Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding.
Dimension of an affine algebraic set
Let K be a field, and L ⊇ K be an algebraically closed extension. An affine algebraic setV is the set of the common zeros in Ln of the elements of an ideal I in a polynomial ring Let A=R/I be the algebra of the polynomials over V. The dimension of V is any of the following integers. It does not change if K is enlarged, if L is replaced by another algebraically closed extension of K and if I is replaced by another ideal having the same zeros (that is having the same radical). The dimension is also independent of the choice of coordinates; in other words is does not change if the xi are replaced by linearly independent linear combinations of them. The dimension of V is
Algebraic varieties are the central objects of study in algebraic geometry. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Modern definitions generalize this concept in several different ways, while attempting to preserve the geometric intuition behind the original definition.
Conventions regarding the definition of an algebraic variety differ slightly. For example, some definitions provide that algebraic variety is irreducible, which means that it is not the union of two smaller sets that are closed in the Zariski topology. Under this definition, non-irreducible algebraic varieties are called algebraic sets. Other conventions do not require irreducibility.
The concept of an algebraic variety is similar to that of an analytic manifold. An important difference is that an algebraic variety may have singular points, while a manifold cannot.
The fundamental theorem of algebra establishes a link between algebra and geometry by showing that a monic polynomial (an algebraic object) in one variable with complex number coefficients is determined by the set of its roots (a geometric object) in the complex plane. Generalizing this result, Hilbert's Nullstellensatz provides a fundamental correspondence between ideals of polynomial rings and algebraic sets. Using the Nullstellensatz and related results, mathematicians have established a strong correspondence between questions on algebraic sets and questions of ring theory. This correspondence is the specificity of algebraic geometry.
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...the dimension of a variety? Or: A space in a space in a space...
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.
Pi...
published: 27 Apr 2024
Bernd Sturmfels (8/28/18): Learning algebraic varieties from samples
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.
published: 29 Aug 2018
9 Dimension
published: 15 Nov 2014
Shigefumi Mori, Recent progress in higher dimensional algebraic geometry I
2007 Clay Research Conference
published: 02 Dec 2020
1.2.2 Questions and intuition about the dimension of affine varieties
In this video, we discuss this idea: Intuition suggests that the dimension of affine variety defined by one equation in n variables would be (n-1), and that each additional equation would add a constraint, so that the dimension of a variety defined by m equations in n unknowns would be (n-m). We look at some examples where this intuition fails and suggest what might be coming regarding how we might determine the dimension of an affine variety.
This video is based on section 1.2 from Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea, and can be purchased here: https://www.amazon.com/gp/product/038797847X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
The numbering is based on the source content. The first two numbers indicate the chapter and section, and the third number indic...
published: 07 Feb 2021
7.1 Dimension (Commutative Algebra and Algebraic Geometry)
How can we define the dimension of an affine variety?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Noetherian topological space, dimension, Krull dimension.
Lecturer: Seidon Alsaody, Uppsala University.
published: 28 Apr 2021
Elliptic Curves - Lecture 4a - Varieties, function fields, dimension
This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves. More information about the course can be found at the course website: https://alozano.clas.uconn.edu/math5020-elliptic-curves/
published: 26 Jan 2021
7.2 Properties of the dimension (Commutative Algebra and Algebraic Geometry)
How does the dimension of an affine variety behave?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Dimension, affine variety, hypersurface, Krull's principal ideal theorem.
Lecturer: Seidon Alsaody, Uppsala University.
published: 28 Apr 2021
An Introduction to Algebraic Geometry : Chapter 1, Section 1 - Affine Varieties
This is the first part of my playlist going over the content in Robin Hartshorne's book Algebraic Geometry - in this part we go over section one (affine varieties) of chapter one (varieties), covering the visualization of zero-sets of polynomials, the Zariski Topology generated by zero sets, the ideal of subsets of affine space, the relation between ideals and zero sets, affine varieties, properties of Noetherian Topological Spaces, basic dimension theory of varieties, and end off with the unofficial concept of potential irreducible to motivate quasi-affine varieties.
The collection of links this lecture is as follows :
https://docs.google.com/document/d/1I2ImYKkun7Rhi21TsPmQWHKnWYnXBGB5NbfbzSFphdY/edit?usp=sharing - Video Script
https://docs.google.com/presentation/d/1U5MQmK9Q-kPGKfN5iI...
published: 16 Aug 2022
algebraic geometry 14 Dimension
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the dimension of a topological space, algebraic set, or ring.
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...the dimension of a variety? Or: A space...
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...the dimension of a variety? Or: A space in a space in a space...
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.
https://en.wikipedia.org/wiki/Dimension#/media/File:Dimension_levels.svg
https://en.wikipedia.org/wiki/Manifold#/media/File:Circle_with_overlapping_manifold_charts.svg
Picture created using https://mathematica.stackexchange.com/questions/34038/plot-points-line-and-plane-in-one-3d-plot
Same picture again
Pictures created using https://reference.wolfram.com/language/ref/ContourPlot3D.html
https://upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Elliptic_curve_on_Z61.svg/2560px-Elliptic_curve_on_Z61.svg.png
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
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...the dimension of a variety? Or: A space in a space in a space...
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.
https://en.wikipedia.org/wiki/Dimension#/media/File:Dimension_levels.svg
https://en.wikipedia.org/wiki/Manifold#/media/File:Circle_with_overlapping_manifold_charts.svg
Picture created using https://mathematica.stackexchange.com/questions/34038/plot-points-line-and-plane-in-one-3d-plot
Same picture again
Pictures created using https://reference.wolfram.com/language/ref/ContourPlot3D.html
https://upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Elliptic_curve_on_Z61.svg/2560px-Elliptic_curve_on_Z61.svg.png
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
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies o...
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.
In this video, we discuss this idea: Intuition suggests that the dimension of affine variety defined by one equation in n variables would be (n-1), and that eac...
In this video, we discuss this idea: Intuition suggests that the dimension of affine variety defined by one equation in n variables would be (n-1), and that each additional equation would add a constraint, so that the dimension of a variety defined by m equations in n unknowns would be (n-m). We look at some examples where this intuition fails and suggest what might be coming regarding how we might determine the dimension of an affine variety.
This video is based on section 1.2 from Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea, and can be purchased here: https://www.amazon.com/gp/product/038797847X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
The numbering is based on the source content. The first two numbers indicate the chapter and section, and the third number indicates the video. The number 1.1.1 indicates that this is the first video covering chapter 1, section 1 from the Cox textbook.
In this video, we discuss this idea: Intuition suggests that the dimension of affine variety defined by one equation in n variables would be (n-1), and that each additional equation would add a constraint, so that the dimension of a variety defined by m equations in n unknowns would be (n-m). We look at some examples where this intuition fails and suggest what might be coming regarding how we might determine the dimension of an affine variety.
This video is based on section 1.2 from Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea, and can be purchased here: https://www.amazon.com/gp/product/038797847X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
The numbering is based on the source content. The first two numbers indicate the chapter and section, and the third number indicates the video. The number 1.1.1 indicates that this is the first video covering chapter 1, section 1 from the Cox textbook.
How can we define the dimension of an affine variety?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: No...
How can we define the dimension of an affine variety?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Noetherian topological space, dimension, Krull dimension.
Lecturer: Seidon Alsaody, Uppsala University.
How can we define the dimension of an affine variety?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Noetherian topological space, dimension, Krull dimension.
Lecturer: Seidon Alsaody, Uppsala University.
This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves...
This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves. More information about the course can be found at the course website: https://alozano.clas.uconn.edu/math5020-elliptic-curves/
This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves. More information about the course can be found at the course website: https://alozano.clas.uconn.edu/math5020-elliptic-curves/
How does the dimension of an affine variety behave?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Dimen...
How does the dimension of an affine variety behave?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Dimension, affine variety, hypersurface, Krull's principal ideal theorem.
Lecturer: Seidon Alsaody, Uppsala University.
How does the dimension of an affine variety behave?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Dimension, affine variety, hypersurface, Krull's principal ideal theorem.
Lecturer: Seidon Alsaody, Uppsala University.
This is the first part of my playlist going over the content in Robin Hartshorne's book Algebraic Geometry - in this part we go over section one (affine varieti...
This is the first part of my playlist going over the content in Robin Hartshorne's book Algebraic Geometry - in this part we go over section one (affine varieties) of chapter one (varieties), covering the visualization of zero-sets of polynomials, the Zariski Topology generated by zero sets, the ideal of subsets of affine space, the relation between ideals and zero sets, affine varieties, properties of Noetherian Topological Spaces, basic dimension theory of varieties, and end off with the unofficial concept of potential irreducible to motivate quasi-affine varieties.
The collection of links this lecture is as follows :
https://docs.google.com/document/d/1I2ImYKkun7Rhi21TsPmQWHKnWYnXBGB5NbfbzSFphdY/edit?usp=sharing - Video Script
https://docs.google.com/presentation/d/1U5MQmK9Q-kPGKfN5iIc-rAOih3m19vD6VHHR2gTLIEM/edit?usp=sharing - Slides
http://abstract.ups.edu/download.html - Abstract Algebra, Theory and applications (Tom Judson)
https://www.math.ru.nl/~mueger/topology.pdf - Topology for the Working Mathematician (Michael Müger)
http://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/ - Link to Blender Library with Topology Cow
https://musescore.com/user/21974371/scores/4220336 - Nickolas Kaedalus' arrangement of Mass Destruction from Persona 3
https://www.youtube.com/c/OliverLugg - Oliver Lugg's channel, go learn category theory there if you haven't already
Chapters :
00:00-02:00 : Introduction
02:00-02:58 : Preliminaries
02:58-05:45 : Shapes from Zeroes
05:45-13:10 : The Topology of Zeroes
13:10-23:08 : Topology ↔ Algebra
23:08-25:50 : Noetherian, Topologically
25:50-33:30 : The Part with Dimension Theory
33:30-37:40 : Moving Beyond the Affine
37:40 : End(ing Remarks)
This is the first part of my playlist going over the content in Robin Hartshorne's book Algebraic Geometry - in this part we go over section one (affine varieties) of chapter one (varieties), covering the visualization of zero-sets of polynomials, the Zariski Topology generated by zero sets, the ideal of subsets of affine space, the relation between ideals and zero sets, affine varieties, properties of Noetherian Topological Spaces, basic dimension theory of varieties, and end off with the unofficial concept of potential irreducible to motivate quasi-affine varieties.
The collection of links this lecture is as follows :
https://docs.google.com/document/d/1I2ImYKkun7Rhi21TsPmQWHKnWYnXBGB5NbfbzSFphdY/edit?usp=sharing - Video Script
https://docs.google.com/presentation/d/1U5MQmK9Q-kPGKfN5iIc-rAOih3m19vD6VHHR2gTLIEM/edit?usp=sharing - Slides
http://abstract.ups.edu/download.html - Abstract Algebra, Theory and applications (Tom Judson)
https://www.math.ru.nl/~mueger/topology.pdf - Topology for the Working Mathematician (Michael Müger)
http://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/ - Link to Blender Library with Topology Cow
https://musescore.com/user/21974371/scores/4220336 - Nickolas Kaedalus' arrangement of Mass Destruction from Persona 3
https://www.youtube.com/c/OliverLugg - Oliver Lugg's channel, go learn category theory there if you haven't already
Chapters :
00:00-02:00 : Introduction
02:00-02:58 : Preliminaries
02:58-05:45 : Shapes from Zeroes
05:45-13:10 : The Topology of Zeroes
13:10-23:08 : Topology ↔ Algebra
23:08-25:50 : Noetherian, Topologically
25:50-33:30 : The Part with Dimension Theory
33:30-37:40 : Moving Beyond the Affine
37:40 : End(ing Remarks)
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the dimension of a topological ...
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the dimension of a topological space, algebraic set, or ring.
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the dimension of a topological space, algebraic set, or ring.
Goal.
Explaining basic concepts in the intersection of geometry and algebra in an intuitive way.
This time.
What is...the dimension of a variety? Or: A space in a space in a space...
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.
https://en.wikipedia.org/wiki/Dimension#/media/File:Dimension_levels.svg
https://en.wikipedia.org/wiki/Manifold#/media/File:Circle_with_overlapping_manifold_charts.svg
Picture created using https://mathematica.stackexchange.com/questions/34038/plot-points-line-and-plane-in-one-3d-plot
Same picture again
Pictures created using https://reference.wolfram.com/language/ref/ContourPlot3D.html
https://upload.wikimedia.org/wikipedia/commons/thumb/7/7e/Elliptic_curve_on_Z61.svg/2560px-Elliptic_curve_on_Z61.svg.png
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
We seek to determine a real algebraic variety from a fixed finite subset of points. Existing methods are studied and new methods are developed. Our focus lies on aspects of topology and algebraic geometry, such as dimension and defining polynomials. All algorithms are tested on a range of datasets and made available in a Julia package.
In this video, we discuss this idea: Intuition suggests that the dimension of affine variety defined by one equation in n variables would be (n-1), and that each additional equation would add a constraint, so that the dimension of a variety defined by m equations in n unknowns would be (n-m). We look at some examples where this intuition fails and suggest what might be coming regarding how we might determine the dimension of an affine variety.
This video is based on section 1.2 from Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea, and can be purchased here: https://www.amazon.com/gp/product/038797847X/ref=ppx_yo_dt_b_search_asin_title?ie=UTF8&psc=1
The numbering is based on the source content. The first two numbers indicate the chapter and section, and the third number indicates the video. The number 1.1.1 indicates that this is the first video covering chapter 1, section 1 from the Cox textbook.
How can we define the dimension of an affine variety?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Noetherian topological space, dimension, Krull dimension.
Lecturer: Seidon Alsaody, Uppsala University.
This video is part of a graduate course on elliptic curves that I taught at UConn in Spring 2021. The course is an introduction to the theory of elliptic curves. More information about the course can be found at the course website: https://alozano.clas.uconn.edu/math5020-elliptic-curves/
How does the dimension of an affine variety behave?
This lecture is part of a master level course on Commutative Algebra and Algebraic Geometry.
Keywords: Dimension, affine variety, hypersurface, Krull's principal ideal theorem.
Lecturer: Seidon Alsaody, Uppsala University.
This is the first part of my playlist going over the content in Robin Hartshorne's book Algebraic Geometry - in this part we go over section one (affine varieties) of chapter one (varieties), covering the visualization of zero-sets of polynomials, the Zariski Topology generated by zero sets, the ideal of subsets of affine space, the relation between ideals and zero sets, affine varieties, properties of Noetherian Topological Spaces, basic dimension theory of varieties, and end off with the unofficial concept of potential irreducible to motivate quasi-affine varieties.
The collection of links this lecture is as follows :
https://docs.google.com/document/d/1I2ImYKkun7Rhi21TsPmQWHKnWYnXBGB5NbfbzSFphdY/edit?usp=sharing - Video Script
https://docs.google.com/presentation/d/1U5MQmK9Q-kPGKfN5iIc-rAOih3m19vD6VHHR2gTLIEM/edit?usp=sharing - Slides
http://abstract.ups.edu/download.html - Abstract Algebra, Theory and applications (Tom Judson)
https://www.math.ru.nl/~mueger/topology.pdf - Topology for the Working Mathematician (Michael Müger)
http://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/ - Link to Blender Library with Topology Cow
https://musescore.com/user/21974371/scores/4220336 - Nickolas Kaedalus' arrangement of Mass Destruction from Persona 3
https://www.youtube.com/c/OliverLugg - Oliver Lugg's channel, go learn category theory there if you haven't already
Chapters :
00:00-02:00 : Introduction
02:00-02:58 : Preliminaries
02:58-05:45 : Shapes from Zeroes
05:45-13:10 : The Topology of Zeroes
13:10-23:08 : Topology ↔ Algebra
23:08-25:50 : Noetherian, Topologically
25:50-33:30 : The Part with Dimension Theory
33:30-37:40 : Moving Beyond the Affine
37:40 : End(ing Remarks)
This lecture is part of an online algebraic geometry course, based on chapter I of "Algebraic geometry" by Hartshorne. It covers the dimension of a topological space, algebraic set, or ring.
Some of these definitions are of geometric nature, while some other are purely algebraic and rely on commutative algebra. Some are restricted to algebraic varieties while others apply also to any algebraic set. Some are intrinsic, as independent of any embedding of the variety into an affine or projective space, while other are related to such an embedding.
Dimension of an affine algebraic set
Let K be a field, and L ⊇ K be an algebraically closed extension. An affine algebraic setV is the set of the common zeros in Ln of the elements of an ideal I in a polynomial ring Let A=R/I be the algebra of the polynomials over V. The dimension of V is any of the following integers. It does not change if K is enlarged, if L is replaced by another algebraically closed extension of K and if I is replaced by another ideal having the same zeros (that is having the same radical). The dimension is also independent of the choice of coordinates; in other words is does not change if the xi are replaced by linearly independent linear combinations of them. The dimension of V is