-
What is algebraic geometry?
Algebraic geometry is often presented as the study of zeroes of polynomial equations. But it's really about something much deeper: the duality between abstract algebra and geometry.
Help fund future projects here: https://www.patreon.com/aleph0
An equally valuable form of support is to simply share the videos.
----
A HUGE HUGE thank you to Faisal Al-Faisal for working with me on the script and storyboard for this video!
And another thank you to Davide Radaelli for helpful conversations when making this video.
----
CORRECTIONS:
At 4:26, I mistakenly wrote that g(1,1)=-2. This is a typo! The corrected version is g(1,-1)=-2.
SOURCES and REFERENCES for Further Reading!
(a) “A guide to plane algebraic curves” by Keith Kendig. It’s written in a very elementary style and has lots o...
published: 17 Oct 2023
-
Algebraic geometry 1 Introduction
This lecture is part of an online algebraic geometry course (Berkeley math 256A fall 2020), based on chapter I of "Algebraic geometry" by Hartshorne. The full set of lectures is in the playlist "Algebraic geometry I: varieties".
(The course continues in the playlist "Algebraic geometry II: schemes" based on chapter II.)
The first lecture describes a birational map from a circle to a line.
published: 21 May 2020
-
Studying Algebraic Geometry (A Dream)
#algebraicgeometry
published: 05 Apr 2022
-
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
-
Ravi Vakil: Algebraic geometry and the ongoing unification of mathematics
Abstract:
I will try to share a glimpse of this strange unification of many different ideas. This talk is aimed at a general audience, and no particular background will be assumed.
When we look carefully at nature, we can discover surprising coincidences, which suggest deeper underlying structure. The centrality of mathematics comes in part from the fact that seemingly unrelated ideas are often unified by some grand theory, which is far more powerful than the sum of its parts. Mathematics is most exciting when different ideas come together unexpectedly to give a new point of view. This is typified in algebraic geometry, and in the work of Deligne in particular, which brings together many themes in mathematics, including geometry, number, shape (topology), algebra, and more. This magic is ...
published: 11 Feb 2020
-
What is...algebraic geometry?
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...algebraic geometry? Or: Polynomials, and more.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
In this course I try to cover my favorite topics in algebraic geometry, from classical ideas such as algebraic varieties, to modern ideas such as schemes, to really modern ideas such as tropical varieties. I give a very biased collection of topics, and not nearly all that can be said will be addressed. Sorry for that.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-algebrageometry-2024.html
Thumbnail.
Picture from the fir...
published: 02 Mar 2024
-
A Swift Introduction to Geometric Algebra
This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realized that this might be useful to the general public, so I also tried to make this useful to others as well.
This video was made using manim (https://github.com/ManimCommunity/manim), a math animation library.
Several things in this video were incorrectly simplified. Please watch the video before reading the rest of the description.
Before I get into the things that were incorrectly simplified, I need to state a few definitions. The grade of a k-vector is defined to be k. A blade is defined to be the ...
published: 18 Aug 2020
-
Introduction To Algebraic Geometry - Course Introduction
Prof. Arijit Dey
IIT Madras
published: 01 Jun 2023
-
How to multiply 2 three digit numbers using the box method - Bonus Lesson
#algebra #Math #Formulas #Mathematics #Education #Learning #Numbers #STEM #Equations #Algebra #Geometry #StudyTips #ProblemSolving #Science #STEMeducation #MathIsFun #MathHelp #Mathletics #AcademicSuccess #StudyInspiration #criticalthinking
published: 30 Oct 2024
-
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
11:50
What is algebraic geometry?
Algebraic geometry is often presented as the study of zeroes of polynomial equations. But it's really about something much deeper: the duality between abstract ...
Algebraic geometry is often presented as the study of zeroes of polynomial equations. But it's really about something much deeper: the duality between abstract algebra and geometry.
Help fund future projects here: https://www.patreon.com/aleph0
An equally valuable form of support is to simply share the videos.
----
A HUGE HUGE thank you to Faisal Al-Faisal for working with me on the script and storyboard for this video!
And another thank you to Davide Radaelli for helpful conversations when making this video.
----
CORRECTIONS:
At 4:26, I mistakenly wrote that g(1,1)=-2. This is a typo! The corrected version is g(1,-1)=-2.
SOURCES and REFERENCES for Further Reading!
(a) “A guide to plane algebraic curves” by Keith Kendig. It’s written in a very elementary style and has lots of really captivating diagrams throughout. If you look at the table of contents, it starts off with lots of examples that only require elementary algebra. And by the end, it actually gets to some pretty deep theorems in algebraic geometry.
(b) "Ideals, Varieties, and Algorithms” by Cox, Little, O’ Shea. This book does not assume any knowledge of abstract algebra and teaches everything from the ground up. It is a very nice book with plenty of computational examples and exercises.
(c) “Algebraic Geometry and Arithmetic Curves” by Qing Liu. This books is all about schemes and Spec. It's a rather terse theorem-proof style book, but it is beautifully written and has lots of exercises.
----
MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.
https://www.vincentrubinetti.com/
Follow me!
Twitter: @00aleph00
Instagram: @00aleph00
What is algebraic geometry?: (0:00)
Coordinate Ring: (3:04)
How algebra detects reducibility: (3:54)
How algebra detects a node: (5:15)
Schemes!: (8:00)
https://wn.com/What_Is_Algebraic_Geometry
Algebraic geometry is often presented as the study of zeroes of polynomial equations. But it's really about something much deeper: the duality between abstract algebra and geometry.
Help fund future projects here: https://www.patreon.com/aleph0
An equally valuable form of support is to simply share the videos.
----
A HUGE HUGE thank you to Faisal Al-Faisal for working with me on the script and storyboard for this video!
And another thank you to Davide Radaelli for helpful conversations when making this video.
----
CORRECTIONS:
At 4:26, I mistakenly wrote that g(1,1)=-2. This is a typo! The corrected version is g(1,-1)=-2.
SOURCES and REFERENCES for Further Reading!
(a) “A guide to plane algebraic curves” by Keith Kendig. It’s written in a very elementary style and has lots of really captivating diagrams throughout. If you look at the table of contents, it starts off with lots of examples that only require elementary algebra. And by the end, it actually gets to some pretty deep theorems in algebraic geometry.
(b) "Ideals, Varieties, and Algorithms” by Cox, Little, O’ Shea. This book does not assume any knowledge of abstract algebra and teaches everything from the ground up. It is a very nice book with plenty of computational examples and exercises.
(c) “Algebraic Geometry and Arithmetic Curves” by Qing Liu. This books is all about schemes and Spec. It's a rather terse theorem-proof style book, but it is beautifully written and has lots of exercises.
----
MUSIC CREDITS:
The song is “Taking Flight”, by Vince Rubinetti.
https://www.vincentrubinetti.com/
Follow me!
Twitter: @00aleph00
Instagram: @00aleph00
What is algebraic geometry?: (0:00)
Coordinate Ring: (3:04)
How algebra detects reducibility: (3:54)
How algebra detects a node: (5:15)
Schemes!: (8:00)
- published: 17 Oct 2023
- views: 223519
20:58
Algebraic geometry 1 Introduction
This lecture is part of an online algebraic geometry course (Berkeley math 256A fall 2020), based on chapter I of "Algebraic geometry" by Hartshorne. The full s...
This lecture is part of an online algebraic geometry course (Berkeley math 256A fall 2020), based on chapter I of "Algebraic geometry" by Hartshorne. The full set of lectures is in the playlist "Algebraic geometry I: varieties".
(The course continues in the playlist "Algebraic geometry II: schemes" based on chapter II.)
The first lecture describes a birational map from a circle to a line.
https://wn.com/Algebraic_Geometry_1_Introduction
This lecture is part of an online algebraic geometry course (Berkeley math 256A fall 2020), based on chapter I of "Algebraic geometry" by Hartshorne. The full set of lectures is in the playlist "Algebraic geometry I: varieties".
(The course continues in the playlist "Algebraic geometry II: schemes" based on chapter II.)
The first lecture describes a birational map from a circle to a line.
- published: 21 May 2020
- views: 123276
2:19
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 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/
- published: 11 Sep 2018
- views: 35584
39:24
Ravi Vakil: Algebraic geometry and the ongoing unification of mathematics
Abstract:
I will try to share a glimpse of this strange unification of many different ideas. This talk is aimed at a general audience, and no particular backgro...
Abstract:
I will try to share a glimpse of this strange unification of many different ideas. This talk is aimed at a general audience, and no particular background will be assumed.
When we look carefully at nature, we can discover surprising coincidences, which suggest deeper underlying structure. The centrality of mathematics comes in part from the fact that seemingly unrelated ideas are often unified by some grand theory, which is far more powerful than the sum of its parts. Mathematics is most exciting when different ideas come together unexpectedly to give a new point of view. This is typified in algebraic geometry, and in the work of Deligne in particular, which brings together many themes in mathematics, including geometry, number, shape (topology), algebra, and more. This magic is the reason I became an algebraic geometer. For example, the theory of Pythagorean triples (such as ) connects geometry to the theory of numbers by way of algebra. This ancient example grows up to be the Weil conjectures, a wondrous prediction whose proof was finally completed by Deligne.
This lecture was given at The University of Oslo, May 22, 2013 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.
Program for the Abel Lectures 2013
1. "Hidden symmetries of algebraic varieties" by Abel Laureate 2013, professor Pierre Deligne, Institute for Advanced Study, Princeton
2. "Life Over Finite Fields" by professor Nicholas Katz, Princeton University
3. "Mixed Hodge structures and the topology of algebraic varieties" by professor Claire Voisin, École Polytechnique and CNRS
4. "Algebraic geometry and the ongoing unification of mathematics", a science lecture by professor Ravi Vakil, Stanford University
https://wn.com/Ravi_Vakil_Algebraic_Geometry_And_The_Ongoing_Unification_Of_Mathematics
Abstract:
I will try to share a glimpse of this strange unification of many different ideas. This talk is aimed at a general audience, and no particular background will be assumed.
When we look carefully at nature, we can discover surprising coincidences, which suggest deeper underlying structure. The centrality of mathematics comes in part from the fact that seemingly unrelated ideas are often unified by some grand theory, which is far more powerful than the sum of its parts. Mathematics is most exciting when different ideas come together unexpectedly to give a new point of view. This is typified in algebraic geometry, and in the work of Deligne in particular, which brings together many themes in mathematics, including geometry, number, shape (topology), algebra, and more. This magic is the reason I became an algebraic geometer. For example, the theory of Pythagorean triples (such as ) connects geometry to the theory of numbers by way of algebra. This ancient example grows up to be the Weil conjectures, a wondrous prediction whose proof was finally completed by Deligne.
This lecture was given at The University of Oslo, May 22, 2013 and was part of the Abel Prize Lectures in connection with the Abel Prize Week celebrations.
Program for the Abel Lectures 2013
1. "Hidden symmetries of algebraic varieties" by Abel Laureate 2013, professor Pierre Deligne, Institute for Advanced Study, Princeton
2. "Life Over Finite Fields" by professor Nicholas Katz, Princeton University
3. "Mixed Hodge structures and the topology of algebraic varieties" by professor Claire Voisin, École Polytechnique and CNRS
4. "Algebraic geometry and the ongoing unification of mathematics", a science lecture by professor Ravi Vakil, Stanford University
- published: 11 Feb 2020
- views: 69943
17:21
What is...algebraic geometry?
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...algebraic geometry? Or: Polynomials, an...
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...algebraic geometry? Or: Polynomials, and more.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
In this course I try to cover my favorite topics in algebraic geometry, from classical ideas such as algebraic varieties, to modern ideas such as schemes, to really modern ideas such as tropical varieties. I give a very biased collection of topics, and not nearly all that can be said will be addressed. Sorry for that.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-algebrageometry-2024.html
Thumbnail.
Picture from the first video slides.
Classical algebraic geometry.
https://en.wikipedia.org/wiki/Algebraic_geometry
https://en.wikipedia.org/wiki/Commutative_algebra
https://en.wikipedia.org/wiki/Multivariate_polynomial
https://en.wikipedia.org/wiki/Algebraic_variety
https://en.wikipedia.org/wiki/Affine_variety
https://en.wikipedia.org/wiki/Projective_variety
https://en.wikipedia.org/wiki/Quasi-projective_variety
https://en.wikipedia.org/wiki/Line_(geometry)
https://en.wikipedia.org/wiki/Circle
https://en.wikipedia.org/wiki/Parabola
https://en.wikipedia.org/wiki/Ellipse
https://en.wikipedia.org/wiki/Hyperbola
https://en.wikipedia.org/wiki/Cubic_plane_curve
https://en.wikipedia.org/wiki/Elliptic_curve
Modern algebraic geometry.
https://en.wikipedia.org/wiki/Algebraic_geometry#Abstract_modern_viewpoint
https://en.wikipedia.org/wiki/Scheme_(mathematics)
https://en.wikipedia.org/wiki/Formal_scheme
https://en.wikipedia.org/wiki/Ind-scheme
https://en.wikipedia.org/wiki/Algebraic_space
https://en.wikipedia.org/wiki/Algebraic_stack
Modern algebraic geometry version 2.
https://en.wikipedia.org/wiki/Algebraic_geometry#Computational_algebraic_geometry
https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis
https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition
https://en.wikipedia.org/wiki/Tropical_geometry
Applications of (algebraic) geometry.
https://math.stackexchange.com/questions/575181/real-life-applications-of-algebraic-geometry
Pictures used.
Pictures created using https://reference.wolfram.com/language/ref/ContourPlot.html
A variation of pictures from https://www.youtube.com/watch?v=cPg62OPdF8s
https://en.wikipedia.org/wiki/Conic_section#/media/File:Conic_Sections.svg
https://mathinstitutes.org/uploads/2021/07/attachments/16_hi-lite-var-of-var-pic-1-caption.png
https://pbelmans.ncag.info/atlas/mumford-curves.png
Picture from https://people.bath.ac.uk/cel34/docs/Updated_CAD_Slides.pdf
https://www.researchgate.net/publication/356841975/figure/fig1/AS:1098748994301952@1638973582355/A-track-curve-of-a-Homotopy-Continuation-algorithm-showing-HX-t-in-black-along-with.ppm
https://www.researchgate.net/publication/326107597/figure/fig1/AS:658529564254208@1534017091242/Homotopy-continuation-method-procedure-IEICE-2018.png
https://hackaday.com/wp-content/uploads/2019/07/Elliptic_Curve_Cryptography_data_points_doughnut.jpg
Some books I am using (I sometimes steal some pictures from there).
https://agag-gathmann.math.rptu.de/class/alggeom-2021/alggeom-2021.pdf
https://www.cambridge.org/core/books/computational-algebraic-geometry/B6E21C8B64D5FF95A88805910B18A006
https://bertini.nd.edu/book.html
https://mathoverflow.net/questions/2446/best-algebraic-geometry-textbook-other-than-hartshorne
Computer talk.
https://magma.maths.usyd.edu.au/magma/handbook/part/15
https://reference.wolfram.com/language/ref/ContourPlot.html
#algebraicgeometry
#geometry
#mathematics
https://wn.com/What_Is...Algebraic_Geometry
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...algebraic geometry? Or: Polynomials, and more.
Disclaimer.
Nobody is perfect, and I might have said something silly. If there is any doubt, then please check the references.
Disclaimer.
In this course I try to cover my favorite topics in algebraic geometry, from classical ideas such as algebraic varieties, to modern ideas such as schemes, to really modern ideas such as tropical varieties. I give a very biased collection of topics, and not nearly all that can be said will be addressed. Sorry for that.
Slides.
http://www.dtubbenhauer.com/youtube.html
Website with exercises.
http://www.dtubbenhauer.com/lecture-algebrageometry-2024.html
Thumbnail.
Picture from the first video slides.
Classical algebraic geometry.
https://en.wikipedia.org/wiki/Algebraic_geometry
https://en.wikipedia.org/wiki/Commutative_algebra
https://en.wikipedia.org/wiki/Multivariate_polynomial
https://en.wikipedia.org/wiki/Algebraic_variety
https://en.wikipedia.org/wiki/Affine_variety
https://en.wikipedia.org/wiki/Projective_variety
https://en.wikipedia.org/wiki/Quasi-projective_variety
https://en.wikipedia.org/wiki/Line_(geometry)
https://en.wikipedia.org/wiki/Circle
https://en.wikipedia.org/wiki/Parabola
https://en.wikipedia.org/wiki/Ellipse
https://en.wikipedia.org/wiki/Hyperbola
https://en.wikipedia.org/wiki/Cubic_plane_curve
https://en.wikipedia.org/wiki/Elliptic_curve
Modern algebraic geometry.
https://en.wikipedia.org/wiki/Algebraic_geometry#Abstract_modern_viewpoint
https://en.wikipedia.org/wiki/Scheme_(mathematics)
https://en.wikipedia.org/wiki/Formal_scheme
https://en.wikipedia.org/wiki/Ind-scheme
https://en.wikipedia.org/wiki/Algebraic_space
https://en.wikipedia.org/wiki/Algebraic_stack
Modern algebraic geometry version 2.
https://en.wikipedia.org/wiki/Algebraic_geometry#Computational_algebraic_geometry
https://en.wikipedia.org/wiki/Gr%C3%B6bner_basis
https://en.wikipedia.org/wiki/Cylindrical_algebraic_decomposition
https://en.wikipedia.org/wiki/Tropical_geometry
Applications of (algebraic) geometry.
https://math.stackexchange.com/questions/575181/real-life-applications-of-algebraic-geometry
Pictures used.
Pictures created using https://reference.wolfram.com/language/ref/ContourPlot.html
A variation of pictures from https://www.youtube.com/watch?v=cPg62OPdF8s
https://en.wikipedia.org/wiki/Conic_section#/media/File:Conic_Sections.svg
https://mathinstitutes.org/uploads/2021/07/attachments/16_hi-lite-var-of-var-pic-1-caption.png
https://pbelmans.ncag.info/atlas/mumford-curves.png
Picture from https://people.bath.ac.uk/cel34/docs/Updated_CAD_Slides.pdf
https://www.researchgate.net/publication/356841975/figure/fig1/AS:1098748994301952@1638973582355/A-track-curve-of-a-Homotopy-Continuation-algorithm-showing-HX-t-in-black-along-with.ppm
https://www.researchgate.net/publication/326107597/figure/fig1/AS:658529564254208@1534017091242/Homotopy-continuation-method-procedure-IEICE-2018.png
https://hackaday.com/wp-content/uploads/2019/07/Elliptic_Curve_Cryptography_data_points_doughnut.jpg
Some books I am using (I sometimes steal some pictures from there).
https://agag-gathmann.math.rptu.de/class/alggeom-2021/alggeom-2021.pdf
https://www.cambridge.org/core/books/computational-algebraic-geometry/B6E21C8B64D5FF95A88805910B18A006
https://bertini.nd.edu/book.html
https://mathoverflow.net/questions/2446/best-algebraic-geometry-textbook-other-than-hartshorne
Computer talk.
https://magma.maths.usyd.edu.au/magma/handbook/part/15
https://reference.wolfram.com/language/ref/ContourPlot.html
#algebraicgeometry
#geometry
#mathematics
- published: 02 Mar 2024
- views: 4737
44:23
A Swift Introduction to Geometric Algebra
This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video ...
This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realized that this might be useful to the general public, so I also tried to make this useful to others as well.
This video was made using manim (https://github.com/ManimCommunity/manim), a math animation library.
Several things in this video were incorrectly simplified. Please watch the video before reading the rest of the description.
Before I get into the things that were incorrectly simplified, I need to state a few definitions. The grade of a k-vector is defined to be k. A blade is defined to be the outer product of vectors. A k-blade is a blade of grade k. Some of the terminology between linear algebra and geometric algebra can be confusing, such as the use of the terms vector space and dimension. Because multivectors form their own vector space, but we don't consider all of them to be vectors, some people use the term linear space instead of vector space. While we think of dimension in terms of spatial degrees of freedom, the mathematical definition means that the 3D geometric algebra shown here is actually eight-dimensional. This is part of the reason for the distinction between the terms grade and dimension.
The following are the things that I purposefully got wrong to make things simpler:
The biggest thing I glossed over is the distinction between blades and vectors. A lot of the ways that I described k-vectors intuitively actually only applied to k-blades, not k-vectors. For example, in four or more dimensions, all 2-blades can be represented as an oriented area, but not all 2-vectors can. I didn't mention this fact for two reasons: first, for a first look, the distinction is not that important. Second, the distinction doesn't even matter until you reach four or more dimensions. In three dimensions, all k-vectors are k-blades.
My use of the term basis was not quite right, and I should have used the term "orthonormal basis". I wanted this video to be understood by people who had learned about vectors in a physics class where they don't go into too much detail about the exact definitions a linear algebra class would give you, so I didn't want to use the term orthonormal. I tried to mitigate this somewhat by always using terms like "the basis" and assuming that it was known that I meant the standard orthonormal basis. Hopefully this doesn't cause any confusion.
I mentioned that the inner product can get complicated when generalized to generic blades. What I didn't mention is that there is a large amount of disagreement on what this generalization is. I've seen at least five different extensions of the inner product to higher-grade blades. I didn't mention this because it was not pertinent to the discussion. However, even though I didn't talk much about the inner and outer products in this video, they are essential to the usage of geometric algebra in many applications.
This wasn't necessarily wrong, but I assumed that space is Euclidean throughout the video. In some applications, especially in relativity, space is not Euclidean, and a few things aren't the same.
In general, the pseudoscalar for a geometric algebra is the highest-grade element, but this does not always square to -1. In the three main applications of geometric algebra (2D Euclidean space, 3D Euclidean space, and 4D Minkowski space), this does happen to be the case, but in general it is not true.
When I mentioned dividing vectors, I didn't mention the fact that when dividing, you have to "divide on the left" or "divide on the right" because multiplication is not commutative. I actually prefer just to divide by multiplying by the inverse, because then you don't have to worry about how you're dividing. I didn't mention this because I never mentioned inverses again after I introduced them.
Another issue with inverses is that while vectors have inverses, not all multivectors have inverses. Also, in other spaces, such as the Minkowski space used in relativity, even some vectors don't have inverses. Again, this was because I didn't really use inverses much after this.
This one is small, but we assumed several properties of the geometric product that we didn't prove or state, such as distributivity and associativity. To be precise, the set of multivectors along with addition and the geometric product forms an associative algebra. I defined the geometric product in a bit of a roundabout way and I couldn't find a good place to insert these facts. The kind of people this video is aiming at are those who don't know a lot of abstract algebra and so I assumed that they would assume associativity and distributivity.
Rotors actually have many different definitions, so don't always think of them as a complex exponential.
https://wn.com/A_Swift_Introduction_To_Geometric_Algebra
This video is an introduction to geometric algebra, a severely underrated mathematical language that can be used to describe almost all of physics. This video was made as a presentation for my lab that I work in. While I had the people there foremost in my mind when making this, I realized that this might be useful to the general public, so I also tried to make this useful to others as well.
This video was made using manim (https://github.com/ManimCommunity/manim), a math animation library.
Several things in this video were incorrectly simplified. Please watch the video before reading the rest of the description.
Before I get into the things that were incorrectly simplified, I need to state a few definitions. The grade of a k-vector is defined to be k. A blade is defined to be the outer product of vectors. A k-blade is a blade of grade k. Some of the terminology between linear algebra and geometric algebra can be confusing, such as the use of the terms vector space and dimension. Because multivectors form their own vector space, but we don't consider all of them to be vectors, some people use the term linear space instead of vector space. While we think of dimension in terms of spatial degrees of freedom, the mathematical definition means that the 3D geometric algebra shown here is actually eight-dimensional. This is part of the reason for the distinction between the terms grade and dimension.
The following are the things that I purposefully got wrong to make things simpler:
The biggest thing I glossed over is the distinction between blades and vectors. A lot of the ways that I described k-vectors intuitively actually only applied to k-blades, not k-vectors. For example, in four or more dimensions, all 2-blades can be represented as an oriented area, but not all 2-vectors can. I didn't mention this fact for two reasons: first, for a first look, the distinction is not that important. Second, the distinction doesn't even matter until you reach four or more dimensions. In three dimensions, all k-vectors are k-blades.
My use of the term basis was not quite right, and I should have used the term "orthonormal basis". I wanted this video to be understood by people who had learned about vectors in a physics class where they don't go into too much detail about the exact definitions a linear algebra class would give you, so I didn't want to use the term orthonormal. I tried to mitigate this somewhat by always using terms like "the basis" and assuming that it was known that I meant the standard orthonormal basis. Hopefully this doesn't cause any confusion.
I mentioned that the inner product can get complicated when generalized to generic blades. What I didn't mention is that there is a large amount of disagreement on what this generalization is. I've seen at least five different extensions of the inner product to higher-grade blades. I didn't mention this because it was not pertinent to the discussion. However, even though I didn't talk much about the inner and outer products in this video, they are essential to the usage of geometric algebra in many applications.
This wasn't necessarily wrong, but I assumed that space is Euclidean throughout the video. In some applications, especially in relativity, space is not Euclidean, and a few things aren't the same.
In general, the pseudoscalar for a geometric algebra is the highest-grade element, but this does not always square to -1. In the three main applications of geometric algebra (2D Euclidean space, 3D Euclidean space, and 4D Minkowski space), this does happen to be the case, but in general it is not true.
When I mentioned dividing vectors, I didn't mention the fact that when dividing, you have to "divide on the left" or "divide on the right" because multiplication is not commutative. I actually prefer just to divide by multiplying by the inverse, because then you don't have to worry about how you're dividing. I didn't mention this because I never mentioned inverses again after I introduced them.
Another issue with inverses is that while vectors have inverses, not all multivectors have inverses. Also, in other spaces, such as the Minkowski space used in relativity, even some vectors don't have inverses. Again, this was because I didn't really use inverses much after this.
This one is small, but we assumed several properties of the geometric product that we didn't prove or state, such as distributivity and associativity. To be precise, the set of multivectors along with addition and the geometric product forms an associative algebra. I defined the geometric product in a bit of a roundabout way and I couldn't find a good place to insert these facts. The kind of people this video is aiming at are those who don't know a lot of abstract algebra and so I assumed that they would assume associativity and distributivity.
Rotors actually have many different definitions, so don't always think of them as a complex exponential.
- published: 18 Aug 2020
- views: 875370
0:58
How to multiply 2 three digit numbers using the box method - Bonus Lesson
#algebra #Math #Formulas #Mathematics #Education #Learning #Numbers #STEM #Equations #Algebra #Geometry #StudyTips #ProblemSolving #Science #STEMeducation #Math...
#algebra #Math #Formulas #Mathematics #Education #Learning #Numbers #STEM #Equations #Algebra #Geometry #StudyTips #ProblemSolving #Science #STEMeducation #MathIsFun #MathHelp #Mathletics #AcademicSuccess #StudyInspiration #criticalthinking
https://wn.com/How_To_Multiply_2_Three_Digit_Numbers_Using_The_Box_Method_Bonus_Lesson
#algebra #Math #Formulas #Mathematics #Education #Learning #Numbers #STEM #Equations #Algebra #Geometry #StudyTips #ProblemSolving #Science #STEMeducation #MathIsFun #MathHelp #Mathletics #AcademicSuccess #StudyInspiration #criticalthinking
- published: 30 Oct 2024
- views: 26
3:01
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 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