Convex geometry | Metric geometry
In mathematics, convex metric spaces are, intuitively, metric spaces with the property any "segment" joining two points in that space has other points in it besides the endpoints. Formally, consider a metric space (X, d) and let x and y be two points in X. A point z in X is said to be between x and y if all three points are distinct, and that is, the triangle inequality becomes an equality. A convex metric space is a metric space (X, d) such that, for any two distinct points x and y in X, there exists a third point z in X lying between x and y. Metric convexity: * does not imply convexity in the usual sense for subsets of Euclidean space (see the example of the rational numbers) * nor does it imply path-connectedness (see the example of the rational numbers) * nor does it imply geodesic convexity for Riemannian manifolds (consider, for example, the Euclidean plane with a closed disc removed). (Wikipedia).
This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
Metric space definition and examples. Welcome to the beautiful world of topology and analysis! In this video, I present the important concept of a metric space, and give 10 examples. The idea of a metric space is to generalize the concept of absolute values and distances to sets more gener
From playlist Topology
Complete metric space: example & proof
This video discusses an example of particular metric space that is complete. The completeness is proved with details provided. Such ideas are seen in branches of analysis.
From playlist Mathematical analysis and applications
Introduction to Metric Spaces - Definition of a Metric. - The metric on R - The Euclidean Metric on R^n - A metric on the set of all bounded functions - The discrete metric
From playlist Topology
What is a metric space? An example
This is a basic introduction to the idea of a metric space. I introduce the idea of a metric and a metric space framed within the context of R^n. I show that a particular distance function satisfies the conditions of being a metric.
From playlist Mathematical analysis and applications
This video is about metric spaces and some of their basic properties.
From playlist Basics: Topology
MAST30026 Lecture 2: Examples of spaces (Part 1)
I started with the definition of a metric space, we briefly discussed the example of Euclidean space (proofs next time) and then I started to explain a few natural metrics on the circle. Lecture notes: http://therisingsea.org/notes/mast30026/lecture2.pdf The class webpage: http://therisin
From playlist MAST30026 Metric and Hilbert spaces
Weird notions of "distance" || Intro to Metric Spaces
Visit https://brilliant.org/TreforBazett/ to get started learning STEM for free, and the first 200 people will get 20% off their annual premium subscription. Check out my MATH MERCH line in collaboration with Beautiful Equations ►https://www.beautifulequation.com/pages/trefor Weird, fun
From playlist Cool Math Series
Metric Spaces: A subset U of Y is open in Y if and only if U = V n Y for some open subset V of X
In this video I do a proof. We have a Metric Space X. I prove that a subset U of Y is open in Y if and only if U = V n Y for some open subset V of X. If you enjoyed this video please consider liking, sharing, and subscribing. Udemy Courses Via My Website: https://mathsorcerer.com Free
From playlist Metric Spaces
Parvaneh Joharinad (7/27/22): Curvature of data
Abstract: How can one determine the curvature of data and how does it help to derive the salient structural features of a data set? After determining the appropriate model to represent data, the next step is to derive the salient structural features of data based on the tools available for
From playlist Applied Geometry for Data Sciences 2022
Jürgen Jost (8/29/21): Geometry and Topology of Data
Data sets are often equipped with distances between data points, and thereby constitute a discrete metric space. We develop general notions of curvature that capture local and global properties of such spaces and relate them to topological concepts such as hyperconvexity. This also leads t
From playlist Beyond TDA - Persistent functions and its applications in data sciences, 2021
Complex geometry of Teichmuller domains (Lecture 1) by Harish Seshadri
PROGRAM CAUCHY-RIEMANN EQUATIONS IN HIGHER DIMENSIONS ORGANIZERS: Sivaguru, Diganta Borah and Debraj Chakrabarti DATE: 15 July 2019 to 02 August 2019 VENUE: Ramanujan Lecture Hall, ICTS Bangalore Complex analysis is one of the central areas of modern mathematics, and deals with holomo
From playlist Cauchy-Riemann Equations in Higher Dimensions 2019
Jürgen Jost (5/13/22): Geometry and Topology of Data
We link the basic concept of topological data analysis, intersection patterns of distance balls, with geometric concepts. The key notion is hyperconvexity, and we also explore some variants. Hyperconvexity in turn leads us to a new concept of generalized curvature for metric spaces. Curvat
From playlist Bridging Applied and Quantitative Topology 2022
An Introduction to Geodesic Convexity - Nisheeth Vishnoi
Optimization, Complexity and Invariant Theory Topic: An Introduction to Geodesic Convexity Speaker: Nisheeth Vishnoi Affiliation: EPFL Date: June 7. 2018 For more videos, please visit http://video.ias.edu
From playlist Mathematics
Complex Brunn–Minkowski theory and positivity of vector bundles – Bo Berndtsson – ICM2018
Geometry | Analysis and Operator Algebras Invited Lecture 5.2 | 8.2 Complex Brunn–Minkowski theory and positivity of vector bundles Bo Berndtsson Abstract: This is a survey of results on positivity of vector bundles, inspired by the Brunn–Minkowski and Prékopa theorems. Applications to c
From playlist Geometry
New Methods in Finsler Geometry - 23 May 2018
http://www.crm.sns.it/event/415 Centro di Ricerca Matematica Ennio De Giorgi The workshop has limited funds to support lodging (and in very exceptional cases, travel) costs of some participants, with priority given to young researchers. When you register, you will have the possibility to
From playlist Centro di Ricerca Matematica Ennio De Giorgi
Set Chasing, with an application to online shortest path - Sébastien Bubeck
Computer Science/Discrete Mathematics Seminar I Topic: Set Chasing, with an application to online shortest path Speaker: Sébastien Bubeck Affiliation: Microsoft Research Lab - Redmond Date: April 18, 2022 Since the late 19th century, mathematicians have realized the importance and genera
From playlist Mathematics
Jürgen Jost (10/29/21): Geometry and Topology of Data
Topological data analysis asks when balls in a metric space (X, d) intersect. Geometric data analysis asks how much balls have to be enlarged to intersect. This is captured by a suitable concept of curvature. And curvature quantifies convexity.
From playlist Vietoris-Rips Seminar
Johnathan Bush (11/5/21): Maps of Čech and Vietoris–Rips complexes into euclidean spaces
We say a continuous injective map from a topological space to k-dimensional euclidean space is simplex-preserving if the image of each set of at most k+1 distinct points is affinely independent. We will describe how simplex-preserving maps can be useful in the study of Čech and Vietoris–Ri
From playlist Vietoris-Rips Seminar
An introduction to the Gromov-Hausdorff distance
Title: An introduction to the Gromov-Hausdorff distance Abstract: We give a brief introduction to the Hausdorff and Gromov-Hausdorff distances between metric spaces. The Hausdorff distance is defined on two subsets of a common metric space. The Gromov-Hausdorff distance is defined on any
From playlist Tutorials