyce/dmanifolds.html We introduce a 2-category dMan of "d-manifolds", new geometric…
A polynomial is presented that models a topological knot in a unique manner. It distinguishes all types of knots including the orientation and has a group theory interpretation. The topologies may be labeled via a number, which upon a base…