Thank you, this was the best video I found on affine varieties. Was there a typo at entrance (2) on 4:45 on the Zariski's topology? isn't the variety of the union the variety of the product instead? V(I)UV(J) = V(IJ)
Your welcome! Thanks for the catch - that's right. I'm pretty sure it was caught and annotated in video, but RUclips removed the annotation feature in 2018. Ack! I also goof the length of chains in a later video in the series, so I'll try to find that before you get there.
One more that says why intersection better: math.stackexchange.com/questions/633256/how-to-define-the-union-of-closed-subschemes-in-an-affine-scheme/633262#633262
5:55 "There are not nearly as many polynomials as there are continuous functions on C^n." Technically, both sets have cardinality 2^Aleph_0 (cardinality of the continuum) so there's actually just as many polynomials as there are continuous functions.
@orbital1337 I think most people watching understand the spirit of the quote and I thought it was a very interesting aside in the video, so I'm glad it was made. I can also tell you there aren't as many even numbers as there are integers and we both simultaneously understand what I mean by that, as well as the fact that the sets have the same cardinality. See the "post-rigorous" rigorous stage of mathematics education, as terrence tao calls it: terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/
orbital1337 it depends on what definition of size you are using. In terms of cardinality the two are of course equal, however the set of polynomials is measure-zero in the set of continuous functions, and you could argue that measure is as valid a definition of set size as cardinality is.
@@________6295 it’s actually very intuitive. All polynomials are differentiable, so we can just see that the set of differentiable functions is measure zero in the set of continuous functions. That requires a little more thinking, but if you imagine that you have a particle that at every instant makes a random decision as to which direction it moves, it becomes clear that it “almost always” moves in a jagged, non-smooth way, and at each non-smooth point, you have a point of non differentiability. This is of course not a rigorous way to show this fact, but it lends some intuition without needing to discuss covering spaces or σ-algebras.
I really appreciate how succinctly you are teaching this material. I have been looking for introductory videos in algebraic geometry for some time now so please continue the series! Do you think you will be lecturing on sheaf theory eventually?
goldi _lox Thanks! Sheaf theory - yes - ultimately I'd like to get some videos on flag manifolds and the Borel-Weil-Bott theorem. It's a long road though.
Are there any associated exercises to accompany these lectures? Thank you so much for these videos by the way! I'm reviewing Group Theory and Point Set Topology as well as learning a bit of Algebraic Geometry and Representation Theory.
Your welcome! The main book I'm using is Cox, Little, and Schenk's Toric Varieties, which has some great problem sets. They (or at least two of the authors) have a beginner's algebraic geometry book which is good also.
+spikeeleslie For vector spaces, we would say that v1,...,vk generates a subspace W if W=Span(v1,...,vk). So some subset which recovers our space of interest using only the basic operations. For rings, we would generate using addition and multiplication. For algebras, addition, multiplication, and scalar multiplication. Consider real polynomials in one variable. As a vector space, a generating set is {1, x, x^2,...}. As a real algebra, a generating set is {1, x}.
A subset of the original space that is sufficient to "build up" the whole space. The meaning of building up depends on the mathematical construction in context.
EvaSlash I spent some time as an actuary, so I'm pro-statistics. A big problem with statistics for pure math people is the language. A good deal of statistics was originally developed by regular scientists, so it does feel like another subject. Pure mathematicians also like structure; statistics is like playing with sand.
You saved my last semester. Thanks
Wow these lectures are surprisingly easy to grasp for a good graduate student....
I'm a third year undergrad that currently taking a course on (basic) commutative algebra and I can follow this easily
The explanation is quite good
Thank you, this was the best video I found on affine varieties. Was there a typo at entrance (2) on 4:45 on the Zariski's topology? isn't the variety of the union the variety of the product instead? V(I)UV(J) = V(IJ)
Your welcome! Thanks for the catch - that's right. I'm pretty sure it was caught and annotated in video, but RUclips removed the annotation feature in 2018. Ack! I also goof the length of chains in a later video in the series, so I'll try to find that before you get there.
According to this, maybe not wrong, but V(IJ) is easier to work with: math.stackexchange.com/questions/814256/what-is-the-union-of-two-varieties
One more that says why intersection better: math.stackexchange.com/questions/633256/how-to-define-the-union-of-closed-subschemes-in-an-affine-scheme/633262#633262
@@MathDoctorBob Thanks! I still couldn't figure that directly, but instead by noticing that (I⋂J)² ⊂ IJ ⊂ (I⋂J)
In fact, I gave a new answer to your first MSE link
5:55 "There are not nearly as many polynomials as there are continuous functions on C^n."
Technically, both sets have cardinality 2^Aleph_0 (cardinality of the continuum) so there's actually just as many polynomials as there are continuous functions.
orbital1337 No one ever accused me of being technical. :)
@orbital1337 I think most people watching understand the spirit of the quote and I thought it was a very interesting aside in the video, so I'm glad it was made.
I can also tell you there aren't as many even numbers as there are integers and we both simultaneously understand what I mean by that, as well as the fact that the sets have the same cardinality.
See the "post-rigorous" rigorous stage of mathematics education, as terrence tao calls it: terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/
orbital1337 it depends on what definition of size you are using. In terms of cardinality the two are of course equal, however the set of polynomials is measure-zero in the set of continuous functions, and you could argue that measure is as valid a definition of set size as cardinality is.
@@dmr11235 May I ask where can i find a proof to your statement?
@@________6295 it’s actually very intuitive. All polynomials are differentiable, so we can just see that the set of differentiable functions is measure zero in the set of continuous functions. That requires a little more thinking, but if you imagine that you have a particle that at every instant makes a random decision as to which direction it moves, it becomes clear that it “almost always” moves in a jagged, non-smooth way, and at each non-smooth point, you have a point of non differentiability. This is of course not a rigorous way to show this fact, but it lends some intuition without needing to discuss covering spaces or σ-algebras.
Greetings from India.
Very concise and to the point video.
Please continue the series and make videos on Commutative Algebra and Homology theory.
I really appreciate how succinctly you are teaching this material. I have been looking for introductory videos in algebraic geometry for some time now so please continue the series! Do you think you will be lecturing on sheaf theory eventually?
goldi _lox Thanks! Sheaf theory - yes - ultimately I'd like to get some videos on flag manifolds and the Borel-Weil-Bott theorem. It's a long road though.
You're an expert in every branch of mathematics. Is there any math you dont know?
I'm not even an expert on the problems I'm working on. I love the game though. :)
Are there any associated exercises to accompany these lectures? Thank you so much for these videos by the way! I'm reviewing Group Theory and Point Set Topology as well as learning a bit of Algebraic Geometry and Representation Theory.
Your welcome! The main book I'm using is Cox, Little, and Schenk's Toric Varieties, which has some great problem sets. They (or at least two of the authors) have a beginner's algebraic geometry book which is good also.
Bob, can you explain to me what a "generating set" is? or can you link me to a video explaining this?
+spikeeleslie For vector spaces, we would say that v1,...,vk generates a subspace W if W=Span(v1,...,vk). So some subset which recovers our space of interest using only the basic operations. For rings, we would generate using addition and multiplication. For algebras, addition, multiplication, and scalar multiplication.
Consider real polynomials in one variable. As a vector space, a generating set is {1, x, x^2,...}. As a real algebra, a generating set is {1, x}.
A subset of the original space that is sufficient to "build up" the whole space. The meaning of building up depends on the mathematical construction in context.
danke schon
Hey Dr. Bob! What are your opinions on the field of statistics? One of my teachers once told me that math majors hate statistics, why is that?
EvaSlash I spent some time as an actuary, so I'm pro-statistics. A big problem with statistics for pure math people is the language. A good deal of statistics was originally developed by regular scientists, so it does feel like another subject. Pure mathematicians also like structure; statistics is like playing with sand.