Binomial Theorem
Basis for dualities?
How does the binomial theorem relate to the triangle of power? (The relationship between a, b, c in {$a=b^c$}?)
Examples
- 2 x 2 square - conceptual factors - "and" and "or"
- eightsome - logical square 2 x 2 vertices and 4 pairs (edges)
- 2 x 2 multiplication
- 3 x 3 factorial experiment - Norman Anderson - multinomial
- girl + boy
- genetics - dominant and recessive
- truth tables
- expand 1.9 to a power = 2 - .01
- |A union B| + |A intersect B| = |A| + |B|
- Russian roulette
- interest rates - and taking the limit - formula for {$e^x$}
- simplexes, cross polytopes, hypercubes, coordinate systems
Infinite Series. Dissecting Hypercubes with Pascal's Triangle.
3 Blue 1 Brown. Derivative formulas through geometry.
Product rule {$u\cdot v + \textrm{d}(u\cdot v) = (u + \textrm{d}u)(v + \textrm{d}v)$}
Proof of the binomial theorem
- By bijection
- By induction
- By calculus
Sum of the row is {$2^n$}.
Inclusion-exclusion principle Counting nondivisible integers. Counting derangements. Number of surjective functions. Stirling numbers of the second kind. Rook polynomials. Main proof and algebraic proof. Combinatorial sieve methods.(Brun.)
The coefficient of {$q^r$} in {$\binom{n}{k}_q$} counts the number of inversions, for example, transforming 00011 to 01100 so that 1 is to the left of 0, here in 4 ways.
The coefficient of {$q^r$} in {$\binom{n+m}{m}_q$} counts the number of partitions of r with m or fewer parts each less than or equal to n.
{$\binom{n}{k}_q$} counts the number of k-dimensional vector subspaces of an n-dimensional vector space over {$F_q$} (a Grassmannian).
When k=1 and q=1 we have the Euler characteristic of the complex Grassmannian, and when q=-1, the Euler characteristic of the real Grassmannian.
The number of k-dimensional affine subspaces of {$F_{q^n}$} is {$q^{n-k}\binom{n}{k}_q$}. This allows an interpretation of the identity {$\binom{m}{r}_q=\binom{m-1}{r}_q + q^{m-r}\binom{m-1}{r-1}_q$}.
- Grassmannian Key to Moore's proof of Bott periodicity.
- God's Binomial Identity Expressing a number {$d=d_1*d_2*d_3$} as the sum of signed binomial coefficients {$d=\binom{d_1+d_2+d_3}{3}-\sum_{i<j}\binom{d_i+d_j}{3}+\sum_{i}\binom{d_i}{3}-\binom{0}{3}$}
Directed simplicial complexes
Finite field with one element
Grassmannians
- Michael Attiyah: Classical projective geometry: the twisted cubic, the quadric surface, or the Klein representation of lines in 3-space. These illustrate, respectively, the theory of rational curves, of homogeneous spaces, and of Grassmannians. The classical conic is a rational curve, a quadric and a Grassmannian all in one.
- Any conic lies on some Gr(2,4).
- Richard Borcherds. Algebraic Geometry 20 Grassmannians.
- Dev Sinha. Algebraic Topology From Geometric Perspective
- Andrei Okounkov. Enumerative geometry and geometric representation theory. (Stable envelopes and K-theory.)
- Positive Grassmanian is the subset of the real Grassmannian where all Plücker coordinates are nonnegative
- Amplituhedron By calculating its volume, in effect you’re calculating the amplitude for the given collision.
- Arkani-Hamed and his collaborators had defined the amplituhedron in relation to the positive Grassmannian. They demonstrated that it’s possible to change a positive Grassmannian into the amplituhedron by multiplying it by a type of matrix, effectively providing a mathematical recipe for moving points on the positive Grassmannian over to points on the amplituhedron. As a result, information about the relatively well-studied positive Grassmannian transfers to the relatively unexplored amplituhedron.
- Nima Arkani-Hamed and Thomas Lam in conversation
- A Mathematician’s Unanticipated Journey Through the Physical World
- Lauren K. Williams. The positive Grassmannian, the amplituhedron, and cluster algebras
- Williams wrote a series of papers with the mathematician Sylvie Corteel that explored an unexpected link between the combinatorics of the positive Grassmannian and statistical physics.
- Solitons and the Kadomtsev-Petviashvili equation. {$\partial_x(\partial_t u+u \partial_x u+\epsilon^2\partial_{xxx}u)+\lambda\partial_{yy}u=0$} where {$\lambda=\pm 1$}. A one-to-one relationship between points on the positive Grassmannian and solutions to the KP equation. The large-scale behavior of a wave formation is entirely determined by which cell your point in the positive Grassmannian lies in.
- Planar bicolored graphs.
Bottom-up and top-down duality