Courses – CULT

2024-2025

Algebraic Topology B

When?

Second semester

Who?

Mario Salvetti

Where?

University of Pisa

For whom?

Master and PhD students

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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.

Groups and Representations

When?

Second semester

Who?

Michele D'Adderio,
Leonardo Patimo

Where?

University of Pisa

For whom?

Bachelor, master and PhD students

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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 bachelor. We will assume also mathematical maturity and curiosity.

Combinatorics and representations of 0-Hecke algebras

When?

Second semester

Who?

Vassilis Moustakas

Where?

University of Pisa

For whom?

Master and PhD students

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We will study the combinatorics and representation theory of 0-Hecke Algebras. Hecke algebras are deformations of the group algebra of the symmetric group, whose representation theory connects those of the symmetric group and the quantum groups. Hecke algebras associated with general Coxeter groups appear in diverse areas such as harmonic analysis, quantum groups, knot theory, algebraic combinatorics and statistical physics. The representation theory of the symmetric group is closely connected to the algebra of symmetric functions, Sym, through the so-called Frobenius characteristic map. Sym admits two notable generalizations: the algebra of quasisymmetric functions, QSym, and the algebra of noncommutative symmetric functions, NSym. We will explore the relationships between these algebras and representations of 0-Hecke algebras through a quasisymmetric characteristic map. These combinatorial Hopf algebras play a central role in contemporary algebraic combinatorics, aspects of which we will discuss throughout the course.

Prerequisites: Familiarity with linear and abstract algebra. Some knowledge of basic group representation theory would be helpful but is not strictly necessary. There are no combinatorial prerequisites for this course.

Polyhedral geometry: at the intersection of combinatorics, geometry, algebra and optimization

When?

Second semester

Who?

Giulia Codenotti

Where?

University of Pisa

For whom?

Bachelor, master and PhD students

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This course can serve as an introduction to fundamental discrete geometric structures; the beauty of the subject is the proximity of the fundamentals to open areas of research, and we will see related open problems and active areas of research throughout the course. We focus first on combinatorial aspects of polytopes: that is, we study the combinatorial structure of faces of polytopes, its face lattice; in the second part of the course we place the emphasis on metric and convex geometric properties, starting with (usual, Lesbegue measure) volume and refining and relating it to other central notions of convex geometry: mixed volumes, Ehrhart polynomials, covering radius. Applications in (linear, polynomial, combinatorial) optimization, algebra, tropical and toric geometry will be seen throughout the course.

2023-2024

Coxeter Groups

When?

First semester

Who?

Michele D'Adderio,
Mario Salvetti

Where?

University of Pisa

For whom?

Bachelor, master and PhD students

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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.

Groups and Representations

When?

Second semester

Who?

Michele D'Adderio,
Alessandro Iraci

Where?

University of Pisa

For whom?

Bachelor, master and PhD students

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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.

Ultrafilters and Nonstandard Methods

When?

First semester

Who?

Mauro Di Nasso

Where?

University of Pisa

For whom?

Master and PhD students

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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

Algebraic Combinatorics

When?

First semester

Who?

Michele D'Adderio

Where?

University of Pisa

For whom?

Bachelor, master and PhD students

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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.

Chromatic Symmetric Functions: Recent Advances

When?

First semester

Who?

Michele D'Adderio,
Alessandro Iraci,
Anna Vanden Wyngaerd

Where?

University of Pisa

For whom?

Bachelor, master and PhD students

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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.

Discrete and Continuous Models in Probability

When?

November 2022 to March 2023

Who?

Alessandra Caraceni,
Franco Flandoli

Where?

Scuola Normale Superiore

For whom?

3rd year bachelor, master and PhD students

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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.

Combinatorics of the Flag Variety

When?

April

Who?

Philippe Nadeau

Where?

University of Pisa

For whom?

Master and PhD students

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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.

2021-2022

Groups and Representations

When?

Second semester

Who?

Michele D'Adderio,
Giovanni Gaiffi

Where?

University of Pisa

For whom?

Bachelor, master and PhD students

CLICK TO EXPAND/COLLAPSE

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 2021 to March 2022

Who?

Alessandra Caraceni,
Franco Flandoli

Where?

Scuola Normale Superiore

For whom?

3rd year bachelor, master and PhD students

CLICK TO EXPAND/COLLAPSE

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?

Mario Salvetti

Where?

University of Pisa

For whom?

Master and PhD students

CLICK TO EXPAND/COLLAPSE

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?

University of Pisa

For whom?

Master and PhD students

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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?

University of Pisa

For whom?

PhD students

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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?

University of Pisa

For whom?

Bachelor, master and PhD students

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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?

University of Pisa

For whom?

Master and PhD students

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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.

Courses in Pisa​

Algebraic Topology B

When?

Who?

Where?

For whom?

Groups and Representations

When?

Who?

Where?

For whom?

Combinatorics and representations of 0-Hecke algebras

When?

Who?

Where?

For whom?

Polyhedral geometry: at the intersection of combinatorics, geometry, algebra and optimization

When?

Who?

Where?

For whom?

Coxeter Groups

When?

Who?

Where?

For whom?

Groups and Representations

When?

Who?

Where?

For whom?

Ultrafilters and Nonstandard Methods

When?

Who?

Where?

For whom?

Algebraic Combinatorics

When?

Who?

Where?

For whom?

Chromatic Symmetric Functions: Recent Advances

When?

Who?

Where?

For whom?

Discrete and Continuous Models in Probability

When?

Who?

Where?

For whom?

Combinatorics of the Flag Variety

When?

Who?

Where?

For whom?

Groups and Representations

When?

Who?

Where?

For whom?

Discrete and Continuous Models in Probability

When?

Who?

Where?

For whom?

Algebraic Topology B

When?

Who?

Where?

For whom?

Combinatorial Topology and Group Theory

When?

Who?

Where?

For whom?

Mutually Enhancing Connections between Ergodic Theory, Combinatorics, and Number Theory

When?

Who?

Where?

Courses in Pisa