Lecture instead of recitation on Wednesday, February 5.
First class on Tuesday, October 15.
lecture  tuesday  1012  Seminarraum Arnimallee 2 (Villa) 
recitation  wednesday  1416  046/T9 Seminarraum (Takustr. 9) 
Homework #9  due February 12  
Homework #8  due January 29  
Homework #7  due January 22  
Homework #5 & #6  due December 18 & January 8  
Homework #4  due December 11  
Homework #3  due November 27  
Homework #2  due November 13  
Homework #1  due October 30 
Feb 11/12 see TeXed notes. 
more details on flips (and what they do to facets);
transition polytope \(T\) between \(P\) and \(Q\);
correspondence between weight spaces via everts; Proof of
the gTheorem: 1. \(\log(P)\) and \(\log(T)\) induces
quad forms on \(\Omega(P)\) with same signature; 2.
\(\log(T)\) and \(\log(Q)\) induce quad form on
\(\Omega(P)\subset\Omega(Q)\); this is done via explicit
calculations with the everts. What else can be done with
gTheorem and weight algebra? Centrallysymmetric
polytopes etc.; Connection to StanleyReisner rings (and
algebra of piecewise polynomials); Wrap up and thanks to
everybody who participated!

Feb 4/5 see TeXed notes. 
HRM in dim \(d1\) imply Lefschetz decomposition in dim
\(d\); \(P,P^\prime\) normally equivalent implies
\(\Omega(P)=\Omega(P^\prime)\); Lemma: corresponding quad.
forms have same signature; strategy of proof: change
righthand side and track changes in signatures; space of
\(A\)polytopes, chamber fans, and type cones; flips:
passing a wall in the type cone; \(Q\) obtained from \(P\)
by a \(m\)flip with \(1 \le m \le \frac{d}{2}\) then
\(g(Q) = g(P) + e_m\).

January 28 see TeXed notes. 
a special product formula; separation and an algebraic
prove of the DehnSommerville equations; isomorphism to
the (real) polytope algebra \(\Pi(P)\); back to the
gConjecture: weak and strong Lefschetz property;
Lefschetz decomposition; nondegenerate quadratic forms;
HodgeRiemannMinkowski inequalities and Minkowski's
second inequality

January 21 see TeXed notes. 
Evaluating frame functionals is computation of mixed
volumes; Recap: Mixed volumes as coefficients of
\(\mathrm{vol}(\lambda_1P_1 + \cdots + \lambda_kP_k)\); a
mixed subdivision formula for multiplying weights (proof
skipped); the full weight algebra \(\Omega(P),+,\cdot)\);
the first weight space via weak Minkowski summands up to
translation; generalized support vectors

January 14 see TeXed notes. 
the Minkowski relation and weights on polytopes; dimension
of the weight spaces
\(\Omega_d(P),\Omega_{d1}(P),\Omega_{0}(P)\);
Theorem: If \(P\) is simple, then
\(\dim \Omega_r(P) = h_r(P)\); Proof via sweeping;
Corollary: If \(P\) simple, then \(\Omega(P) \rightarrow
\Omega(F)\) surjective for all faces \(F \subseteq P\);
how to compute frame functionals

January 7 see TeXed notes. 
separation in the polytope algebra and frame functionals;
(weak) Minkowski summands; Shephard's theorem on Minkowski
summands; graded subrings \(\Pi(P)\) associated to
polytopes; the polytope algebra as an inverse limit of the
subrings \(\Pi(P)\); (finite) separation for subrings.

December 17 see TeXed notes. 
the nilradical of \(\Pi\);
the polytope algebra as a graded ring; motivation and
intuition via homogeneous valuations; \(\Pi\) as a
\(\mathbb{Q}\)vector space (torsion and unique
divisibility); logarithms and exponentials of polytopes;
the graded structure via `taylor expansion'; explicit
description of the first graded piece \(\Pi_1\) and
structure of \(\mathbb{R}\)vector space; translation
invariant group of polytopes and piecewise linear
functions modulo linears

December 10 see TeXed notes. 
translation invariant valuations and the translation
ideal; the polytope algebra \(\Pi^d =
\mathrm{S}\mathcal{P}_d/\mathcal{T}\); Theorem:
\(([P]1)^{\dim P +1} = 0\) in \(\Pi^d\); polyhedral
complexes, subdivisions, and triangulations; Hilbert's 3rd
problem and the translation ideal; the polytope algebra on
the real line; Main result: polytope algebra is a graded
ring; First step: \(\Pi = \mathbb{Z} \oplus \Pi_+\) where \(\Pi_+
= \ker \chi\)

December 3 see TeXed notes. 
a gallery of separating homomorphisms: projections to
subspaces, generalized support functions, and face maps
separate; face map: \([P] \mapsto [P^c]\) for some
direction \(c\); examples of valuations with `polynomial'
behaviour; What are polynomials?; some calculus of finite
differences; Prop: \(g : \mathbb{N} \rightarrow G\)
polynomial of degree \(< m\) iff \(\Delta^mg \equiv 0\);
translation invariant valuations give rise to polynomials;
what is a `universal' polynomial valuation?

November 26 
Euler map as extension of valuation; the group
\(\mathbf{S}\mathcal{P}_d^\times\) of invertible polytopal
functions; Theorem: \(f\) invertible iff \(f = \pm
[P]\star[Q]^{1}\); context: nonempty polytopes form
cancellative monoid under Minkowski sum; can form
Grothendieck group: \(G(\tilde{\mathcal{P}}_d) =
(\tilde{\mathcal{P}}_d \times \tilde{\mathcal{P}}_d) /
\sim\); Cor: \(G(\tilde{\mathcal{P}}_d)\) is exactly
\(\mathbf{S}\mathcal{P}_d^\times\); Observation:
\(\ell_*f\) invertible whenever \(f\) is; consider
invertibles in subalgebra \(\mathbf{S}\mathcal{P}_d^0\);
support functions of polytopal simple functions;
separating/distinguishing sets of homomorphisms

November 19 
Operations on polytopal simple functions; dilation
operator \( \Delta_\lambda [P] = [\lambda P]\); Pullback:
\( T : \mathbb{R}^d \rightarrow \mathbb{R}^e \Rightarrow
T^* \mathbf{S}\mathcal{Q}_e \rightarrow
\mathbf{S}\mathcal{Q}_d\) with \( T^*[P] = [T^{1}(P)]\);
Pushforward (nontrivial!):
\( T : \mathbb{R}^d \rightarrow \mathbb{R}^e \Rightarrow
T_* \mathbf{S}\mathcal{Q}_d \rightarrow
\mathbf{S}\mathcal{Q}_e\) with \( T_*[P] = [T(P)]\);
yields new (and powerful) multiplication with \([P] \star
[Q] = [P+Q]\); Unit: \([0]\); Invertible elements?
Theorem: \( [P]^{1} = (1)^{\dim P}
[\mathrm{relint}(P)]\)

November 12 
polyhedral simple functions; (intersection) product: \([P]
\cdot [Q] = [P \cap Q]\); ring of polyhedral simple
functions \((\mathbf{S}\mathcal{Q}_d,+,\cdot)\);
valuations on polytopes; Theorem (Volland, PerlesSallee,
Grömer): Correspondence \(\phi :
\mathcal{P}_d \rightarrow G\) and \(\phi:
\mathbf{S}\mathcal{Q}_d \rightarrow G\); Cor: \(P \mapsto
[P]\) is the universal valuation; Cor: Valuations on
polytopes have inclusionexclusion property; polyconvex
sets and extension of valuations; the Euler characteristic
\(\chi\); extension of \(\chi\) to polyconvex sets and
polyhedral simple functions

November 5 
fvectors, hvectors and Hilbert series of StanleyReisner
rings; pure and partitionable complexes; Q: When is \(H:
\mathbb{Z}_{\ge0} \rightarrow \mathbb{Z}_{\ge0} \) Hilbert
function of standard kalgebra?; multicomplexes / order
ideals and Osequences / Msequences; Thm(Macaulay):
\(H(n)\) Hilbert function of standard kalgebra iff
Osequence; binomial representation; Macaulay conditions;
Thm: \((h_0,\dots,h_d)\) hvector of shellable
\((d1)\)dim simplicial complex iff Macaulay conditions
hold iff Hilbert function of standard artinian algebra;
Proof of Upper Bound Theorem; Idea for GLBC: Find
kalgebra as above and Lefschetz element; the
gConjecture/Theorem (finally!)

October 29 
simplicial complexes, fvectors, hvectors; Generalized
Lower Bound conjecture and equality cases; convex hull of
simplicial fvectors; gvectors; commutative algebra:
rings, ring maps, ideals, quotients; finitely generated
kalgebras, graded rings/algebras and ideals, homogeneous
ring maps;finitely generated standard graded algebras;
Hilbert function, Hilbert series; Thm: standard
\(\Rightarrow (1t)^dH(R,t)\) polynomial; StanleyReisner
rings/ideals

October 22 
Euler characteristic, EulerPoincaré formula; only linear relation on all face numbers; more for simplicial polytopes; dual setting: simple polytopes; summing Euler characteristics of kfaces; general linear functions, sweeping, and hvectors; DehnSummerville equations and consequences; UBT and LBT in terms of hnumbers

October 15 
towards the gTheorem; polyhedra, polytopes, cones; convex hull, conical hull; MinkowskiWeyl, affine hull, dimension, support function, faces, fvectors; Steinitz characterization in dimension 3; the misery in dimension 4; linear programming and combinatorial complexity; Upper Bound Conjecture (Motzkin); kneighbortly polytopes, neighborliness and neighborly polytopes; cyclic polytopes; Upper Bound Theorem (McMullen 1970); simplicial polytopes; lower bounds on face numbers; simplicial polytopes and stacking; Lower Bound Conjecture (Klee/Grünbaum); proved by Barnette also in 1970
