Categoria: seminari ALGANT

ALGANT student seminar – Representation Theory

Date and time: Wednesday, March 21st, at 13:30

Room: Aula Chisini

Dott. ELIA CHINELLATO
Title: Roots systems
Abstract: The goal of this talk is to give a brief introduction to roots systems in an euclidean space, explaining some fundamental concepts, such as reflections in an euclidean space, the axioms for roots systems and the Weyl group.  Examples will be given in euclidean spaces of dimension 1 and 2. The final part is dedicated to the definition and the construction of a base of a roots system.

Dott. CHIARA DAMIOLINI
Title: Projective representations
Abstract: The lecture will regard some applications of projective representations of finite groups. The main result will be the Schur’s theorem concerning central extensions of finite groups with projective lifting property, which is a very useful instrument to deduce information about the group. If there is more time we will see, using character triples, other results about characters induced by normal subgroups.

Dott. FRANCESCO MONOPOLI
Title: Levi’s and Ado’s theorems
Abstract: In this seminar I will prove the two theorems mentioned in the title. Levi’s theorem states that every Lie algebra can be written as the direct sum of its radical and a semisimple subalgebra, while Ado’s theorem says that every Lie algebra has a finite-dimensional faithful representation.

Dott. GIANPIETRO COLATO
title: Groups in categories
abstract: in the first part of the seminar we introduce the basic categorical language needed to define Lie groups and Lie algebras categories. In the second part we describe some results of the theory of smooth manifolds in terms of functors, natural transformations and universal constructions, focusing in particular on the Lie functor, which associates a Lie group to its Lie algebra of smooth left-invariant  vector fields.

Anyone interested is invited.

ALGANT STUDENT SEMINAR – NUMBER THEORY COURSE

Date and time: Monday, June 13th, at 12.30

Room: aula 9

Andrea Berti

Title: Adeles, Ideles and a first look at Global Class Field Theory.

Abstract: This lecture is divided into two parts.
In the first part we will develop tools for an adelic/idelic approach to number theory. More in detail, we will introduce restrict topological product to define Adeles/Ideles, we will state and discuss Strong Approximation Theorem and give an Idelic interpretation of the Ray Class Groups of a number field. Time permitting, we will also give, as application, either a proof of the finiteness of the Class Group of a number field or of the Dirichlet’s Unit Theorem.
In the second part we will introduce the Global case of Class Field Theory, trying to understand (at least the statement of) the Global Reciprocity Law.

References:
[1] Cassels, J. W. S., and A. Froehlich, Algebraic Number Theory, Academic Press (1967).
[2] J.S. Milne Class Field Theory, Online notes avaible at http://www.jmilne.org/math/CourseNotes/cft.html
[3] GJ Janusz, “Algebraic Number Fields,” Academic Press, New York (1973).

ALGANT Student seminar
Date and time: Wednesday, June 8th, at 2pm
Room: Aula Dottorato
Lecturer: Ziyang Gao

Title: The loop and suspension functors & Fibration sequences (PDF)
Abstract: In this lecture, we’ll discuss at first the analogy of fundamental group in abstract homotopy theory and represent it in terms of loop and suspension functors. Then we will go on discussing the fibration sequences in the derived model category. We will basically follow Quillen’s Homotopical Algebra.

Everyone is invited.

ALGANT Student seminar – Commutative Algebra

Date and time: Thrusday, May 26th, at 2pm
Room: Aula 6

Lecture 1: Riccardo Ghezzi
Title: Cohen-Macaulay rings and Gorenstein rings.
Abstract: In this lecture we will show the main properties of Cohen-Macaulay rings, as their good behaviour in terms of dimension, and of Gorenstein rings. In particular, introducing the definition of injective hull, we will characterize Gorenstein rings as Cohen-Macaulay rings of type 1.

Lecture 2: Francesco Monopoli
Title: Unique factorization in regular local rings
Abstract: The goal of this seminar is to prove that any local regular ring is a unique factorization domain. To prove this theorem I will use methods and results coming from commutative and homological algebra.

Anyone interested is invited.

ALGANT Student seminar – Commutative Algebra

Date and time: Wednesday, May 25th, at 4pm

Room: Aula Dottorato

Lecture 1: Federico Binda
Title: the Intersection formula (PDF)
The goal of this lecture is to present the minimum background material that is necessary to understand Serre’s definition of intersection multiplicity, at least in the case of two irreducible subvarieties of an n-dimensional affine space. Following [2], we will quickly review Samuel’s definition of multiplicity of an ideal. Then we will see how the homology modules of the Koszul complex are related to the Tor functor and in which way they can be used to give a “new” definition of multiplicity, as in [1]. We will conclude the talk proving that Serre’s definition is the “right one”, at least in the case of plane curves, and, if time permits, we will also give an example that should convince that the higher Tor are actually useful.
References:
[1] J. P. Serre (2000), Local Agebra, revised edition. Springer Monographs in Mathematics.
[2] P. Roberts (1998), Multiplicities and Chern Classes in Local Algebra. Cambridge Univ. Press. Cambridge.

Lecture 2: Ziyang Gao
Title: Grothendieck-Serre Duality (PDF)
Abstract: This lecture can be viewed as a preliminary part for the next lecture I will give about global Grothendieck duality. In this seminar, we will talk about Koszul complexes of both local and global versions. At the end of the talk, a non-trivial theorem about the canonical bundle of a regular closed immersion will be presented.

Anyone interested is invited.