Categoria: seminari

STUDENT SEMINAR, NUMBER THEORY COURSE

Date and time: Thursday, May 5th, from 14.30

Room: Aula 4

1) Daniele Casazza

Title: Analytic and geometric tecniques for Dirichlet series.

Abstract: First we find analytical properties of Dirichlet series as region of convergence and residue in a simple pole, then we define zeta function for algebraic number fields and for ray classes of a modulus. Using geometric tools as lattices and the geometric view of the action of automorphism group we estimate the number of integral ideals of given norm in a ray class.

2) Fabio Malanchini

Title: Residue formula for zeta functions on number fields.

Abstract: Calculating the volume of a particular solid using lattice point counting we find the residue of a zeta function of a ray class and an explicit formula relating class number, regulator and discriminant of an algebraic number fields to the residue at s=1 of its zeta function. Using these informations we can obtain a slightly larger region of convergence for the zeta function of a number field.

3) Gao Ziyang (Jerry).

Title: An Introduction to Arakelov Geometry: Slope Theory (PDF note of the seminar)

Abstract:
In this lecture, I aim to introduce the slope theory, which is an important tool in Arakelov Geometry. I will start from the basic definitions of Arakelov Geometry and ends in the statement and proof of the slope inequality. At last, a proof of Siegel’s Lemma, which is one of the fundamental theorems of Arithmetic Algebraic Geometry, in flavor of the slope theory will be presented as an application.

Federico Binda and Ziyang Gao
will present two seminars on Lie Algebras – Representation theory

Here is a small note of the seminar.

Binda’s abstract:
Root systems provide a relatively uncomplicated way of completely characterizing simple and semisimple Lie algebras. We follow the axiomatic approach (as in Serre [2], Humphreys [1]). We introduce bases, the Weyl group and we explain its action on the set of bases (or, equivalently, on the Weyl chambers). Finally we introduce the classification theorem, using the Cartan matrix, Coxeter graphs and Dynkin diagrams.
References:
[1] J. Humphreys (1972), Introduction to Lie Algebras and Representation Theory, GTM, Springer-Verlag, NY.
[2] J. P. Serre (1966), Algébres de Lie semi-simples complexes, W.A. Benjamin, NY.

Gao’s abstract:
In this lecture, our main subject is the root space decomposition, a useful method to describe the representations of a Lie algebra. We will mainly focus on the case of . We will first characterize all irreducible representations of in terms of highest weight, then study the general root space decomposition. The notion of root system will be introduced here. Finally, we will generalize this notion and give the definition and a few basic properties of abstract root system.
References:
[1] J.Humphreys(1972), Introduction to Lie Algebras and Representation Theory, GTM-9, Springer-Verlag, NY.
[2] Jinpeng An, Lecture Notes, on-line version: ftp://162.105.69.120/teachers/anjp/李群与齐性空间