Related papers: Lectures on Differential Topology
Let P be a connected smooth p-manifold. We describe the group of all cobordism classes of smooth maps of n-manifolds to P with singularities of a given $cal K$-invariant class in terms of certain stable homotopy groups by applying the…
Topological Data Analysis (TDA) is a rigorous framework that borrows techniques from geometric and algebraic topology, category theory, and combinatorics in order to study the "shape" of such complex high-dimensional data. Research in this…
This article has been written for an educational magazine whose target audience consists of students and teachers of mathematics in universities, colleges and schools. It concerns a notion of duality between rectangles. A proof is given…
These notes are based on a series of five lectures given at the 2009 Villa de Leyva Summer School on Geometric and Topological Methods for Quantum Field Theory. The purpose of the lectures was to give an introduction to…
In this informal expository note, we quickly introduce and survey compactifications of strata of holomorphic 1-forms on Riemann surfaces, i.e. spaces of translation surfaces. In the last decade, several of these have been constructed,…
This lecture notes cover a Part III (first year graduate) course that was given at Cambridge University over several years on pseudo-differential operators. The calculus on manifolds is developed and applied to prove propagation of…
Toposes can be pictured as mathematical universes. Besides the standard topos, in which most of mathematics unfolds, there is a colorful host of alternate toposes in which mathematics plays out slightly differently. For instance, there are…
The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in recent work (collaboration with H.…
Recently, considerable research effort has been devoted to developing deep architectures for topic models to learn topic structures. Although several deep models have been proposed to learn better topic proportions of documents, how to…
This article provides a gentle, visual introduction to the basic concepts of differential geometry appropriate for students familiar with special relativity. Visual methods are used to explain basics of differential geometry and build…
These are lecture notes expanding upon a set of lectures given by G.M. at the TASI 2023 School. Part I is an introduction to topological field theory, including extended topological field theory. Part II is an introduction to generalized…
Diffeology extends differential geometry to spaces beyond smooth manifolds. This paper explores diffeology's key features and illustrates its utility with examples including singular and quotient spaces, and applications in symplectic…
Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation…
This is a slightly revised version of lectures notes for a course in Summer 2022 joint between Bonn and Copenhagen, intended as a stable citable version. The goal of this course is to make our general approach to analytic geometry via…
A notion of general manifolds is introduced. It covers all usual manifolds in mathematics. Essentially, it is a way how to get a bigger 'fibration' over a site which locally coincides with a given one. An enrichment with generalized…
Many important theorems in differential topology relate properties of manifolds to properties of their underlying homotopy types -- defined e.g. using the total singular complex or the \v{C}ech nerve of a good open cover. Upon embedding the…
Techniques from higher categories and higher-dimensional rewriting are becoming increasingly important for understanding the finer, computational properties of higher algebraic theories that arise, among other fields, in quantum…
Scholars who want to research a scientific topic must take time to read, extract meaning, and identify connections across many papers. As scientific literature grows, this becomes increasingly challenging. Meanwhile, authors summarize prior…
These lectures on supersymmetry and extra dimensions are aimed at finishing undergraduate and beginning postgraduate students with a background in quantum field theory and group theory. Basic knowledge in general relativity might be…
These notes reproduce the content of a short, 50-minutes, survey talk given at the Nice University in September, 2004. We added a few topics that have not been touched on in the lecture by lack of time.