Simplices and simplicial complexes | Algebraic Topology | NJ Wildberger
58.3 هزار بار بازدید -
12 سال پیش
-
Simplices are higher dimensional analogs
Simplices are higher dimensional analogs of line segments and triangle, such as a tetrahedron. We begin this lecture by discussing convex combinations and convex hulls, and showing a natural hierarchy from point to line segment to triangle to tetrahedron. Each of these also has a standard representation as the convex hull of the unit basis vectors in a vector space.
Simplicial complexes are arrangements of simplices where any two are either disjoint or meet at a face, where by face we mean the convex hull of any subset of the vertices of a simplex.
Each simplex has two possible orientations, corresponding to a particular ordering of the vertices. We show how an orientation of a tetrahedron induces orientations on each of its four triangular faces. The boundary of a simplex is then an alternating combination of its oriented faces of dimension one less. A key theorem is that if we apply the boundary to a boundary, we always get zero. This is the basis of the definition of homology.
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Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at Patreon: njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
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WildTrig: Intro to Rational Trigonometry
Math Foundations
Wild Linear Algebra
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Elementary Mathematics (K-6) Explained
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Here are the Wild Egg Maths Playlists (some available only to Members!)
Algebraic Calculus One
Classical to Quantum
Algebraic Calculus Two
Advice for mathematics exploration
Solving Polynomial Equations | Explor...
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Simplicial complexes are arrangements of simplices where any two are either disjoint or meet at a face, where by face we mean the convex hull of any subset of the vertices of a simplex.
Each simplex has two possible orientations, corresponding to a particular ordering of the vertices. We show how an orientation of a tetrahedron induces orientations on each of its four triangular faces. The boundary of a simplex is then an alternating combination of its oriented faces of dimension one less. A key theorem is that if we apply the boundary to a boundary, we always get zero. This is the basis of the definition of homology.
************************
Screenshot PDFs for my videos are available at the website http://wildegg.com. These give you a concise overview of the contents of the lectures for various Playlists: great for review, study and summary.
My research papers can be found at my Research Gate page, at https://www.researchgate.net/profile/...
My blog is at http://njwildberger.com/, where I will discuss lots of foundational issues, along with other things.
Online courses will be developed at openlearning.com. The first one, already underway is Algebraic Calculus One at https://www.openlearning.com/courses/... Please join us for an exciting new approach to one of mathematics' most important subjects!
If you would like to support these new initiatives for mathematics education and research, please consider becoming a Patron of this Channel at Patreon: njwildberger Your support would be much appreciated.
Here are the Insights into Mathematics Playlists:
MathHistory: A course in the History ...
WildTrig: Intro to Rational Trigonometry
Math Foundations
Wild Linear Algebra
Famous Math Problems
Box Arithmetic
Elementary Mathematics (K-6) Explained
Year9Maths
Ancient Mathematics
Wild West Banking
Sociology and Pure Mathematics
Sociology and Pure Physics
Old Babylonian Mathematics
Probability and Statistics: an introd...
Boole's Logic and Circuit Analysis
Universal Hyperbolic Geometry
Differential Geometry
Algebraic Topology
MathSeminars
Playing Go
Diffusion Symmetry: A bridge between ...
Here are the Wild Egg Maths Playlists (some available only to Members!)
Algebraic Calculus One
Classical to Quantum
Algebraic Calculus Two
Advice for mathematics exploration
Solving Polynomial Equations | Explor...
The Hexagrammum Mysticum (Members only)
Algebraic Calculus and dCB curves | E...
Dynamics on Graphs
Playlist
Triangle Geometry
Explorations with q-series
Six: An elementary course in Pure Mat...
Maxel Inverses and Orthogonal Polynom...
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12 سال پیش
در تاریخ 1391/07/16 منتشر شده
است.
58,307
بـار بازدید شده