250164 VO Discrete & Convex Geometry and Singular Homology (2023W)

In this course we will study mathematical objects which can be defined by finitely many data and which have a strong geometric flavor. Whereas the first instance allows for explicit computations and constructions, the second serves as a guideline for designing proofs, as we can use our geometric intuition and perception. As such, the theory is very concrete (though often highly non-trivial). In addition, the problems we study pop up frequently in other circumstances, e.g., algebra, combinatorics, number theory or even analysis. As for the concepts and topics we will look at, we will choose an attractive selection out of the following list:

Lattices and lattice points, Pick's theorem about the number of lattice points in planar polygons; Pommersheim's theorem about the number of lattice points in tetrahedra; Ehrhart's polynomial about the dilatation of polytopes and the lattice points count; the LLL-algorithm.

Polytopes, slicing and decomposition, Hilbert's third problem and Dehn's solution using the Dehn-invariant.

Convex and non-convex polyhedra, rigidity and flexibility, Cauchy's theorem computing the surface area of a polytope via projections on hyperplanes; the flexidron of Connelly.

Brion's theorem about the generating function of polyhedra.

The Euler formula for polytopes: number of faces minus number of edges plus number of vertices; the Euler characteristic of triangulated objects; the f- and h-vectors counting these numbers; the upper bound conjecture and its proof using commutative algebra.

Simplicial complexes, singular homology, homological algebra.

Convex geometry, Helly's theorem on the intersection of convex sets; mixed volumes; Hadwiger's theorems; the theorem of Bernstein-Koushnirenko about the number of solutions of polynomial equations using convex geometry; Minkowski's and isoperimetric inequalities, the theorem of Steiner-Minkowski about epsilon neighborhoods of polytopes, Rothe's theorem about non-convex polygons; regular and archimedian solids, Schläfli-symbols.

Billiards and their trajectories, sphere packings.

Cristallographic groups; tilings and wallpaper groups.

Lattice walks and generating functions.

Toric geometry, monomially generated algebras and binomials ideals.

Knot theory, invariants of knots, Reidemeister moves, higher dimensional knots.

Phylogenetic trees and their five characterizations.

Sperner's Lemma to prove Brouwer's fix point theorem.

Schubert-calculus and enumerative geometry.

Note: We recommend to interested students the complementary course "Convex Analysis" by Radu Bot, focussing more on the analytic side of convexity.