Corso di laurea in Fisica

The course aims at providing the advanced mathematical tools necessary for later advanced physics courses. The focus is on complex analysis and harmonic analysis, with a short introduction to functional spaces. We aim also at developing students' familiarity with tha abstract way of arguing of modern mathematics utilizing in a systematic way the fundamental algebraic structures: group, ring, algebra, vector space.

Knowledge and understanding
The course introduces students to the main concepts of complex and harmonic analysis and to their use to solve integrals and differential equations. These tools are important for understanding advanced topics in all branches of physics, from quantum mechanics to electromagnetism. The student should also have assimilated the topological and algebraic concepts at the basis of the theory of complex analytic functions.

Applying knowledge and understanding

At the end of the course the student should be able to solve integrals and differential equations in the field of complex numbers, to find the Fourier decomposition of a given function, as well as to compute Fourier transforms. He/she should also be able to apply the main concepts of Hilbert spaces. Hability in calculus is a goal, but an equally important goal is the absoption of general basic mathematical concepts, regarding mathematics not as a technique to calculate things rather as an integral part of any physical theory.

Traditional classroom lectures based on the projections of the slides associated with the Course and available on this site. The teacher will illustrate the content of such slides developing some points on the board when necessary. Emphasis in lecturing is not on doing actual computations on the board, rather on explainingn fundamental concepts, highliting the crucial assumptions and illustrating the meaning of the results. We aim at enlightnening the conncetions among different topics promoting a critical and polyhedric vision of the mathematics object of study. Furthermore we try to put into a historical perspective the developments of ideas.

The exam consists of a written examination lasting 4 hours, evaluated over 30 points, based on exercises on the topics developed dutring the course. The written exam contains also some short theoretical questions partly separated partly mixed with the development of the exercises. A minimum grade of 18/30 in the Written examination is necessary to attend the Oral examination. The Oral examination is not compulsary, since the student is allowed to accept (if he/she wishes) the grade of the written text as final grade.Written and Oral examinations (if any) must be attended in the same examination period. During the examination period when two examination calls are scheduled (March-April), a written examination passed at the first examination call is still valid also for the Oral Examination at the second call; if the students attend the writtem examination at both calls, the latter call result is graded. When attending the written examination session the student is allowed and invited to bring along a copy of the Lecture Notes and of the Slides, both available at this website.

1 FUNDAMENTAL NOTIONS of ALGEBRA
1.1 Historical Remarks 1.1.1 Galois and the advent of Group Theory 1.2 Mathematical Summary 
1.3 Groups 1.3.1 Some examples 1.3.2 The group commutator 1.3.3 Conjugate elements 
1.3.4 Order and dimension of a group 1.3.5 Order of an element 1.3.6 The multiplication table of a finite group 
1.3.7 Homomorphisms 1.3.8 Isomorphisms 1.3.9 Subgroups 1.3.10 Conjugate subgroups 1.3.11 Invariant subgroups
1.3.12 Left and right cosets 1.3.13 Factor groups 1.4 Rings 1.5 Fields 1.6 Vector Spaces 1.6.1 Dual vector spaces 
1.6.2 Inner products 1.7 Algebras 1.8 Lie Algebras 1.8.1 Homomorphism 1.8.2 Subalgebras and Ideals 
1.8.3 Representations 1.8.4 Isomorphism 1.8.5 Adjoint representation 1.9 Moduli and representations.

2 DIVISION ALGEBRAS AND THE COMPLEX NUMBERS 2.1 History of Algebra and Complex Numbers.
2.1.1 The Middle Age conception of Algebra 2.1.2 The cubic and quartic equation 2.1.3 The imaginary numbers froms Descartes to Euler 2.1.4 Fields, Algebraic closure and Division Algebras 2.2 There are only two other division algebras: the quaternions and the octonions 2.2.1 Frobenius and his theorem. 2.2.2 Galois and field extensions.

HOLOMORPHIC FUNCTIONS AND THE CAUCHY INTEGRAL 
3.1 Introduction 3.2 Differential forms and integrals in the plane 3.2.1 Curvilinear Integrals in the R2 plane 3.2.2 The primitive of a differential 1-form 3.2.3 The Green-Riemann formula 3.2.4 Generalization and intepretation 3.2.5 Homotopy 
3.2.6 A fundamental theorem of algebraic topology and its intuitive consequences. 3.3 Holomorphic Functions 
3.3.1 The notion of complex structure .  3.3.2 Holomorphicity condition .  3.3.3 Cauchy theorem 3.4 Introducing Punctures and the Cauchy Integral Formula . 3.4.1 The index of a cycle in a punctured domain D  3.4.2 Cauchy's integral formula . . . . . . . . . 
3.4.3 Generalization of Cauchy integral formula when D is not simply connected. 3.4.4 Taylor series expansion of a holomorphic function. 3.5 Singular Points of Holomorphic Functions and the Laurent
expansion. 3.5.1 General consequences of the Taylor expansion of holomorphic functions. 
3.5.2 Laurent series development of a holomorphic function. 3.5.3 The singular points of holomorphic functions and their
classification  3.6 The point at infinity and the Riemann sphere 3.6.1 The notion of differentiable manifold 
3.6.2 The two-sphere and the stereographic projection 3.6.3 The annulus and its preimage on the Riemann sphere 
3.7 The automorphism group of the Riemann sphere and PSL(2;C) 3.7.1 The automorphism group of the complex plane C
3.7.2 The group PSL(2;C) and the automorphisms of the Riemann sphere 3.8 The Residue Theorem.

4 EVALUATION of INTEGRALS with the RESIDUE METHOD 
4.1 Introduction 4.2 The gallery of typologies  4.2.1 Type 1 4.2.2 Type 2 4.2.3 Type 3 4.2.4 Type 4 
4.2.5 Type 5  4.3 Conclusions 

5 LINEAR DIFFERENTIAL EQUATIONS of THE SECOND ORDER 125
5.1 Introduction 5.2 Ordinary Differential Equations 
5.3 Second order differential equations with singular points 5.3.1 Solutions at regular singular points and the indicial equation 5.3.2 Fuchsian equations with three regular singular points and the P-symbol 5.3.3 The Hypergeometric Equation and its Solutions 5.4 Topology, Groups and Differential Equations  5.4.1 The proper tetrahedral group 5.4.2 The tetrahedral group in two dimensions and the torus 5.4.3 An algebraic representation of the torus by means of a cubic 5.4.4 A differential equation enters the play and brings in a new group.

6 FUNCTIONAL and HARMONIC ANALYSIS: an Introduction 
6.1 Introduction to an Introduction 6.2 The Idea of Functional Spaces 6.2.1 Limits of successions of continuous functions 
6.2.2 A very short introduction to measure theory and to the Lebesgue integral 6.2.3 Space of square summable functions 
6.2.4 Hilbert space 6.2.5 Infinite orthonormal bases and Weierstrass theorem 6.2.6 Consequence of Weierstrass theorem 6.2.7 The Schimdt orthogonalization algorithm 6.3 Orthogonal Polynomials 6.3.1 The classical orthogonal polynomials 6.3.2 The differential equation satisfied by orthogonal polynomials 6.4 The Heisenberg group, Hermite polynomials and the harmonic oscillator 6.4.1 The Heisenberg group and its Lie algebra 6.5 Self Adjoint Operators 6.6 Fourier Series 6.6.1 Uniform convergence 6.6.2 Some Examples 6.7 Fourier Series as Harmonic Expansions

All the Mathematical Analysis courses offered during the first and second year of Bachelor Degree are preparatory to attend this course. Attendance at the course is strongly recommended. 

Students with the first letter of the surname from L to Z