Courses in Pisa
Here you will find the courses with a major combinatorial content given in Pisa.
Algebraic Combinatorics
When?
First semester
Who?
Michele D'Adderio
Where?
University of Pisa
For whom?
Bachelor, master and PhD students
The course starts with an introduction to enumerative combinatorics, focusing on generating functions. 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. In the second part of the course we will study some fundamental combinatorial structures like posets, simplicial complexes, graphs and matroids, associate algebraic objects to them, and illustrate some of the many connections to other fields of mathematics, like representation theory, algebraic topology and algebraic geometry.
Elements of Representation Theory
When?
Second semester
Who?
Leonardo Patimo
Where?
University of Pisa
For whom?
Bachelor, master and PhD students
Algebraic and Combinatorial aspects of Tropical Geometry
When?
Second semester
Who?
Maria Angélica Cueto
Where?
University of Pisa
For whom?
Master and PhD students
This is a 16-hour course on algebraic and combinatorial aspects of Tropical Geometry, an emerging field bridging combinatorics, algebraic geometry, and non-Archimedean analytic geometry, with applications to many other areas. Classical algebraic varieties become polyhedral complexes which remember certain invariant of their algebraic counterparts. In the first part of the course, we will study “embedded” tropical varieties: we will tropicalize a subvariety of the algebraic n-dimensional torus over a non-Archimedean valued field. Topics include: structure of tropical varieties, the fundamental theorem of tropical geometry, tropical linear algebra and matroid theory (including the spaces of phylogenetic trees), and applications to computations like elimination and implicitization. We will revisit many classical enumerative results and see how tropical geometry methods can rediscover these classical count. We will loosely follow the textbook [19] as well as several research articles on the subject, listed in the references. In the second part of the course we will study tropicalizations from an abstract perspective and in connection with Berkovich non-Archimedean analytic spaces and classical moduli spaces. In the curve case, abstract tropical curves become vertex-weighted metric graphs. We will discuss their relation to algebraic and non-Archimedean analytic curves. Our goal would be to construct the moduli spaces of tropical curves and study the theory of divisors on such curves, known to combinatorialists as chip-firing on graphs or sandpile models.
The Geometry and Topology of Matroids
When?
Second semester
Who?
Luis Ferroni, Lorenzo Venturello
Where?
University of Pisa
For whom?
Master and PhD students
In this course we will study matroids from the perspective of combinatorial topology and discrete geometry. To a matroid one can associate several geometric objects: shellable simplicial complexes, convex polytopes and, under certain assumptions, hyperplane arrangements or algebraic varieties. We will study the essentials of the interplay among these objects, paving the way for anyone who is interested in pursuing later a more detailed study of matroid Hodge theory.
Probability on graphs
When?
Second semester
Who?
Alessandra Caraceni
Where?
Scuola Normale Superiore
For whom?
Master and PhD students
The course will explore various aspects of the behavior of random processes on graphs, emphasizing the synergy between probabilistic and combinatorial arguments in yielding surprising results with rich applications. The main topics covered will be:
– Random walks on graphs and their long-term behavior; the analogy with electrical network theory, Pólya’s theorem, and the type problem;
– A dialogue between discrete and continuous: Donsker’s theorem, Green’s functions, harmonic measure, and Brownian motion;
– The discrete Gaussian free field;
– Uniform spanning trees and uniform spanning forests: Wilson’s algorithm, matrix-tree theorem, local limits;
– Percolation and related topics;
– Mixing time of Markov chains.