Practice

Here we list some exercises that we suggest for students to do to learn the material.

Lecture 1: From Ziegler, problems 0.0, 0.1 (second question seems tricky), 0.3, 0.5, 0.6 (first question), 0.8.
Lecture 2: From Ziegler, problems 0.12, 1.1(i), (ii), 1.3, Prove that $LaTeX: P\times Q\:is\:affinely\:isomorphic\:to\:P\times Q',\:where\:Q'\:is\:a\:translation\:of\:Q.$ $P\times Q\:is\:affinely\:isomorphic\:to\:P\times Q',\:where\:Q'\:is\:a\:translation\:of\:Q.$
Lecture 3: From Ziegler, problems 2.3, 2.4, 2.7, Determine the f-vector of a d-dim cube and of a d-dim crosspolytope.
Determine the f-vector $LaTeX: f\left(bipyr\left(P\right)\right)$ $f\left(bipyr\left(P\right)\right)$ in terms of $LaTeX: f\left(P\right)$ $f\left(P\right)$ .
Determine the f-vector $LaTeX: f\left(P\times Q\right)$ $f\left(P\times Q\right)$ given $LaTeX: f\left(P\right)\:and\:f\left(Q\right)$ $f\left(P\right)\:and\:f\left(Q\right)$ .

Lecture 4: From Ziegler, problems 2.5,2.10,2.12.
Determine the face lattices L(pyr(P)), L(prism(P)) in terms of L(P).
Determine the face lattice $LaTeX: L\left(P\times Q\right)$ $L\left(P\times Q\right)$ in terms of L(P) and L(Q). (This might need more knowledge of poset theory than some of you have)
Is the polar of $LaTeX: \Delta_2\times\Delta_2$ $\Delta_2\times\Delta_2$ combinatorially equivalent to the cylcic polytope $LaTeX: C_4\left(6\right)$ $C_4\left(6\right)$ ? Can you use the face lattice to solve this?
Lecture 5: From Ziegler, problems 8.1.iii, 8.1.iv, 8.6
Lecture 6: From Ziegler, problems 8.2, 8.12
Lecture 7: From Ziegler, problem 8.18
Lecture 8: From Ziegler, problems 8.13, 8.14, 8.15
Lecture 9-16: In this pdf. Download In this pdf.