A simplicial complex is a set of simplices that satisfies the following conditions:
Note that the empty set is a face of every simplex. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.
A simplicial k-complex is a simplicial complex where the largest dimension of any simplex in equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices.
A pure or homogeneous simplicial k-complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices.
Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What are...simplicial complexes? Or: Triangles everywhere.
Nonsense.
At 8:11, the one object sets correspond to dimension zero, not one. This always confuses me and I messed that up, sorry! That is also why I messed up the notation on the triangle - one should better start counting at zero because of this "of-by-one-error".
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
http://www.dtubbenhauer.com/y...
published: 28 Aug 2021
Simplicial Complexes - Your Brain as Math Part 2 | Infinite Series
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi
What is a Simplicial Complex and how can it help us decode the brain’s neurological structure? This is Part 2 in our Your Brain as Math mini-series.
Check out Part 1 here: https://www.youtube.com/watch?v=M0M3srBoTkY
Check out Part 3 here: https://www.youtube.com/watch?v=akgU8nRNIp0
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
Your Brain as Math - Part 1
https://youtu.be/M0M3srBoTkY
Network Mathematics and Rival Factions
https://www.youtube.com/watch?v=qEKNFOaGQcc
Last episode we saw that your neural network can be modeled as a graph, which -- we’ll show in this episode -- can b...
published: 23 Aug 2017
Simplices and simplicial complexes | Algebraic Topology | NJ Wildberger
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 al...
published: 07 Oct 2012
Week 1 Lecture 1A Simplicial Complexes
This first lecture is a leisurely introduction to abstract simplicial complexes, which form basic building blocks of algebraic topology in general and topological data analysis in particular. https://people.maths.ox.ac.uk/nanda/cat/
published: 16 Jan 2021
Introduction to Simplicial Complexes [Adenike Yeside Adetowubo]
Definition of n-dimensional simplex and simplicial complexes with examples
This video is part of the Fall 2021 Tutorial-a-thon hosted by AATRN and WinCompTop https://sites.google.com/view/aatrn-tutorial-a-thon
published: 17 Nov 2021
Simplicial complexes
published: 25 Dec 2020
Simplicial complexes
This video discusses some ideas and examples related to simplicial complexes.
published: 19 Mar 2020
GTAC 8.1: Simplicial Complexes and Maps
We see a generalization of graphs to simplicial complexes. This will give us a new perspective on graphs.
published: 25 Sep 2020
Applied topology 14: Cech and Vietoris-Rips simplicial complexes
Applied topology 14: Cech and Vietoris-Rips simplicial complexes
Abstract: We describe Cech and Vietoris-Rips simplicial complexes, which are the two most common ways to build an increasing sequence of simplicial complexes on top of point cloud data, in order to measure the shape of that dataset via persistent homology.
This video accompanies the class "Topological Data Analysis" at Colorado State University:
https://www.math.colostate.edu/~adams/teaching/dsci475spr2021/
Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What are...simplicial complexes? Or: Triangles everywhere.
Nonsense.
A...
Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What are...simplicial complexes? Or: Triangles everywhere.
Nonsense.
At 8:11, the one object sets correspond to dimension zero, not one. This always confuses me and I messed that up, sorry! That is also why I messed up the notation on the triangle - one should better start counting at zero because of this "of-by-one-error".
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-algtop-2021.html
Simplicial complexes.
https://en.wikipedia.org/wiki/Simplicial_complex
Abstract simplicial complex.
https://en.wikipedia.org/wiki/Abstract_simplicial_complex
https://community.wolfram.com/groups/-/m/t/2119747
Pictures used.
https://en.wikipedia.org/wiki/Simplicial_complex#/media/File:Simplicial_complex_example.svg
https://quantdare.com/understanding-the-shape-of-data/
https://www.cs.columbia.edu/~suman/avik_slides.pdf
https://www.routledgehandbooks.com/doi/10.1201/9781584888215-c3
Hatcher’s book (I sometimes steal some pictures from there).
https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
Always useful.
https://en.wikipedia.org/wiki/Counterexamples_in_Topology
Mathematica.
https://demonstrations.wolfram.com/SimplicialHomologyOfTheAlphaComplex/
#algebraictopology
#topology
#mathematics
Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What are...simplicial complexes? Or: Triangles everywhere.
Nonsense.
At 8:11, the one object sets correspond to dimension zero, not one. This always confuses me and I messed that up, sorry! That is also why I messed up the notation on the triangle - one should better start counting at zero because of this "of-by-one-error".
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-algtop-2021.html
Simplicial complexes.
https://en.wikipedia.org/wiki/Simplicial_complex
Abstract simplicial complex.
https://en.wikipedia.org/wiki/Abstract_simplicial_complex
https://community.wolfram.com/groups/-/m/t/2119747
Pictures used.
https://en.wikipedia.org/wiki/Simplicial_complex#/media/File:Simplicial_complex_example.svg
https://quantdare.com/understanding-the-shape-of-data/
https://www.cs.columbia.edu/~suman/avik_slides.pdf
https://www.routledgehandbooks.com/doi/10.1201/9781584888215-c3
Hatcher’s book (I sometimes steal some pictures from there).
https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
Always useful.
https://en.wikipedia.org/wiki/Counterexamples_in_Topology
Mathematica.
https://demonstrations.wolfram.com/SimplicialHomologyOfTheAlphaComplex/
#algebraictopology
#topology
#mathematics
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi
What is a Simplicial Complex and how c...
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi
What is a Simplicial Complex and how can it help us decode the brain’s neurological structure? This is Part 2 in our Your Brain as Math mini-series.
Check out Part 1 here: https://www.youtube.com/watch?v=M0M3srBoTkY
Check out Part 3 here: https://www.youtube.com/watch?v=akgU8nRNIp0
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
Your Brain as Math - Part 1
https://youtu.be/M0M3srBoTkY
Network Mathematics and Rival Factions
https://www.youtube.com/watch?v=qEKNFOaGQcc
Last episode we saw that your neural network can be modeled as a graph, which -- we’ll show in this episode -- can be viewed as a higher-dimensional simplicial complex. So… what is a simplicial complex??
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
http://journal.frontiersin.org/article/10.3389/fncom.2017.00048/full
The Blue Brain Project:
http://bluebrain.epfl.ch/
Barcodes: The Persistent Topology of Data
https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf
Network Neuroscience
http://www.nature.com/neuro/journal/v20/n3/full/nn.4502.html
Algebraic Topology
https://www.math.cornell.edu/~hatcher/AT/ATpage.html
Special thanks to Kathryn Hess and Florian Frick!
Special thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!
And thanks to Mauricio Pacheco who is supporting us at the Lemma level!
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi
What is a Simplicial Complex and how can it help us decode the brain’s neurological structure? This is Part 2 in our Your Brain as Math mini-series.
Check out Part 1 here: https://www.youtube.com/watch?v=M0M3srBoTkY
Check out Part 3 here: https://www.youtube.com/watch?v=akgU8nRNIp0
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
Your Brain as Math - Part 1
https://youtu.be/M0M3srBoTkY
Network Mathematics and Rival Factions
https://www.youtube.com/watch?v=qEKNFOaGQcc
Last episode we saw that your neural network can be modeled as a graph, which -- we’ll show in this episode -- can be viewed as a higher-dimensional simplicial complex. So… what is a simplicial complex??
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
http://journal.frontiersin.org/article/10.3389/fncom.2017.00048/full
The Blue Brain Project:
http://bluebrain.epfl.ch/
Barcodes: The Persistent Topology of Data
https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf
Network Neuroscience
http://www.nature.com/neuro/journal/v20/n3/full/nn.4502.html
Algebraic Topology
https://www.math.cornell.edu/~hatcher/AT/ATpage.html
Special thanks to Kathryn Hess and Florian Frick!
Special thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!
And thanks to Mauricio Pacheco who is supporting us at the Lemma level!
Simplices are higher dimensional analogs of line segments and triangle, such as a tetrahedron. We begin this lecture by discussing convex combinations and conve...
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.
************************
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/Norman_Wildberger
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/algebraic-calculus-one/ 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 https://www.patreon.com/njwildberger Your support would be much appreciated.
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м
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.
************************
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/Norman_Wildberger
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/algebraic-calculus-one/ 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 https://www.patreon.com/njwildberger Your support would be much appreciated.
***********************
Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
************************
And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
м
This first lecture is a leisurely introduction to abstract simplicial complexes, which form basic building blocks of algebraic topology in general and topologic...
This first lecture is a leisurely introduction to abstract simplicial complexes, which form basic building blocks of algebraic topology in general and topological data analysis in particular. https://people.maths.ox.ac.uk/nanda/cat/
This first lecture is a leisurely introduction to abstract simplicial complexes, which form basic building blocks of algebraic topology in general and topological data analysis in particular. https://people.maths.ox.ac.uk/nanda/cat/
Definition of n-dimensional simplex and simplicial complexes with examples
This video is part of the Fall 2021 Tutorial-a-thon hosted by AATRN and WinCompTop h...
Definition of n-dimensional simplex and simplicial complexes with examples
This video is part of the Fall 2021 Tutorial-a-thon hosted by AATRN and WinCompTop https://sites.google.com/view/aatrn-tutorial-a-thon
Definition of n-dimensional simplex and simplicial complexes with examples
This video is part of the Fall 2021 Tutorial-a-thon hosted by AATRN and WinCompTop https://sites.google.com/view/aatrn-tutorial-a-thon
Applied topology 14: Cech and Vietoris-Rips simplicial complexes
Abstract: We describe Cech and Vietoris-Rips simplicial complexes, which are the two most comm...
Applied topology 14: Cech and Vietoris-Rips simplicial complexes
Abstract: We describe Cech and Vietoris-Rips simplicial complexes, which are the two most common ways to build an increasing sequence of simplicial complexes on top of point cloud data, in order to measure the shape of that dataset via persistent homology.
This video accompanies the class "Topological Data Analysis" at Colorado State University:
https://www.math.colostate.edu/~adams/teaching/dsci475spr2021/
Applied topology 14: Cech and Vietoris-Rips simplicial complexes
Abstract: We describe Cech and Vietoris-Rips simplicial complexes, which are the two most common ways to build an increasing sequence of simplicial complexes on top of point cloud data, in order to measure the shape of that dataset via persistent homology.
This video accompanies the class "Topological Data Analysis" at Colorado State University:
https://www.math.colostate.edu/~adams/teaching/dsci475spr2021/
Goal.
Explaining basic concepts of algebraic topology in an intuitive way.
This time.
What are...simplicial complexes? Or: Triangles everywhere.
Nonsense.
At 8:11, the one object sets correspond to dimension zero, not one. This always confuses me and I messed that up, sorry! That is also why I messed up the notation on the triangle - one should better start counting at zero because of this "of-by-one-error".
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
These videos are concerned with algebraic topology, and not general topology. (These two are not to be confused.) I assume that you know bits and pieces about general topology, but not too much, I hope.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-algtop-2021.html
Simplicial complexes.
https://en.wikipedia.org/wiki/Simplicial_complex
Abstract simplicial complex.
https://en.wikipedia.org/wiki/Abstract_simplicial_complex
https://community.wolfram.com/groups/-/m/t/2119747
Pictures used.
https://en.wikipedia.org/wiki/Simplicial_complex#/media/File:Simplicial_complex_example.svg
https://quantdare.com/understanding-the-shape-of-data/
https://www.cs.columbia.edu/~suman/avik_slides.pdf
https://www.routledgehandbooks.com/doi/10.1201/9781584888215-c3
Hatcher’s book (I sometimes steal some pictures from there).
https://pi.math.cornell.edu/~hatcher/AT/AT.pdf
Always useful.
https://en.wikipedia.org/wiki/Counterexamples_in_Topology
Mathematica.
https://demonstrations.wolfram.com/SimplicialHomologyOfTheAlphaComplex/
#algebraictopology
#topology
#mathematics
Viewers like you help make PBS (Thank you 😃) . Support your local PBS Member Station here: https://to.pbs.org/donateinfi
What is a Simplicial Complex and how can it help us decode the brain’s neurological structure? This is Part 2 in our Your Brain as Math mini-series.
Check out Part 1 here: https://www.youtube.com/watch?v=M0M3srBoTkY
Check out Part 3 here: https://www.youtube.com/watch?v=akgU8nRNIp0
Tweet at us! @pbsinfinite
Facebook: facebook.com/pbsinfinite series
Email us! pbsinfiniteseries [at] gmail [dot] com
Previous Episode
Your Brain as Math - Part 1
https://youtu.be/M0M3srBoTkY
Network Mathematics and Rival Factions
https://www.youtube.com/watch?v=qEKNFOaGQcc
Last episode we saw that your neural network can be modeled as a graph, which -- we’ll show in this episode -- can be viewed as a higher-dimensional simplicial complex. So… what is a simplicial complex??
Written and Hosted by Kelsey Houston-Edwards
Produced by Rusty Ward
Graphics by Ray Lux
Assistant Editing and Sound Design by Mike Petrow
Made by Kornhaber Brown (www.kornhaberbrown.com)
Resources:
Cliques of Neurons Bound into Cavities Provide a Missing Link between Structure and Function
http://journal.frontiersin.org/article/10.3389/fncom.2017.00048/full
The Blue Brain Project:
http://bluebrain.epfl.ch/
Barcodes: The Persistent Topology of Data
https://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf
Network Neuroscience
http://www.nature.com/neuro/journal/v20/n3/full/nn.4502.html
Algebraic Topology
https://www.math.cornell.edu/~hatcher/AT/ATpage.html
Special thanks to Kathryn Hess and Florian Frick!
Special thanks to Matthew O'Connor and Yana Chernobilsky who are supporting us on Patreon at the Identity level!
And thanks to Mauricio Pacheco who is supporting us at the Lemma level!
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.
************************
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/Norman_Wildberger
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/algebraic-calculus-one/ 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 https://www.patreon.com/njwildberger Your support would be much appreciated.
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Here are all the Insights into Mathematics Playlists:
Elementary Mathematics (K-6) Explained: https://www.youtube.com/playlist?
list=PL8403C2F0C89B1333
Year 9 Maths: https://www.youtube.com/playlist?list=PLIljB45xT85CcGpZpO542YLPeDIf1jqXK
Ancient Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85Aqe2b4FBWUGJdYROT6-o4e
Wild West Banking: https://www.youtube.com/playlist?list=PLIljB45xT85DB7CzoFWvA920NES3g8tJH
Sociology and Pure Mathematics: https://www.youtube.com/playlist?list=PLIljB45xT85A-qCypcmZqRvaS1pGXpTua
Old Babylonian Mathematics (with Daniel Mansfield): https://www.youtube.com/playlist?
list=PLIljB45xT85CdeBmQZ2QiCEnPQn5KQ6ov
Math History: https://www.youtube.com/playlist?list=PL55C7C83781CF4316
Wild Trig: Intro to Rational Trigonometry: https://www.youtube.com/playlist?list=PL3C58498718451C47
MathFoundations: https://www.youtube.com/playlist?list=PL5A714C94D40392AB
Wild Linear Algebra: https://www.youtube.com/playlist?list=PLIljB45xT85BhzJ-oWNug1YtUjfWp1qAp
Famous Math Problems: https://www.youtube.com/playlist?list=PLIljB45xT85Bfc-S4WHvTIM7E-ir3nAOf
Probability and Statistics: An Introduction: https://www.youtube.com/playlist?list=PLIljB45xT85AMigTyprOuf__daeklnLse
Boole's Logic and Circuit Analysis: https://www.youtube.com/playlist?list=PLIljB45xT85CnIGIWb7tH1F_S2PyOC8rb
Universal Hyperbolic Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85CN9oJ4gYkuSQQhAtpIucuI
Differential Geometry: https://www.youtube.com/playlist?list=PLIljB45xT85DWUiFYYGqJVtfnkUFWkKtP
Algebraic Topology: https://www.youtube.com/playlist?list=PL6763F57A61FE6FE8
Math Seminars: https://www.youtube.com/playlist?list=PLBF39AFBBC3FB30AF
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And here are the Wild Egg Maths Playlists:
Triangle Centres: https://www.youtube.com/watch?v=iLBGXDSUohM&list=PLzdiPTrEWyz6VcJQ5xcuqY6g4DWjvpmjM
Six: An elementary course in pure mathematics: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4KD007Ge10dfrDVc4YwlYS
Algebraic Calculus One: https://www.youtube.com/playlist?list=PLzdiPTrEWyz4rKFN541wFKvKPSg5Ea6XB
Algebraic Calculus Two: https://www.youtube.com/playlist?list=PLzdiPTrEWyz5VLVr-0LPPgm4T1mtU_DG-
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This first lecture is a leisurely introduction to abstract simplicial complexes, which form basic building blocks of algebraic topology in general and topological data analysis in particular. https://people.maths.ox.ac.uk/nanda/cat/
Definition of n-dimensional simplex and simplicial complexes with examples
This video is part of the Fall 2021 Tutorial-a-thon hosted by AATRN and WinCompTop https://sites.google.com/view/aatrn-tutorial-a-thon
Applied topology 14: Cech and Vietoris-Rips simplicial complexes
Abstract: We describe Cech and Vietoris-Rips simplicial complexes, which are the two most common ways to build an increasing sequence of simplicial complexes on top of point cloud data, in order to measure the shape of that dataset via persistent homology.
This video accompanies the class "Topological Data Analysis" at Colorado State University:
https://www.math.colostate.edu/~adams/teaching/dsci475spr2021/
A simplicial complex is a set of simplices that satisfies the following conditions:
Note that the empty set is a face of every simplex. See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.
A simplicial k-complex is a simplicial complex where the largest dimension of any simplex in equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices.
A pure or homogeneous simplicial k-complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex of dimension exactly k. Informally, a pure 1-complex "looks" like it's made of a bunch of lines, a 2-complex "looks" like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices.
The new measure, called simplicial homophily, separates the homophily seen in one-on-one interactions from those in larger group interactions and is based on the mathematical concept of a simplicial complex.
hypergraphs and simplicial complexes ... Simplicial complexes are more structured, using triangles (and higher-dimensional surfaces analogous to triangles) to represent those connections. Simplicial ...
Taylor and Kilic propose expanding these graphs into simplicial complexes—or hypergraphs—that can map connections between vast networks of neurons ... A simplicial complex model, or equivalently, a ...