Related papers: Manifolds and Disc-presheaves
These notes are designed for those who either plan to work in differential geometry, or at least want to have a good reason not to do it. We discuss smooth curves and surfaces -- the main gate to differential geometry. We focus on the…
The product of smooth valuations on manifolds is described in terms of differential forms, Gelfand transforms and blow-up spaces. It is shown that the product extends partially to generalized valuations and corresponds geometrically to…
To study embeddings of tangles in knots, we use quandle cocycle invariants. Computations are carried out for the tables of knots and tangles, to investigate which tangles may or may not embed in knots in the tables.
Graph embeddings, wherein the nodes of the graph are represented by points in a continuous space, are used in a broad range of Graph ML applications. The quality of such embeddings crucially depends on whether the geometry of the space…
We study configuration spaces of framed points on oriented closed smooth manifolds. Such configuration spaces admit natural actions of the framed little discs operads, that play an important role in the study of embedding spaces of…
In this paper we give a complete description about normal monohedral tilings of a convex disc with smooth boundary where we have at most three topological discs as tiles. This result is a far-reaching generalization of the results of…
A subclass of complex-valued close-to-convex harmonic functions that are univalent and sense-preserving in the open unit disc is investigated. The coefficient estimates, growth results, area theorem, boundary behavior, convolution and…
We use mass-transportation as a tool to compare surfaces (2-manifolds). In particular, we determine the "similarity" of two given surfaces by solving a mass-transportation problem between their conformal densities. This mass transportation…
Smooth parametrization consists in a subdivision of the mathematical objects under consideration into simple pieces, and then parametric representation of each piece, while keeping control of high order derivatives. The main goal of the…
We study families of submanifolds in symmetric spaces of compact type arising as exponential images of s-orbits of variable radii. Special attention is given to the cases where the s-orbits are symmetric.
Stratifolds are considered from a categorical point of view. We show among others that the category of stratifolds fully faithfully embeds into the category of ${\mathbb R}$-algebras as does the category of smooth manifolds. We prove that a…
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete…
Generic polyhedra are interesting mathematical objects to study in their own right. In this paper, we initialize a systematic study of two-dimensional generic polyhedra with an eye towards applications to low-dimensional topology,…
We give an overview of how calculus of the embedding functor can be used for the study of long knots and summarize various results connecting the calculus approach to the rational homotopy type of spaces of long knots, collapse of the…
Refining the notion of an ideal triangulation of a compact three-manifold, we provide in this paper a combinatorial presentation of the set of pairs (M,a), where M is a three-manifold and a is a collection of properly embedded arcs. We also…
This master thesis looks at the gradient flow of the length functional on embedded loops. The space of embedded loops is endowed with a scale structure so that the length functional becomes scale smooth. For certain underlying manifolds,…
We present an unsupervised approach for discovering semantic representations of mathematical equations. Equations are challenging to analyze because each is unique, or nearly unique. Our method, which we call equation embeddings, finds good…
Variational analysis presents a unified theory encompassing in particular both smoothness and convexity. In a Euclidean space, convex sets and smooth manifolds both have straightforward local geometry. However, in the most basic hybrid case…
This paper contains a set of lecture notes on manifolds with boundary and corners, with particular attention to the space of quantum states. A geometrically inspired way of dealing with these kind of manifolds is presented,and explicit…
A simplified, user-friendly repackaging of the curvature estimates implied by the Seiberg-Witten equations is formulated in terms of the convex hull of the set of monopole classes. New results are also obtained concerning boundary cases of…