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Binomial theorem

多胞形

Consider polytopes - does the relations between the components give an algebra of perspectives?

• In three dimensions,
  • the 3-simplex - a triangle (1) is mapped to a point (0), yielding 1+3+0=4 triangles
  • the octahedron - a triangle (1) is mapped to an upside-down triangle (1), yielding 1+3+3+1 triangles, the large, 1 edge from large and 1 vertex from small, 1 vertex from large and 1 edge from small, the small
  • the icosahedron - a triangle (1) is related to an upside-down triangle (1) by way of 20=1+3+6+6+3+1 triangles
• In four dimensions, the tetrahedron, as a building block, yields
  • the 4-simplex - a tetrahedron (1) is mapped to a point (0), yielding 1+4+0=5 tetrahedra, the original and one for each face
  • the 4-orthoplex - a tetrahedron (1) is mapped to an upside-down tetrahedron, yielding 1+4+6+4+1=16 tetrahedra, the large, 1 face from large and 1 vertex from small, 1 edge from large and 1 from small, 1 vertex from large and 1 face from small, and the small tetrahedra.
  • the 600-cell - a tetrahedron is mapped to an icosahedron?
  • the 24-cell maps an octahedron to an octahedron
  • the 8-cell tessarect maps a cube to a cube 8=1+6+1

Readings

• The g-conjecture about an h-vectors of a simplicial polytope and the triangulation of a sphere.
  • Amazing: Karim Adiprasito proved the g-conjecture for spheres!

Federico Ardila

• Videos: Polytopes
• Teaching - online courses
  • Lectures: Polytopes
  • Lecture notes: Polytopes Δ
  • Texts: Polytopes
    • G. Ziegler. Lectures on polytopes 8 USD.
• Research talks

Lecture 1: Course is a combinatorial focus on convex polytopes and hyperplane arrangements.

• Euler's theorem: 1 - v + e - f + 1 = 0
• Steinitz's theorem: A 3-polytope exists iff 1 - v + e - f + 1 = 0, v <= 2f - 4, f <= 2v - 4.
• Regular polytopes.

Lecture 2: Definition of polytope P as convex hull of vertices. Sum of lambda x vertex where lambdas are nonnnegative and sum to 1.

Lecture 3: Intersections and products of polytopes are also polytopes.

• Main theorem of polytopes: Polytopes = convex hulls of finitely many points = bounded intersections of halfspaces.

Lecture 4: V-polyhedron intersected with affine plane is a V-polyhedron.

• Dual polytope consists of a where a<=1-x for all x in P.
• Caratheodory's theorem

Lecture 5 and 6: Farkas' lemma versions 1 to 4.

Lecture 7: Faces of polytopes. Face of P in direction c is the set of all x in P where c-x is maximal.

• Affine spaces.
• Dim(face) = Dim(Aff(face))
• f-vector gives the number of faces in each dimension
• f-poly the generated function where the lowest coefficient gives the number of vertices and the highest coefficient gives the volume

Lecture 8: Construction of faces

• Building polytopes: pyramids (adding the center).
• Vertex figures (converting the vertices to faces).
• Face lattice

Lecture 9: Face lattice

• P and Q are combinatorially isomorphic if their face lattices are isomorphic.
• Polar (Dual) polytopes. Dual = c in dual space where c x <=1 for all x in P.
• If 0 is in P, then P equals its dual's dual.

Lecture 10:

• Dual faces.
• The face lattices of P and its dual are opposites.
• Simple and simplicial polytopes.
• P is simple iff its dual is simplicial.

Lecture 11: The cyclic polytope.

Lecture 12: Graphs of polytopes

Lecture 13: How good is linear programming?

• Hirsch conjecture is false.

Lecture 14: Balinski's theorem: P is a d-polytope implies G(P) is d-connected.

Lecture 15: If P is simple, then G(P) determines P combinatorially.

Lecture 16: Complexes, subdivisions, triangulations.

• Every P has a triangulation.

Lecture 17: Triangulation of d-cross-polytope.

Lecture 18: Counting lattice points in polytopes.

Lecture 19: Partition functions.

Lecture 20: Generating functions for cones.