Courses in Pisa
Here you will find the courses with a major combinatorial content given in Pisa.
Algebraic Topology B
When?
Second semester
Who?
Luis Ferroni
Where?
University of Pisa
For whom?
Master and PhD students
Broadly speaking the goal of algebraic topology is to apply algebraic techniques to study topological spaces and associated invariants. This course focuses on combinatorial aspects of the subject, which emerge for instance when we model topological spaces with finite objects such as cell complexes, simplicial complexes or posets, or when we study the topology of spaces arising from discrete data. We will present several results and techniques used in this setting.
Topics covered in this course will include the following:
– Hyperplane arrangements: intersection poset, regions count, Zaslavsky’s Theorem.
Depending on time constraints and audience we will treat some of the following topics:
– Simplicial complexes, Cohen–Macaulayness, shellability and vertex decomposability;
– Poset topology and combinatorial applications;
– Discrete Morse theory.
Groups and Representations
When?
Second semester
Who?
Giovanni Gaiffi, Andrea Maffei
Where?
University of Pisa
For whom?
Bachelor, master and PhD students
Lie algebras are the infinitesimal version of Lie groups and find applications whenever a mathematical problem exhibits a continuous symmetry group, and therefore in numerous areas of mathematics as well as physics. The most important class of Lie algebras, both from a mathematical standpoint and for applications, is that of simple algebras (these are Lie algebras that have no proper ideals) or, more generally, semisimple algebras (direct products of simple algebras). Over an algebraically closed field of characteristic zero, the classification of these algebras is intimately connected to the classification of finite groups generated by reflections in a Euclidean space—an object seemingly of a very different nature. This allows one to classify Lie algebras using combinatorial objects called Dynkin diagrams, and many aspects of the theory of semisimple Lie algebras and their representations can be described starting from these diagrams.
After introducing the main examples and foundational results, the course will focus on the study of semisimple algebras and their classification. Time permitting, the final part of the course will cover the finite-dimensional representations of such algebras. A fundamental aspect that will be emphasized throughout the course is the interaction between abstract theory and the study of specific examples.
Random trees and random graphs
When?
November 26 - February 27
Who?
Alessandra Caraceni
Where?
Scuola Normale Superiore
For whom?
Master and PhD students
The course will cover various aspects of the theory of random graphs. First, we will discuss the Bienaimé–Galton–Watson tree model and their local and scaling limits. We will then move on to the Erdős–Rényi model, studying its phase transitions, before arriving at various related examples and generalizations, such as the configuration model and inhomogeneous random graphs. Finally, we will delve into the theory of random planar maps and their limits, highlighting combinatorial enumeration techniques and connections to different areas of Mathematics and Physics.