B9048 - TOPOLOGIA COMBINATORIA

Learning outcomes

By the end of the course, students will be familiar with lattices, polytopes, simplicial complexes, and other fundamental structures in combinatorics, and will be able to associate certain notable algebras with them. They will be able to apply these techniques to the computation of topological and geometric invariants, for instance in the case of hyperplane arrangements.

Course contents

• Partially ordered sets (posets), incidence algebra, lattices, Möbius function, polynomial invariants associated with posets.
• Simplicial complexes and CW complexes: geometric realization, triangulations, barycentric subdivision, order complex of a poset.
• Polytopes.
• Shellability of posets, simplicial complexes, and polytopes.
• Hyperplane arrangements: intersection poset, characteristic polynomial, complements of real and complex arrangements, Salvetti complex.
• Graphic arrangements and the chromatic polynomial of a graph.
• Matroids: geometric lattices, characteristic polynomial, and connections with hyperplane arrangements.
• The braid arrangement, the braid group, and an introduction to Coxeter groups.

Prerequisites: basic notions of topology and algebra.

Recommended prerequisite courses: the compulsory courses of the Bachelor's degree are sufficient. Having taken Algebraic Topology may be useful, but is not necessary.

Readings/Bibliography

D. Kozlov, Combinatorial Algebraic Topology

R. P. Stanley, Enumerative Combinatorics

G. M. Ziegler, Lectures on Polytopes

P. Orlik, H. Terao, Arrangements of Hyperplanes

J. Oxley, Matroid Theory

A. Björner, F. Brenti, Combinatorics of Coxeter Groups

M. L. Wachs, Poset Topology: Tools and Applications

O. Papini, Appunti di Matematica Discreta

Teaching methods

Lectures at the blackboard.

Assessment methods

Oral exam on theory and exercises.

During the exam, the use of AI tools and any electronic device is prohibited.

Office hours

See the website of Giovanni Paolini