This lecture is on Introduction to Higher Mathematics (Proofs). For more see http://calculus123.com.
From playlist Proofs
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
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
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
Metric Spaces Proof: The Interior of Y is Open
Let X be a metric space and Y a subset of X. In this video I prove that the interior of Y, denoted int(Y), is an open subset of X. Recall that the interior of Y is the set of all of the interior points of Y. Also recall that we say that x in X is an interior point of Y if there is an open
From playlist Metric Spaces
Abstract Algebra | Injective Functions
We give the definition of an injective function, an outline of proving that a given function is injective, and a few examples. http://www.michael-penn.net http://www.randolphcollege.edu/mathematics/
From playlist Abstract Algebra
Definition of an Injective Function and Sample Proof
We define what it means for a function to be injective and do a simple proof where we show a specific function is injective. Injective functions are also called one-to-one functions. Useful Math Supplies https://amzn.to/3Y5TGcv My Recording Gear https://amzn.to/3BFvcxp (these are my affil
From playlist Injective, Surjective, and Bijective Functions
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
What is an Injective Function? Definition and Explanation
An explanation to help understand what it means for a function to be injective, also known as one-to-one. The definition of an injection leads us to some important properties of injective functions! Subscribe to see more new math videos! Music: OcularNebula - The Lopez
From playlist Functions
Urs Lang (2/3/23): Combinatorial dimension and higher-rank hyperbolicity
Dress characterized metric spaces of combinatorial dimension at most n in terms of a 2(n+1)-point inequality. We investigate a relaxed version of this inequality, which in the case n = 1 reduces to Gromov's quadruple definition of δ-hyperbolicity and which we experimentally call (n,δ)-hype
From playlist Vietoris-Rips Seminar
Elchanan Solomon (10/16/18): An intrinsic persistent homology transform
The Persistent Homology Transform (PHT) and Euler Characteristic Transform (ECT), first proposed by Turner, Mukherjee, and Boyer, were the first TDA invariants shown to be injective on the space of shapes embedded in Euclidean space. A number of recent papers have presented new, elegant ne
From playlist AATRN 2018
Facundo Mémoli: Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius
Title: Vietoris-Rips Persistent Homology, Injective Metric Spaces, and The Filling Radius Abstract: The persistent homology induced by the Vietoris-Rips simplicial filtration is a standard method for capturing topological information from metric spaces. We consider a different, more geome
From playlist Vietoris-Rips Seminar
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
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
Singularities of Teichmueller harmonic map flow - Melanie Rupflin
Workshop on Geometric Functionals: Analysis and Applications Topic: Singularities of Teichmueller harmonic map flow Speaker: Melanie Rupflin Affiliation: University of Oxford Date: March 4, 2019 For more video please visit http://video.ias.edu
From playlist Workshop on Geometric Functionals: Analysis and Applications
A. Song - What is the (essential) minimal volume? 2
I will discuss the notion of minimal volume and some of its variants. The minimal volume of a manifold is defined as the infimum of the volume over all metrics with sectional curvature between -1 and 1. Such an invariant is closely related to "collapsing theory", a far reaching set of resu
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
A. Song - What is the (essential) minimal volume? 2 (version temporaire)
I will discuss the notion of minimal volume and some of its variants. The minimal volume of a manifold is defined as the infimum of the volume over all metrics with sectional curvature between -1 and 1. Such an invariant is closely related to "collapsing theory", a far reaching set of resu
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
Melanie Rupflin: Singularities of Teichmüller harmonic map flow
We discuss singularities of Teichmüller harmonic map flow, which is a geometric flow that changes maps from surfaces into branched minimal immersions, and explain in particular how winding singularities of the map component can lead to singular behaviour of the metric component. Recording
From playlist Geometry
T. Richard - Advanced basics of Riemannian geometry 1 (version temporaire)
We will present some of the tools used by the more advanced lectures. The topics discussed will include : Gromov Hausdorff distance, comparison theorems for sectional and Ricci curvature, the Bochner formula and basics of Ricci flow.
From playlist Ecole d'été 2021 - Curvature Constraints and Spaces of Metrics
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