- published: 11 Sep 2018
- views: 35584
Create your page here
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory.
Differences between low-dimensional and high-dimensional topology
Manifolds differ radically in behavior in high and low dimension.
High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.
Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Geometric_topology
Geometric topology (object)
In mathematics, the geometric topology is a topology one can put on the set H of hyperbolic 3-manifolds of finite volume.
Use
Convergence in this topology is a crucial ingredient of hyperbolic Dehn surgery, a fundamental tool in the theory of hyperbolic 3-manifolds.
Definition
The following is a definition due to Troels Jorgensen:
converging to 0, and 
-bi-Lipschitz diffeomorphisms 
Alternate definition
There is an alternate definition due to Mikhail Gromov. Gromov's topology utilizes the Gromov-Hausdorff metric and is defined on pointed hyperbolic 3-manifolds. One essentially considers better and better bi-Lipschitz homeomorphisms on larger and larger balls. This results in the same notion of convergence as above as the thick part is always connected; thus, a large ball will eventually encompass all of the thick part.
On framed manifolds
As a further refinement, Gromov's metric can also be defined on framed hyperbolic 3-manifolds. This gives nothing new but this space can be explicitly identified with torsion-free Kleinian groups with the Chabauty topology.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Geometric_topology_(object)
Geometric topology (disambiguation)
In mathematics, the phrase geometric topology may refer to:
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Geometric_topology_(disambiguation)
Podcasts:
-
Relating Topology and Geometry - 2 Minute Math with Jacob Lurie
Many believe the mathematical fields of Algebraic Topology and Algebraic Geometry are totally unrelated, but Harvard Professor Jacob Lurie delights in finding the connections. Hear about his work at the leading edge of mathematics. Professor Jacob Lurie visited the Fields Institute for the Séminaire de mathématiques supérieures (SMS) 2018 from June 11 to June 15, 2018. The Fields Institute is a centre for mathematical research activity - a place where mathematicians from Canada and abroad, from academia, business, industry and financial institutions, can come together to carry out research and formulate problems of mutual interest. Our mission is to provide a supportive and stimulating environment for mathematics innovation and education. For more on the amazing mathematical research we...
published: 11 Sep 2018 -
Algebra, Geometry, and Topology: What's The Difference?
This Math-Dance video aims to describe how the fields of mathematics are different. Focusing on Algebra, Geometry, and Topology, we use dance to describe how each one of these fields would study a circle. The geometer dances with a rigid hula hoop, the topologist dances with a loop of fabric, and the algebraist dances with a circle of lasers. Written and Created by: Nancy Scherich Filmed and edited by : Alex Nye Produced by: Steven Deeble And Nancy Scherich First Assistant Camera: Chitoh Yung Choreographed by: Nancy Scherich, in collaboration with Katelyn Carano, Erika Walther, Steve Trettel Original Music by Whetzel "This Is What Topology Sounds Like" jameswhetzel.com Cast: Eric Boesser, Nic Brody, Christian Bueno, Katelyn Carano, Michelle Chu, Olivia Davi, Ken Millett, Erin Morg...
published: 15 Feb 2019 -
Who cares about topology? (Inscribed rectangle problem)
An unsolved conjecture, and a clever topological solution to a similar question. Help fund future projects: https://www.patreon.com/3blue1brown An equally valuable form of support is to simply share some of the videos. Special thanks to these supporters: http://3b1b.co/topology-thanks Home page: https://www.3blue1brown.com/ This video is based on a proof from H. Vaughan, 1977. To learn more, take a look at this survey: https://pure.mpg.de/rest/items/item_3120610/component/file_3120611/content Thanks to these viewers for their contributions to translations Hebrew: Omer Tuchfeld ------------------ 3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the b...
published: 04 Nov 2016 -
What is...geometric topology?
Goal. Explaining basic concepts of geometric topology in an intuitive way. This time. What is...geometric topology? Or: It’s a disc! Disclaimer. Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references. Disclaimer. Geometric topology is usually the study of manifolds. This can mean manifold things ;-) So to be precise, this video series is mostly about knots, surfaces, three and four manifolds. Slides. http://www.dtubbenhauer.com/youtube.html Website with exercises. http://www.dtubbenhauer.com/lecture-geotop-2022.html Thumbnail. https://upload.wikimedia.org/wikipedia/commons/4/47/Torus.svg Geometric topology. https://en.wikipedia.org/wiki/Geometric_topology https://en.wikipedia.org/wiki/List_of_geometric_topology_topics ht...
published: 06 Aug 2022 -
The Biggest Ideas in the Universe | 13. Geometry and Topology
The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody. This is Idea #13, "Geometry and Topology." Yes that's two ideas, and furthermore they're from math more than from science, but we'll put them to good use. In particular we look at Riemannian (non-Euclidean) geometry, and a kind of topological invariants called "homotopy groups." My web page: http://www.preposterousuniverse.com/ My YouTube channel: https://www.youtube.com/c/seancarroll Mindscape podcast: http://www.preposterousuniverse.com/podcast The Biggest Ideas playlist: https://www.youtube.com/playlist?list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX Blo...
published: 16 Jun 2020 -
Cellular decomposition | Geometric topology
Suborno Isaac is the World's Youngest AIME Qualifier in US Math Olympiad. Link, https://www.brooklyn.edu/bc-news/president-michelle-j-anderson-meets-11-year-old-brooklyn-college-student-suborno-isaac-bari/ Email: suborno.bari@stonybrook.edu
published: 05 Mar 2024 -
Infinite Groups in Geometric Topology, Part 1
This is the first in a series of three one-hour talks delivered by Principal Speaker Kevin Whyte of the University of Illinois at Chicago at the 31st Annual Workshop in Geometric Topology held at the University of Wisconsin-Milwaukee on the dates of June 12-14, 2014.
published: 16 Jul 2014 -
Geometry Topology
published: 23 Sep 2020 -
What is a manifold?
A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability. If you want to learn more, check out one of these (or any other basic differential geometry or topology book): - M. Spivak: "A Comprehensive Introduction to Differential Geometry" - M. Nakahara: "Geometry, Topology and Physics" - J. W. Milnor: "Topology from the differentiable viewpoint"
published: 11 Oct 2015 -
Clifford Torus || Geometric Topology
Clifford Torus || Geometric Topology
published: 02 Jan 2024
developed with YouTube
2:19
Relating Topology and Geometry - 2 Minute Math with Jacob Lurie
- Order: Reorder
- Duration: 2:19
- Uploaded Date: 11 Sep 2018
- views: 35584
Many believe the mathematical fields of Algebraic Topology and Algebraic Geometry are totally unrelated, but Harvard Professor Jacob Lurie delights in finding t...
Many believe the mathematical fields of Algebraic Topology and Algebraic Geometry are totally unrelated, but Harvard Professor Jacob Lurie delights in finding the connections. Hear about his work at the leading edge of mathematics.
Professor Jacob Lurie visited the Fields Institute for the Séminaire de mathématiques supérieures (SMS) 2018 from June 11 to June 15, 2018.
The Fields Institute is a centre for mathematical research activity - a place where mathematicians from Canada and abroad, from academia, business, industry and financial institutions, can come together to carry out research and formulate problems of mutual interest. Our mission is to provide a supportive and stimulating environment for mathematics innovation and education.
For more on the amazing mathematical research we support,
Subscribe to our channel - https://www.youtube.com/c/FieldsInstitute
Follow us on Twitter - https://twitter.com/FieldsInstitute
Visit our website - http://www.fields.utoronto.ca/about
Music:
The Walk by Split Phase
Licensed under Creative Commons: By Attribution 3.0 License
http://creativecommons.org/licenses/by/3.0/
https://wn.com/Relating_Topology_And_Geometry_2_Minute_Math_With_Jacob_Lurie
Many believe the mathematical fields of Algebraic Topology and Algebraic Geometry are totally unrelated, but Harvard Professor Jacob Lurie delights in finding the connections. Hear about his work at the leading edge of mathematics.
Professor Jacob Lurie visited the Fields Institute for the Séminaire de mathématiques supérieures (SMS) 2018 from June 11 to June 15, 2018.
The Fields Institute is a centre for mathematical research activity - a place where mathematicians from Canada and abroad, from academia, business, industry and financial institutions, can come together to carry out research and formulate problems of mutual interest. Our mission is to provide a supportive and stimulating environment for mathematics innovation and education.
For more on the amazing mathematical research we support,
Subscribe to our channel - https://www.youtube.com/c/FieldsInstitute
Follow us on Twitter - https://twitter.com/FieldsInstitute
Visit our website - http://www.fields.utoronto.ca/about
Music:
The Walk by Split Phase
Licensed under Creative Commons: By Attribution 3.0 License
http://creativecommons.org/licenses/by/3.0/
3:01
Algebra, Geometry, and Topology: What's The Difference?
- Order: Reorder
- Duration: 3:01
- Uploaded Date: 15 Feb 2019
- views: 46484
This Math-Dance video aims to describe how the fields of mathematics are different. Focusing on Algebra, Geometry, and Topology, we use dance to describe how e...
This Math-Dance video aims to describe how the fields of mathematics are different. Focusing on Algebra, Geometry, and Topology, we use dance to describe how each one of these fields would study a circle. The geometer dances with a rigid hula hoop, the topologist dances with a loop of fabric, and the algebraist dances with a circle of lasers.
Written and Created by: Nancy Scherich
Filmed and edited by : Alex Nye
Produced by: Steven Deeble And Nancy Scherich
First Assistant Camera: Chitoh Yung
Choreographed by: Nancy Scherich, in collaboration with Katelyn Carano, Erika Walther, Steve Trettel
Original Music by Whetzel
"This Is What Topology Sounds Like"
jameswhetzel.com
Cast: Eric Boesser, Nic Brody, Christian Bueno, Katelyn Carano, Michelle Chu, Olivia Davi, Ken Millett, Erin Morgan, Viki Papadakis, Abe Pressman, Nancy Scherich, Steve Trettel, Erika Walther
Special Thanks to
UCSB Mathematics Dept. and Darren Long
UCSB Theater and Dance Dept.
This material is based upon work supported by the National Science Foundation under Grant No. 1045292
http://nas.edu/ElevatingMath
https://wn.com/Algebra,_Geometry,_And_Topology_What's_The_Difference
This Math-Dance video aims to describe how the fields of mathematics are different. Focusing on Algebra, Geometry, and Topology, we use dance to describe how each one of these fields would study a circle. The geometer dances with a rigid hula hoop, the topologist dances with a loop of fabric, and the algebraist dances with a circle of lasers.
Written and Created by: Nancy Scherich
Filmed and edited by : Alex Nye
Produced by: Steven Deeble And Nancy Scherich
First Assistant Camera: Chitoh Yung
Choreographed by: Nancy Scherich, in collaboration with Katelyn Carano, Erika Walther, Steve Trettel
Original Music by Whetzel
"This Is What Topology Sounds Like"
jameswhetzel.com
Cast: Eric Boesser, Nic Brody, Christian Bueno, Katelyn Carano, Michelle Chu, Olivia Davi, Ken Millett, Erin Morgan, Viki Papadakis, Abe Pressman, Nancy Scherich, Steve Trettel, Erika Walther
Special Thanks to
UCSB Mathematics Dept. and Darren Long
UCSB Theater and Dance Dept.
This material is based upon work supported by the National Science Foundation under Grant No. 1045292
http://nas.edu/ElevatingMath
- published: 15 Feb 2019
- views: 46484
18:16
Who cares about topology? (Inscribed rectangle problem)
- Order: Reorder
- Duration: 18:16
- Uploaded Date: 04 Nov 2016
- views: 3216079
An unsolved conjecture, and a clever topological solution to a similar question.
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valua...
An unsolved conjecture, and a clever topological solution to a similar question.
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/topology-thanks
Home page: https://www.3blue1brown.com/
This video is based on a proof from H. Vaughan, 1977. To learn more, take a look at this survey:
https://pure.mpg.de/rest/items/item_3120610/component/file_3120611/content
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown
https://wn.com/Who_Cares_About_Topology_(Inscribed_Rectangle_Problem)
An unsolved conjecture, and a clever topological solution to a similar question.
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/topology-thanks
Home page: https://www.3blue1brown.com/
This video is based on a proof from H. Vaughan, 1977. To learn more, take a look at this survey:
https://pure.mpg.de/rest/items/item_3120610/component/file_3120611/content
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown
- published: 04 Nov 2016
- views: 3216079
16:38
What is...geometric topology?
- Order: Reorder
- Duration: 16:38
- Uploaded Date: 06 Aug 2022
- views: 4625
Goal.
Explaining basic concepts of geometric topology in an intuitive way.
This time.
What is...geometric topology? Or: It’s a disc!
Disclaimer.
Nobody is ...
Goal.
Explaining basic concepts of geometric topology in an intuitive way.
This time.
What is...geometric topology? Or: It’s a disc!
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
Geometric topology is usually the study of manifolds. This can mean manifold things ;-) So to be precise, this video series is mostly about knots, surfaces, three and four manifolds.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-geotop-2022.html
Thumbnail.
https://upload.wikimedia.org/wikipedia/commons/4/47/Torus.svg
Geometric topology.
https://en.wikipedia.org/wiki/Geometric_topology
https://en.wikipedia.org/wiki/List_of_geometric_topology_topics
https://en.wikipedia.org/wiki/Manifold
https://en.wikipedia.org/wiki/Knot_(mathematics)
https://en.wikipedia.org/wiki/Surface_(topology)
https://en.wikipedia.org/wiki/3-manifold
Concepts in geometric topology.
https://en.wikipedia.org/wiki/Knot_invariant
https://en.wikipedia.org/wiki/Braid_group
https://en.wikipedia.org/wiki/Euler_characteristic
https://en.wikipedia.org/wiki/Mapping_class_group
https://en.wikipedia.org/wiki/Exotic_sphere
https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
https://en.wikipedia.org/wiki/Thurston%27s_geometrization_conjecture
https://en.wikipedia.org/wiki/Cobordism
https://en.wikipedia.org/wiki/Unknotting_problem
Applications of geometric topology.
https://physics.stackexchange.com/questions/27051/applications-of-geometric-topology-to-theoretical-physics
https://mathoverflow.net/questions/48222/applications-of-knot-theory
https://math.mit.edu/research/highschool/primes/circle/documents/2019/Lim_Martin_2019.pdf
Pictures used.
Picture from https://www.youtube.com/watch?v=cPg62OPdF8s
https://en.wikipedia.org/wiki/Manifold#/media/File:Sphere_with_chart.svg
https://w7.pngwing.com/pngs/80/162/png-transparent-torus-mathematics-topology-klein-bottle-geometry-mathematics-shape-space-topology.png
https://cdna.lystit.com/1200/630/tr/photos/lanecrawford/da4e36fc/stella-mccartney-Blue-Heart-Knee-Patch-Cropped-Jeans.jpeg
https://www.math.unl.edu/~mbrittenham2/ldt/poincare/poinc_conj.jpg
Picture from https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
Pictures from https://www.youtube.com/watch?v=valxSdpgNXM
https://www.open.edu/openlearn/pluginfile.php/90214/mod_oucontent/oucontent/751/604e7db1/e1b914fb/m338_1_041i.jpg
Some books I am using (I sometimes steal some pictures from there).
https://www.math.cuhk.edu.hk/course_builder/1920/math4900e/Adams--The%20Knot%20Book.pdf
https://arxiv.org/abs/1610.02592?context=math
https://www.degruyter.com/document/doi/10.1515/9781400865321/html?lang=en
https://bookstore.ams.org/fourman
https://www.degruyter.com/document/doi/10.1515/9783110250367/html?lang=en
https://bookstore.ams.org/surv-55
https://bookstore.ams.org/gsm-20
KnotAtlas.
http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table
SnapPy.
https://snappy.math.uic.edu/
Mathematica.
http://katlas.org/wiki/The_Mathematica_Package_KnotTheory%60
https://demonstrations.wolfram.com/ChartForATorus/
SageMath.
https://doc.sagemath.org/html/en/reference/knots/sage/knots/knot.html
#geometrictopology
#topology
#mathematics
https://wn.com/What_Is...Geometric_Topology
Goal.
Explaining basic concepts of geometric topology in an intuitive way.
This time.
What is...geometric topology? Or: It’s a disc!
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
Geometric topology is usually the study of manifolds. This can mean manifold things ;-) So to be precise, this video series is mostly about knots, surfaces, three and four manifolds.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-geotop-2022.html
Thumbnail.
https://upload.wikimedia.org/wikipedia/commons/4/47/Torus.svg
Geometric topology.
https://en.wikipedia.org/wiki/Geometric_topology
https://en.wikipedia.org/wiki/List_of_geometric_topology_topics
https://en.wikipedia.org/wiki/Manifold
https://en.wikipedia.org/wiki/Knot_(mathematics)
https://en.wikipedia.org/wiki/Surface_(topology)
https://en.wikipedia.org/wiki/3-manifold
Concepts in geometric topology.
https://en.wikipedia.org/wiki/Knot_invariant
https://en.wikipedia.org/wiki/Braid_group
https://en.wikipedia.org/wiki/Euler_characteristic
https://en.wikipedia.org/wiki/Mapping_class_group
https://en.wikipedia.org/wiki/Exotic_sphere
https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
https://en.wikipedia.org/wiki/Thurston%27s_geometrization_conjecture
https://en.wikipedia.org/wiki/Cobordism
https://en.wikipedia.org/wiki/Unknotting_problem
Applications of geometric topology.
https://physics.stackexchange.com/questions/27051/applications-of-geometric-topology-to-theoretical-physics
https://mathoverflow.net/questions/48222/applications-of-knot-theory
https://math.mit.edu/research/highschool/primes/circle/documents/2019/Lim_Martin_2019.pdf
Pictures used.
Picture from https://www.youtube.com/watch?v=cPg62OPdF8s
https://en.wikipedia.org/wiki/Manifold#/media/File:Sphere_with_chart.svg
https://w7.pngwing.com/pngs/80/162/png-transparent-torus-mathematics-topology-klein-bottle-geometry-mathematics-shape-space-topology.png
https://cdna.lystit.com/1200/630/tr/photos/lanecrawford/da4e36fc/stella-mccartney-Blue-Heart-Knee-Patch-Cropped-Jeans.jpeg
https://www.math.unl.edu/~mbrittenham2/ldt/poincare/poinc_conj.jpg
Picture from https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
Pictures from https://www.youtube.com/watch?v=valxSdpgNXM
https://www.open.edu/openlearn/pluginfile.php/90214/mod_oucontent/oucontent/751/604e7db1/e1b914fb/m338_1_041i.jpg
Some books I am using (I sometimes steal some pictures from there).
https://www.math.cuhk.edu.hk/course_builder/1920/math4900e/Adams--The%20Knot%20Book.pdf
https://arxiv.org/abs/1610.02592?context=math
https://www.degruyter.com/document/doi/10.1515/9781400865321/html?lang=en
https://bookstore.ams.org/fourman
https://www.degruyter.com/document/doi/10.1515/9783110250367/html?lang=en
https://bookstore.ams.org/surv-55
https://bookstore.ams.org/gsm-20
KnotAtlas.
http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table
SnapPy.
https://snappy.math.uic.edu/
Mathematica.
http://katlas.org/wiki/The_Mathematica_Package_KnotTheory%60
https://demonstrations.wolfram.com/ChartForATorus/
SageMath.
https://doc.sagemath.org/html/en/reference/knots/sage/knots/knot.html
#geometrictopology
#topology
#mathematics
- published: 06 Aug 2022
- views: 4625
1:26:09
The Biggest Ideas in the Universe | 13. Geometry and Topology
- Order: Reorder
- Duration: 1:26:09
- Uploaded Date: 16 Jun 2020
- views: 156682
The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world...
The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody.
This is Idea #13, "Geometry and Topology." Yes that's two ideas, and furthermore they're from math more than from science, but we'll put them to good use. In particular we look at Riemannian (non-Euclidean) geometry, and a kind of topological invariants called "homotopy groups."
My web page: http://www.preposterousuniverse.com/
My YouTube channel: https://www.youtube.com/c/seancarroll
Mindscape podcast: http://www.preposterousuniverse.com/podcast
The Biggest Ideas playlist: https://www.youtube.com/playlist?list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX
Blog posts for the series: http://www.preposterousuniverse.com/blog/category/biggest-ideas-in-the-universe/
Background image by RyoThorn at DeviantArt: https://www.deviantart.com/ryothorn/art/Steamworks-to-my-own-oblivion-147929840
#science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology
https://wn.com/The_Biggest_Ideas_In_The_Universe_|_13._Geometry_And_Topology
The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody.
This is Idea #13, "Geometry and Topology." Yes that's two ideas, and furthermore they're from math more than from science, but we'll put them to good use. In particular we look at Riemannian (non-Euclidean) geometry, and a kind of topological invariants called "homotopy groups."
My web page: http://www.preposterousuniverse.com/
My YouTube channel: https://www.youtube.com/c/seancarroll
Mindscape podcast: http://www.preposterousuniverse.com/podcast
The Biggest Ideas playlist: https://www.youtube.com/playlist?list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX
Blog posts for the series: http://www.preposterousuniverse.com/blog/category/biggest-ideas-in-the-universe/
Background image by RyoThorn at DeviantArt: https://www.deviantart.com/ryothorn/art/Steamworks-to-my-own-oblivion-147929840
#science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology
- published: 16 Jun 2020
- views: 156682
2:47
Cellular decomposition | Geometric topology
- Order: Reorder
- Duration: 2:47
- Uploaded Date: 05 Mar 2024
- views: 27362
Suborno Isaac is the World's Youngest AIME Qualifier in US Math Olympiad.
Link, https://www.brooklyn.edu/bc-news/president-michelle-j-anderson-meets-11-year-old...
Suborno Isaac is the World's Youngest AIME Qualifier in US Math Olympiad.
Link, https://www.brooklyn.edu/bc-news/president-michelle-j-anderson-meets-11-year-old-brooklyn-college-student-suborno-isaac-bari/
Email: suborno.bari@stonybrook.edu
https://wn.com/Cellular_Decomposition_|_Geometric_Topology
Suborno Isaac is the World's Youngest AIME Qualifier in US Math Olympiad.
Link, https://www.brooklyn.edu/bc-news/president-michelle-j-anderson-meets-11-year-old-brooklyn-college-student-suborno-isaac-bari/
Email: suborno.bari@stonybrook.edu
- published: 05 Mar 2024
- views: 27362
58:08
Infinite Groups in Geometric Topology, Part 1
- Order: Reorder
- Duration: 58:08
- Uploaded Date: 16 Jul 2014
- views: 2220
This is the first in a series of three one-hour talks delivered by Principal Speaker Kevin Whyte of the University of Illinois at Chicago at the 31st Annual Wor...
This is the first in a series of three one-hour talks delivered by Principal Speaker Kevin Whyte of the University of Illinois at Chicago at the 31st Annual Workshop in Geometric Topology held at the University of Wisconsin-Milwaukee on the dates of June 12-14, 2014.
https://wn.com/Infinite_Groups_In_Geometric_Topology,_Part_1
This is the first in a series of three one-hour talks delivered by Principal Speaker Kevin Whyte of the University of Illinois at Chicago at the 31st Annual Workshop in Geometric Topology held at the University of Wisconsin-Milwaukee on the dates of June 12-14, 2014.
- published: 16 Jul 2014
- views: 2220
21:35
Geometry Topology
- Order: Reorder
- Duration: 21:35
- Uploaded Date: 23 Sep 2020
- views: 78
- published: 23 Sep 2020
- views: 78
3:51
What is a manifold?
- Order: Reorder
- Duration: 3:51
- Uploaded Date: 11 Oct 2015
- views: 211842
A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability.
If you want to learn...
A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability.
If you want to learn more, check out one of these (or any other basic differential geometry or topology book):
- M. Spivak: "A Comprehensive Introduction to Differential Geometry"
- M. Nakahara: "Geometry, Topology and Physics"
- J. W. Milnor: "Topology from the differentiable viewpoint"
https://wn.com/What_Is_A_Manifold
A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability.
If you want to learn more, check out one of these (or any other basic differential geometry or topology book):
- M. Spivak: "A Comprehensive Introduction to Differential Geometry"
- M. Nakahara: "Geometry, Topology and Physics"
- J. W. Milnor: "Topology from the differentiable viewpoint"
- published: 11 Oct 2015
- views: 211842
0:11
Clifford Torus || Geometric Topology
- Order: Reorder
- Duration: 0:11
- Uploaded Date: 02 Jan 2024
- views: 1097
Clifford Torus || Geometric Topology
Clifford Torus || Geometric Topology
https://wn.com/Clifford_Torus_||_Geometric_Topology
Clifford Torus || Geometric Topology
- published: 02 Jan 2024
- views: 1097
2:19
Relating Topology and Geometry - 2 Minute Math with Jacob Lurie
Many believe the mathematical fields of Algebraic Topology and Algebraic Geometry are tota...
published: 11 Sep 2018
Relating Topology and Geometry - 2 Minute Math with Jacob Lurie
Relating Topology and Geometry - 2 Minute Math with Jacob Lurie
- Report rights infringement
- published: 11 Sep 2018
- views: 35584
Many believe the mathematical fields of Algebraic Topology and Algebraic Geometry are totally unrelated, but Harvard Professor Jacob Lurie delights in finding the connections. Hear about his work at the leading edge of mathematics.
Professor Jacob Lurie visited the Fields Institute for the Séminaire de mathématiques supérieures (SMS) 2018 from June 11 to June 15, 2018.
The Fields Institute is a centre for mathematical research activity - a place where mathematicians from Canada and abroad, from academia, business, industry and financial institutions, can come together to carry out research and formulate problems of mutual interest. Our mission is to provide a supportive and stimulating environment for mathematics innovation and education.
For more on the amazing mathematical research we support,
Subscribe to our channel - https://www.youtube.com/c/FieldsInstitute
Follow us on Twitter - https://twitter.com/FieldsInstitute
Visit our website - http://www.fields.utoronto.ca/about
Music:
The Walk by Split Phase
Licensed under Creative Commons: By Attribution 3.0 License
http://creativecommons.org/licenses/by/3.0/
3:01
Algebra, Geometry, and Topology: What's The Difference?
This Math-Dance video aims to describe how the fields of mathematics are different. Focusi...
published: 15 Feb 2019
Algebra, Geometry, and Topology: What's The Difference?
Algebra, Geometry, and Topology: What's The Difference?
- Report rights infringement
- published: 15 Feb 2019
- views: 46484
This Math-Dance video aims to describe how the fields of mathematics are different. Focusing on Algebra, Geometry, and Topology, we use dance to describe how each one of these fields would study a circle. The geometer dances with a rigid hula hoop, the topologist dances with a loop of fabric, and the algebraist dances with a circle of lasers.
Written and Created by: Nancy Scherich
Filmed and edited by : Alex Nye
Produced by: Steven Deeble And Nancy Scherich
First Assistant Camera: Chitoh Yung
Choreographed by: Nancy Scherich, in collaboration with Katelyn Carano, Erika Walther, Steve Trettel
Original Music by Whetzel
"This Is What Topology Sounds Like"
jameswhetzel.com
Cast: Eric Boesser, Nic Brody, Christian Bueno, Katelyn Carano, Michelle Chu, Olivia Davi, Ken Millett, Erin Morgan, Viki Papadakis, Abe Pressman, Nancy Scherich, Steve Trettel, Erika Walther
Special Thanks to
UCSB Mathematics Dept. and Darren Long
UCSB Theater and Dance Dept.
This material is based upon work supported by the National Science Foundation under Grant No. 1045292
http://nas.edu/ElevatingMath
18:16
Who cares about topology? (Inscribed rectangle problem)
An unsolved conjecture, and a clever topological solution to a similar question.
Help fund...
published: 04 Nov 2016
Who cares about topology? (Inscribed rectangle problem)
Who cares about topology? (Inscribed rectangle problem)
- Report rights infringement
- published: 04 Nov 2016
- views: 3216079
An unsolved conjecture, and a clever topological solution to a similar question.
Help fund future projects: https://www.patreon.com/3blue1brown
An equally valuable form of support is to simply share some of the videos.
Special thanks to these supporters: http://3b1b.co/topology-thanks
Home page: https://www.3blue1brown.com/
This video is based on a proof from H. Vaughan, 1977. To learn more, take a look at this survey:
https://pure.mpg.de/rest/items/item_3120610/component/file_3120611/content
Thanks to these viewers for their contributions to translations
Hebrew: Omer Tuchfeld
------------------
3blue1brown is a channel about animating math, in all senses of the word animate. And you know the drill with YouTube, if you want to stay posted about new videos, subscribe, and click the bell to receive notifications (if you're into that).
If you are new to this channel and want to see more, a good place to start is this playlist: https://www.youtube.com/playlist?list=PLZHQObOWTQDPHP40bzkb0TKLRPwQGAoC-
Various social media stuffs:
Twitter: https://twitter.com/3Blue1Brown
Facebook: https://www.facebook.com/3blue1brown/
Reddit: https://www.reddit.com/r/3Blue1Brown
16:38
What is...geometric topology?
Goal.
Explaining basic concepts of geometric topology in an intuitive way.
This time.
W...
published: 06 Aug 2022
What is...geometric topology?
What is...geometric topology?
- Report rights infringement
- published: 06 Aug 2022
- views: 4625
Goal.
Explaining basic concepts of geometric topology in an intuitive way.
This time.
What is...geometric topology? Or: It’s a disc!
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
Geometric topology is usually the study of manifolds. This can mean manifold things ;-) So to be precise, this video series is mostly about knots, surfaces, three and four manifolds.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-geotop-2022.html
Thumbnail.
https://upload.wikimedia.org/wikipedia/commons/4/47/Torus.svg
Geometric topology.
https://en.wikipedia.org/wiki/Geometric_topology
https://en.wikipedia.org/wiki/List_of_geometric_topology_topics
https://en.wikipedia.org/wiki/Manifold
https://en.wikipedia.org/wiki/Knot_(mathematics)
https://en.wikipedia.org/wiki/Surface_(topology)
https://en.wikipedia.org/wiki/3-manifold
Concepts in geometric topology.
https://en.wikipedia.org/wiki/Knot_invariant
https://en.wikipedia.org/wiki/Braid_group
https://en.wikipedia.org/wiki/Euler_characteristic
https://en.wikipedia.org/wiki/Mapping_class_group
https://en.wikipedia.org/wiki/Exotic_sphere
https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
https://en.wikipedia.org/wiki/Thurston%27s_geometrization_conjecture
https://en.wikipedia.org/wiki/Cobordism
https://en.wikipedia.org/wiki/Unknotting_problem
Applications of geometric topology.
https://physics.stackexchange.com/questions/27051/applications-of-geometric-topology-to-theoretical-physics
https://mathoverflow.net/questions/48222/applications-of-knot-theory
https://math.mit.edu/research/highschool/primes/circle/documents/2019/Lim_Martin_2019.pdf
Pictures used.
Picture from https://www.youtube.com/watch?v=cPg62OPdF8s
https://en.wikipedia.org/wiki/Manifold#/media/File:Sphere_with_chart.svg
https://w7.pngwing.com/pngs/80/162/png-transparent-torus-mathematics-topology-klein-bottle-geometry-mathematics-shape-space-topology.png
https://cdna.lystit.com/1200/630/tr/photos/lanecrawford/da4e36fc/stella-mccartney-Blue-Heart-Knee-Patch-Cropped-Jeans.jpeg
https://www.math.unl.edu/~mbrittenham2/ldt/poincare/poinc_conj.jpg
Picture from https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture
Pictures from https://www.youtube.com/watch?v=valxSdpgNXM
https://www.open.edu/openlearn/pluginfile.php/90214/mod_oucontent/oucontent/751/604e7db1/e1b914fb/m338_1_041i.jpg
Some books I am using (I sometimes steal some pictures from there).
https://www.math.cuhk.edu.hk/course_builder/1920/math4900e/Adams--The%20Knot%20Book.pdf
https://arxiv.org/abs/1610.02592?context=math
https://www.degruyter.com/document/doi/10.1515/9781400865321/html?lang=en
https://bookstore.ams.org/fourman
https://www.degruyter.com/document/doi/10.1515/9783110250367/html?lang=en
https://bookstore.ams.org/surv-55
https://bookstore.ams.org/gsm-20
KnotAtlas.
http://katlas.math.toronto.edu/wiki/The_Rolfsen_Knot_Table
SnapPy.
https://snappy.math.uic.edu/
Mathematica.
http://katlas.org/wiki/The_Mathematica_Package_KnotTheory%60
https://demonstrations.wolfram.com/ChartForATorus/
SageMath.
https://doc.sagemath.org/html/en/reference/knots/sage/knots/knot.html
#geometrictopology
#topology
#mathematics
1:26:09
The Biggest Ideas in the Universe | 13. Geometry and Topology
The Biggest Ideas in the Universe is a series of videos where I talk informally about some...
published: 16 Jun 2020
The Biggest Ideas in the Universe | 13. Geometry and Topology
The Biggest Ideas in the Universe | 13. Geometry and Topology
- Report rights infringement
- published: 16 Jun 2020
- views: 156682
The Biggest Ideas in the Universe is a series of videos where I talk informally about some of the fundamental concepts that help us understand our natural world. Exceedingly casual, not overly polished, and meant for absolutely everybody.
This is Idea #13, "Geometry and Topology." Yes that's two ideas, and furthermore they're from math more than from science, but we'll put them to good use. In particular we look at Riemannian (non-Euclidean) geometry, and a kind of topological invariants called "homotopy groups."
My web page: http://www.preposterousuniverse.com/
My YouTube channel: https://www.youtube.com/c/seancarroll
Mindscape podcast: http://www.preposterousuniverse.com/podcast
The Biggest Ideas playlist: https://www.youtube.com/playlist?list=PLrxfgDEc2NxZJcWcrxH3jyjUUrJlnoyzX
Blog posts for the series: http://www.preposterousuniverse.com/blog/category/biggest-ideas-in-the-universe/
Background image by RyoThorn at DeviantArt: https://www.deviantart.com/ryothorn/art/Steamworks-to-my-own-oblivion-147929840
#science #physics #ideas #universe #learning #cosmology #philosophy #math #geometry #topology
2:47
Cellular decomposition | Geometric topology
Suborno Isaac is the World's Youngest AIME Qualifier in US Math Olympiad.
Link, https://ww...
published: 05 Mar 2024
Cellular decomposition | Geometric topology
Cellular decomposition | Geometric topology
- Report rights infringement
- published: 05 Mar 2024
- views: 27362
Suborno Isaac is the World's Youngest AIME Qualifier in US Math Olympiad.
Link, https://www.brooklyn.edu/bc-news/president-michelle-j-anderson-meets-11-year-old-brooklyn-college-student-suborno-isaac-bari/
Email: suborno.bari@stonybrook.edu
58:08
Infinite Groups in Geometric Topology, Part 1
This is the first in a series of three one-hour talks delivered by Principal Speaker Kevin...
published: 16 Jul 2014
Infinite Groups in Geometric Topology, Part 1
Infinite Groups in Geometric Topology, Part 1
- Report rights infringement
- published: 16 Jul 2014
- views: 2220
This is the first in a series of three one-hour talks delivered by Principal Speaker Kevin Whyte of the University of Illinois at Chicago at the 31st Annual Workshop in Geometric Topology held at the University of Wisconsin-Milwaukee on the dates of June 12-14, 2014.
3:51
What is a manifold?
A visual explanation and definition of manifolds are given. This includes motivations for ...
published: 11 Oct 2015
What is a manifold?
What is a manifold?
- Report rights infringement
- published: 11 Oct 2015
- views: 211842
A visual explanation and definition of manifolds are given. This includes motivations for topology, Hausdorffness and second-countability.
If you want to learn more, check out one of these (or any other basic differential geometry or topology book):
- M. Spivak: "A Comprehensive Introduction to Differential Geometry"
- M. Nakahara: "Geometry, Topology and Physics"
- J. W. Milnor: "Topology from the differentiable viewpoint"
0:11
Clifford Torus || Geometric Topology
Clifford Torus || Geometric Topology
published: 02 Jan 2024
Clifford Torus || Geometric Topology
Clifford Torus || Geometric Topology
- Report rights infringement
- published: 02 Jan 2024
- views: 1097
Clifford Torus || Geometric Topology
Geometric topology
In mathematics, geometric topology is the study of manifolds and maps between them, particularly embeddings of one manifold into another.
History
Geometric topology as an area distinct from algebraic topology may be said to have originated in the 1935 classification of lens spaces by Reidemeister torsion, which required distinguishing spaces that are homotopy equivalent but not homeomorphic. This was the origin of simple homotopy theory.
Differences between low-dimensional and high-dimensional topology
Manifolds differ radically in behavior in high and low dimension.
High-dimensional topology refers to manifolds of dimension 5 and above, or in relative terms, embeddings in codimension 3 and above. Low-dimensional topology is concerned with questions in dimensions up to 4, or embeddings in codimension up to 2.
Dimension 4 is special, in that in some respects (topologically), dimension 4 is high-dimensional, while in other respects (differentiably), dimension 4 is low-dimensional; this overlap yields phenomena exceptional to dimension 4, such as exotic differentiable structures on R4. Thus the topological classification of 4-manifolds is in principle easy, and the key questions are: does a topological manifold admit a differentiable structure, and if so, how many? Notably, the smooth case of dimension 4 is the last open case of the generalized Poincaré conjecture; see Gluck twists.
This page contains text from Wikipedia, the Free Encyclopedia - https://wn.com/Geometric_topology
Relating Topology and Geometry - 2 Minute Math wit...
Algebra, Geometry, and Topology: What's The Differ...
Who cares about topology? (Inscribed rectangle p...
What is...geometric topology?...
The Biggest Ideas in the Universe | 13. Geometry a...
Cellular decomposition | Geometric topology...
Infinite Groups in Geometric Topology, Part 1...
Geometry Topology...
What is a manifold?...
Clifford Torus || Geometric Topology...
Latest News for: geometric topology
Edit
Webinar on Popularizing Mathematics and Science to be held on Jan 15
The Man Who Refused a Million Dollar Award.”</p><p>Gregori Perelman, a Russian mathematician, made history in 2006 by solving the century-old Poincaré Conjecture, a milestone in the field of topology.
Edit
The world's most fascinating transport maps: From London to Paris and New York to Mexico, ...
Edit
The Morning Call
09 Jul 2024
Weeks after retirement, Lehigh professor leads teens to national math win
The Morning Call
09 Jul 2024
Most of his work focuses on topology, which is the section of math involved with deformed properties of geometric objects ... He also studies differential topology, topological robotics and combinatorial number theory in his work.
Edit
New theory links quantum geometry to electron-phonon coupling
Yu said, "My motivation is to go beyond the common wisdom and find out how the geometric and topological properties of wavefunctions affect interactions in quantum materials.
Edit
Researchers engineer new approach for controlling thermal emission
Professor Coskun Kocabas, professor of 2D device materials at The University of Manchester, explains, "We have demonstrated a new class of thermal devices using concepts from topology—a branch of ...
Edit
Tailoring electron vortex beams with customizable intensity patterns by electron diffraction holography
... by a single integer describing their global topological invariance, microscopically, they are actually a superposition of different eigenstates resulting from locally varying geometric structures.
Edit
Physicists discover a novel quantum state in an elemental solid
This area of study combines quantum physics with topology—a branch of theoretical mathematics that explores geometric properties that can be deformed but not intrinsically changed.
Edit
Research team establishes synthetic dimension dynamics to manipulate light
In the realm of physics, synthetic dimensions (SDs) have emerged as one of the frontiers of active research, offering a pathway to explore phenomena in higher-dimensional spaces, beyond our conventional 3D geometrical space.
Edit
When the music changes, so does the dance: Controlling cooperative electronic states in kagome metals
The system is therefore geometrically frustrated to reach a magnetically ordered state, normally referred as magnetic frustration ... topological excitations and strong electronic correlations.
Edit
Scientists discover exotic quantum interference effect in a topological insulator device
Topological states of matter and coherence ... This area of study combines quantum physics with topology—a branch of theoretical mathematics that explores geometric properties that can be deformed but not intrinsically changed.
Edit
Sub-wavelength confinement of light demonstrated in indium phosphide nanocavity
In the new work, the researchers first designed an EDC cavity in the III-V semiconductor indium phosphide (InP) using a systematic mathematical approach that optimized the topology while relaxing geometric constraints.
Edit
Widely accepted Weyl semimetal shown to be a magnetic semiconductor
First predicted in the early 2010s, they belong to the class of topological materials that owe their unique transport, optical and thermoelectric behavior to distinct geometric and topological features, rather than to their chemical composition.
Edit
Team discovers thousands of new transformable knots
The team conducted further analysis across knot types that revealed new geometric and topological patterns with constructive principles not seen in previously tabulated knot types, showing how ...
Edit
Pills From The 3D Printer
Complex shapes can be created with relatively little effort, including tablets and pills in a variety of geometric shapes ... For the first time, the team is applying an inverse design and so-called topology optimization.
Edit
Global symmetry found to be not completely necessary for the protection of topological boundary states
The concept of topology in physics is inherited from mathematics, where topology is employed to study geometric properties of an object concerning quantities that are preserved under continuous deformation.
- 1
- 2
- Next page »