Courses in Pisa
Here you will find the courses with a major combinatorial content given in Pisa. Click to get abstracts.
2023–2024
Groups and Representations
When?
Second semester
Who?
Alessandro Iraci
Michele D'Adderio
Where?
Unipi
For whom?
Bachelor, master and PhD students
After a first part covering representation theory of finite groups, including character theory, we explore in more details the representation theory of the symmetric groups and its beautiful algebraic combinatorics, via the theory of symmetric functions. The prerequisites are linear algebra and group theory, as they are studied in the first two years of the bachelor. We will assume also mathematical maturity and curiosity.
Coxeter Groups
When?
First semester
Who?
Michele D'Adderio
Mario Salvetti
Where?
Unipi
For whom?
Bachelor, master and PhD students
In 1934 the great geometer H. S. M. Coxeter introduced the groups that now go under his name as an abstraction of groups generated by reflections through hyperplanes. A prototypical example of these groups is given by the symmetric groups. In fact a lot of the related mathematical activity can be described as an attempt to generalize known results for symmetric groups to a larger family of Coxeter groups. Other classical examples are the groups of symmetries of regular polyhedra and of regular tassellations of Euclidean spaces of dimension ≤3. The theory of Coxeter groups sits naturally at the intersection of geometry, algebra, combinatorics and topology. In this course we will introduce these objects and start their study from all these four points of view.
We will prove the paradigmatic classification of finite Coxeter groups. In fact, it turns out that these are precisely the finite groups generated by reflections. Furthermore, we will study the topology of the so called Coxeter complex associated to a Coxeter group, providing an important application of the notion of shellability.
Ultrafilters and Nonstandard Methods
When?
First semester
Who?
Mauro Di Nasso
Where?
Unipi
For whom?
Master and PhD students
This course focuses on Arithmetic Ramsey Theory and on combinatorial properties of sets of integers. The two fundamental tools that will be used are the ones that give the course its title: ultrafilters and methods of nonstandard analysis. These are two techniques originated from mathematical logic: ultrafilters are fundamental objects of infinite combinatorics studied in set theory; while nonstandard analysis was developed within model theory. We will see that those two tools are actually closely related, and can be seen as two sides of the same technique.
We will prove fundamental results in this area of combinatorics, namely Ramsey’s Theorem, Hindman’s Theorem Van der Waerden’s Theorem, the Rado’s Theorem, the Jin’s Theorem and their main consequences. Basic aspects will also be presented of discrete topological dynamics on the space nonstandard natural numbers, and its applications in combinatorics of numbers. Finally, some topics at the research frontier will be presented. In fact, the main purpose of this course is to provide technique skills sufficient to address even open problems.
2022–2023
Combinatorics of the Flag Variety
When?
April
Who?
Philippe Nadeau
Where?
Unipi
For whom?
Master and PhD students
In this course I will present various combinatorial aspects of the complex Grassmannian (briefly) and flag variety (in more detail). We will first recall the classical path going from intersection problems in these varieties to algebraic and combinatorial techniques. In the Grassmannian case, integer partitions and symmetric polynomials (in particular Schur polynomials) play a fundamental role. In the case of the complete flag variety, the combinatorics is that of permutations and Schubert polynomials, which form a basis of multivariate polynomials. We will present these objects in detail, leading to current research questions about them.
Algebraic Combinatorics
When?
First semester
Who?
Michele D'Adderio
Where?
Unipi
For whom?
Bachelor, master and PhD students
After an introduction to enumerative and algebraic combinatorics, in this master course we will discuss basic methods to compute ordinary and exponential generating functions, including several fundamental examples, as well as more advanced combinatorial and algebraic tools to solve enumerative problems. While the prerequisites are minimal (nothing that is not covered in a first year bachelor), we will assume mathematical maturity and curiosity.
Discrete and Continuous Models in Probability
When?
November 2022 to March 2023
Who?
Alessandra Caraceni
Franco Flandoli
Where?
Scuola Normale
For whom?
3rd year bachelor, master and PhD students
The course will consist of two parts, relatively independent of each other.
The first 20 hours, taught by Prof. Franco Flandoli, will give a general introduction to interactive particle systems, discrete- and continuous-time Markov chains, and various probabilistic tools to analyse their behaviour.
The subsequent 20 hours, probably starting in January or February, will be taught by Dr Alessandra Caraceni and focus more specifically on discrete models in probability, developing a blend of probabilistic and combinatorial techniques. How long does it take to properly shuffle a deck of cards? We will discuss the notion of mixing time and introduce different ways to estimate it for a variety of discrete-time Markov chains. In the final part of the course, we give an introduction to branching processes and use them to prove several basic results about a celebrated model of a random graph, i.e. the so-called Erdős-Renyi graph.
Chromatic Symmetric Functions: Recent Advances
When?
First semester
Who?
Alessandro Iraci
Anna Vanden Wyngaerd
Michele D'Adderio
Where?
Unipi
For whom?
Bachelor, master and PhD students
Chromatic symmetric functions were introduced in the nineties by Stanley as an extension of chromatic polynomials of graphs, and they immediately attracted a lot of attention as they were shown to be related to Hecke algebras and Kazhdan-Lusztig polynomials. In the last few years, in an attempt to make progress on the so called Stanley-Stembridge conjecture (the most important open problem in this area, but probably also in the whole of algebraic combinatorics), a burst of activity led to the discovery of new interesting connections among chromatic symmetric functions, Hessenberg varieties and LLT polynomials.
In this course we present some of the most interesting developments that occurred in the last decade.
The prerequisites are little to none, so the course will be accessible to any student with mathematical maturity and curiosity.
2021–2022
Groups and Representations
When?
Second semester
Who?
Michele D'Adderio
Giovanni Gaiffi
Where?
Unipi
For whom?
Bachelor, master and PhD students
After a first part covering representation theory of finite groups, including character theory, we explore in more details the representation theory of the symmetric groups and its beautiful algebraic combinatorics, via the theory of symmetric functions. The prerequisites are linear algebra and group theory, as they are studied in the first two years of the bachelor. We will assume also mathematical maturity and curiosity.
Discrete and Continuous Models in Probability
When?
November 2022 to March 2023
Who?
Alessandra Caraceni
Franco Flandoli
Where?
Scuola Normale
For whom?
3rd year bachelor, master and PhD students
The course will consist of two parts, relatively independent of each other.
The first 20 hours, taught by Prof. Franco Flandoli, will give a general introduction to interactive particle systems, discrete- and continuous-time Markov chains, and various probabilistic tools to analyse their behaviour.
The subsequent 20 hours, probably starting in January or February, will be taught by Dr Alessandra Caraceni and focus more specifically on discrete models in probability, developing a blend of probabilistic and combinatorial techniques. How long does it take to properly shuffle a deck of cards? We will discuss the notion of mixing time and introduce different ways to estimate it for a variety of discrete-time Markov chains. In the final part of the course, we give an introduction to branching processes and use them to prove several basic results about a celebrated model of a random graph, i.e. the so-called Erdős-Renyi graph.
Algebraic Topology B
When?
Second semester
Who?
Filippo Callegaro
Mario Salvetti
Where?
Unipi
For whom?
Master and PhD students
Simplicial complexes are one of the simplest models of topological spaces. They allow study interaction between combinatorial data and topology. We focus the first part of the course on these objects, investigating results like Quillen Theorem A, nerve lemma and properties like shallability. Discrete Morse theory also plays an important role, leading to some applications.
Another part of the course is related to Garside theory, which generalizes relations between permutation groups and braid groups. This led to the study of many interesting families of Coxeter and Artin groups.
The final part of the course deals with hyperplane arrangements and the special relation between combinatorics and topology of these objects. Among the key topics we investigate Mobius function, deletion-restriction arguments, Zaslavsky Theorem, graphical arrangements and chromatic polynomial, Orlik-Solomon algebra and its relation with the complement of a complex arrangement.
Combinatorial Topology and Group Theory
When?
First semester
Who?
Giovanni Paolini
Where?
Unipi
For whom?
Master and PhD students
In this course (10 hours), we will cover combinatorial techniques to study topological spaces and groups. On the topological side, we are going to see how discrete Morse theory can be used to prove homotopical and homological statements through the language of partially ordered sets (posets). On the group-theoretic side, we are going to introduce Garside theory and apply it to the study of symmetric groups and braid groups (as well as their generalizations, Coxeter and Artin groups). Using discrete Morse theory, we will prove one of the main results of Garside theory, namely the construction of a classifying space (or K(π,1)) for an arbitrary Garside group. We will also mention algorithmic aspects in discrete Morse theory (related to collapsibility) and in Garside theory (the word problem). If time permits, we will also touch on current research directions in the theory of Garside groups, Coxeter groups, and Artin groups.
Mutually Enhancing Connections between Ergodic Theory, Combinatorics, and Number Theory
When?
Second semester
Who?
Vitaly Bergelson
Where?
Unipi
For whom?
PhD students
Here is the syllabus of the course (14 hours):
- Recurrences and multiple recurrences in topological and measurable dynamics.
- Furstenberg’s Principle of Correspondence and Ramsey’s Ergodic Theory.
- Problems and results involving prime numbers.
- A look at Sarnak’s conjecture.
- Some open problems and conjectures.
2020–2021
Combinatorics of Diagonal Coinvariants
When?
Second semester
Who?
Michele D'Adderio
Where?
Unipi
For whom?
Bachelor, master and PhD students
Since the pioneering works of Frobenius, Young and Schur, more than a century ago, the interplay between representations of the symmetric group, symmetric functions and the endless combinatorics arising in this context, has been a central paradigm in algebraic combinatorics, which we like to call “the threefold way”. In the last three decades, since Macdonald introduced his famous symmetric functions, there has been a plethora of new developments, all apparently related to so called “diagonal coinvariants”. The goal of this course is to provide a survey of some recent and surprising breakthroughs and conjectures, which gave rise to quite a bit of excitement in the community. The course has very little prerequisites, and should be accessible to any student with basic background in algebra (bachelor level), mathematical curiosity and open-mindedness.
Combinatorial Methods in Topology
When?
Second semester
Who?
Emanuele Delucchi
Where?
Unipi
For whom?
Master and PhD students
The study of topological objects associated to combinatorial structures is a relatively recent but fast developing two-way success story. Topological obstructions have played a major role in the solution of famous combinatorial problems (e.g., the Kneser conjecture) and, conversely, combinatorial techniques (e.g., discrete Morse theory) have led to new results in topology, both of computational and theoretical nature. This lecture takes mainly the second point of view and aims at offering an introduction to some techniques for the study of topological spaces with a strong combinatorial structure. Our main examples will include matroid complexes, order complexes of posets as well as subspace arrangements and their complements. A tentative list of the techniques that will be covered includes Quillen-type theorems, shellings, discrete Morse theory, acyclic categories. The syllabus can be adapted to the interests and the background of the participants.