Istituzioni di Didattica, Logica e Storia della Matematica

Tema 1. Didattica: ( docente benedetto Scoppola , sem II)
- algoritmo di euclide in aritmetica
(massimo comun divisore) e nella teoria
dei numeri (frazioni continue)
- proporzioni ad elementi interi: i risultati
del libro VII
- potenze e proporzioni continue
- incommensurabili.
Tema 2. Logica:( docente Lorenzo Tortora de Falco, sem II)
- Soddisfacibilità e dimostrabilità. Il
teorema fondamentale dell'analisi
canonica.
- Gentzen e l'eliminazione del taglio
- Dimostrazioni e programmi: la
corrispondenza di Curry-Howard
SI
- Introduzione alla Logica Lineare.
Tema 3. Storia:( docente Enrico Rogora, sem I)
- Lagrange
- Il teorema di Ruffini Abel
- Il lavoro di Abel sulle equazioni e sulle
funzioni ellittiche
- Il contributo di Galois
- Le soluzioni analitiche delle equazioni di
quinto grado
- La teoria di Galois-Klein

Time series analysis

1. We will recall the basic principles of applied and numerical Fourier analysis: Fourier series and transform, energy and power spectrum, mutual and autocorrelation and their numerical computation.
2. Impulse and harmonic response of a system.
3. Filtering of a time series.
4. Time series as sampling of a continuous signal.

Advanced Topics in Mathematical Physics and Probability: Renormalization Group and Critical Phenomena in Statistical Mechanics

The goal of the course is to provide an introduction to perturbative Renormalization Group in the context of $phi^4_d$ scalar field models in dimensions $dge 4$. We will discuss the notions of: scaling dimensions; relevant, marginal and irrelevant operators; effective potentials; flow equation and beta function. We will show that the quartic interaction is irrelevant in dimensions $dge 5$ and marginal with negative beta function (sometimes called `marginally irrelevant'), in dimension $d=4$. Time permitting, we will discuss the definition and construction of the non-trivial fixed point for the $phi^4_d$ model in dimension $d=4-epsilon$ (Wilson-Fisher fixed point).
NOTE: It is recommended to attend the course in presence. Nevertheless, there will be the option to attend it remotely. The interested students are invited to contact Prof. Alessandro Giuliani (at alessandro.giuliani@uniroma3.it) to enroll to the course and/or for additional informations.

Geometry and Mechanics

The goal of these lessons is to show the relashionships between Continuum Physics and Differential Geometry, starting from the fundamentals of Mechanics. In particular, we show how any typical theory of mathematical-physics is based on two layers: the physical layer – the phenomenon under investigation, and the mathematical model used to represent the physics. We shall discuss in detail some model problems, starting from the theoretical point of view, up to some noteworthy solutions; for each model, we shall show the dual role, mathematical and physical, of the notions that are used. Model problems will be selected together with the students, and chosen among:  
·       Active Soft Matter;
·       Liquid Crystals;
·       Solid - Fluid interactions.
 
Synopsis
- A continuum body as a differentiable manifold.
- Geometric elements; change of densities.-
- Geometric meaning of Divergence and Gradient
- Tell the difference between tensors: strain tensor versus stress tensor.
- Pull back & push forward of scalar, vector and tensor fields.
- Principle of virtual power:  the notion of force as a power gauge.
- Power versus Energy.
- Dissipation principle.
- Frame invariance.