@@sirabusch858 Powerful : she knows it’s about simplexes … that ‘ s what her smooth geometry is about , most literally . @6:40 : Obviously the projective relationship between (1) Green ‘ s theorem in the plane (2) The Stoke ‘ s ( and Divergence) theorem in 3 space Would be one’s need for speaking of the polar nature of a projective plane as it sits in space : A plane of projection … If it may have more than 1 relationship with the grassmanian
When I hear the word "polar" what I think of is like a dartboard. Let's say the _m,n_ open dartboard is the subset of the plane whose lines are the circles centered at the origin of radius i ≤ m for positive integer i and open rays from the origin at angles 2πj/n for j integer and whose points are intersections between them (so technically the lines are finite subsets of the lines I mentioned). Well, I guess there's no reason I couldn't rewrite it as cross-sections and lines of a cylinder. Good word, then, for for m ≥ 3 and n ≥ 3 this space is a polar space: given a point and either a circle or a ray not through it, that circle has exactly one point at the same angle (resp., that ray has exactly one point at the same radius); that is the lone collinear point. A projective subspace of this space is by def'n. a collection of lines in this space where all points are collinear (and such that whenever AB and CD intersect, AC and BD intersect). Well, no two distinct lines in this polar space have all points collinear. So the only projective subspaces are lines, and thus this polar space is of rank 1+1=2. Let's say a dartspace is like the dartboard but in 3D. The lines are now rays and spheres. Well, it's still rank 2. I also tried to do something with circles centered at the origin and/or planes through the origin minus the origin in the quotient space of ℝ³ that identifies antipodal points, but I couldn't make it work, because two such planes have too big an intersection, and arbitrary points and circles don't have something to do with each other. Maybe if one were judicious as to precisely which such circles and planes to admit and/or precisely which such intersections to admit at points. I wonder if you know of an easy example of rank 3.
There are six families of polar spaces of rank 3 or higher. These are 1: the Symplectic polar spaces, connected to alternating forms, C_n series and the Quaternions; 2: the Hermitian polar spaces of even dimension, connected to Hermitian forms, A_2n series and the Complexes; 3: the Hermitian polar spaces of odd dimension, connected to Hermitian forms, A_2n+1 series and the Complexes; 4: the Hyperbolic polar spaces, connected to Quadratic forms, D_n series and the Reals; 5: the Parabolic polar spaces, connected to Quadratic forms, B_n series and the Reals; 6: the Elliptic polar spaces, connected to Quadratic forms, 2^D_n series and the Reals.
Excellent, wish there were more videos like this explaining proofs clearly. As someone from a quasigroup theory background, I also find Moufang planes interesting because they're in bijection with alternative division rings - idk, I just think it's really cool that (plane satisfies Moufang property) (coordinate division ring satisfies Moufang identities)
Yeah, and the multiplicative Moufang loops of division algebras (aka alternative division rings) generalise symmetry groups, and their natural left and right loop actions (which do not commute when combined) generalise group actions. So we get twisted composition of symmetries, among a set of symmetries that do not close group wise, but closes loop wise, and where certain compound twisted compositions (like g o h o g) coincide with ordinary composition and therefore close. Also, just like projective planes are coordinatized by division algebras, so polar planes (generalized quadrangles) are coordinatized by quadrangular algebras, and generalized hexagons are coordinatized by Jordan algebras. It seems that these rank 2 buildings all have instances that exist over alternative non-associative division algebras (such as octonions), in the sense that there are quadrangular algebras and Jordan algebras "coming from" these. With rank 3 and higher rank buildings however, it seems they all exist only over associative division rings, and have Desarguesian projective subspaces. I may be wrong though.
Really insightful and interesting videos! Thank you for sharing your research and love for mathematics 👍🏻
Thank you! ☺
@@sirabusch858
Powerful : she knows it’s about simplexes … that ‘ s what her smooth geometry is about , most literally .
@6:40 :
Obviously the projective relationship between
(1) Green ‘ s theorem in the plane
(2) The Stoke ‘ s
( and Divergence) theorem in 3 space
Would be one’s need for speaking of the polar nature of a projective plane as it sits in space :
A plane of projection …
If it may have more than 1 relationship with the grassmanian
When I hear the word "polar" what I think of is like a dartboard. Let's say the _m,n_ open dartboard is the subset of the plane whose lines are the circles centered at the origin of radius i ≤ m for positive integer i and open rays from the origin at angles 2πj/n for j integer and whose points are intersections between them (so technically the lines are finite subsets of the lines I mentioned). Well, I guess there's no reason I couldn't rewrite it as cross-sections and lines of a cylinder.
Good word, then, for for m ≥ 3 and n ≥ 3 this space is a polar space: given a point and either a circle or a ray not through it, that circle has exactly one point at the same angle (resp., that ray has exactly one point at the same radius); that is the lone collinear point.
A projective subspace of this space is by def'n. a collection of lines in this space where all points are collinear (and such that whenever AB and CD intersect, AC and BD intersect). Well, no two distinct lines in this polar space have all points collinear. So the only projective subspaces are lines, and thus this polar space is of rank 1+1=2.
Let's say a dartspace is like the dartboard but in 3D. The lines are now rays and spheres. Well, it's still rank 2.
I also tried to do something with circles centered at the origin and/or planes through the origin minus the origin in the quotient space of ℝ³ that identifies antipodal points, but I couldn't make it work, because two such planes have too big an intersection, and arbitrary points and circles don't have something to do with each other. Maybe if one were judicious as to precisely which such circles and planes to admit and/or precisely which such intersections to admit at points.
I wonder if you know of an easy example of rank 3.
There are six families of polar spaces of rank 3 or higher. These are 1: the Symplectic polar spaces, connected to alternating forms, C_n series and the Quaternions; 2: the Hermitian polar spaces of even dimension, connected to Hermitian forms, A_2n series and the Complexes; 3: the Hermitian polar spaces of odd dimension, connected to Hermitian forms, A_2n+1 series and the Complexes; 4: the Hyperbolic polar spaces, connected to Quadratic forms, D_n series and the Reals; 5: the Parabolic polar spaces, connected to Quadratic forms, B_n series and the Reals; 6: the Elliptic polar spaces, connected to Quadratic forms, 2^D_n series and the Reals.
I can't believe the guy's name is Jack Tits. I want that last name.
Great content! Please make more videos about incidence geometry and buildings.
Oh thank you! I will! 😊 It just always takes surprisingly much time. (~ 2 work days)
Excellent, wish there were more videos like this explaining proofs clearly. As someone from a quasigroup theory background, I also find Moufang planes interesting because they're in bijection with alternative division rings - idk, I just think it's really cool that (plane satisfies Moufang property) (coordinate division ring satisfies Moufang identities)
Yeah, and the multiplicative Moufang loops of division algebras (aka alternative division rings) generalise symmetry groups, and their natural left and right loop actions (which do not commute when combined) generalise group actions. So we get twisted composition of symmetries, among a set of symmetries that do not close group wise, but closes loop wise, and where certain compound twisted compositions (like g o h o g) coincide with ordinary composition and therefore close.
Also, just like projective planes are coordinatized by division algebras, so polar planes (generalized quadrangles) are coordinatized by quadrangular algebras, and generalized hexagons are coordinatized by Jordan algebras.
It seems that these rank 2 buildings all have instances that exist over alternative non-associative division algebras (such as octonions), in the sense that there are quadrangular algebras and Jordan algebras "coming from" these.
With rank 3 and higher rank buildings however, it seems they all exist only over associative division rings, and have Desarguesian projective subspaces. I may be wrong though.
Line wizardry
Pointless
@@digbysirchickentf2315
Grassmanian
@5:30
Thanks!
My pleasure 😊
I don't understand any of these words. cool eyeshadow, though.
IKR hahaha. maybe if we keep on watching we'll make some sense one day. Bomb eye looks indeed tho.