PYTHON COURSE

The course is designed to provide an introduction to the main features on Python programming language like I/O, visualization, data structures and manipulation. Mainly we want to point on how we can easily and fast manage the most common techniques that we use on our scientific work.

After the introduction  we’ll discuss about data post processing, Montecarlo, DB interface, libraries, IDE available, IA and machine learning.

 

 

Around Random Walks, Interacting or Not (Mini-course)

Abstract

We plan to explain in six hours how to treat two phenomena involving simple random walks, and using simple estimates about them.
The treatment is going to be mathematical even though the problems arise in physics: the actual models are going
to be highly idealized, using basically random walks...and we hope to have self-contained lectures (no pre-requisites).

1. The difficulty of building long fingers if random walks move according to internal diffusion limited aggregation which is a celebrated model of erosion.
2. The difficulty of hitting colored lattice sites, if their density is low. This is rather linked with the phenomena of avoiding an acqueous solvant for long hydrophobic polymer.

GIBBS MEASURES IN CLASSICAL STATISTICAL MECHANICS AND STOCHASTIC PROCESSES MINI- COURSE

Gibbs measures are the centerpiece of rigorous studies in classical statistical mechanics.  Based on the initial ansatz of Boltzmann, Maxwell and Gibbs, these measures are at present ubiquitous probabilistic objects endowed with a rich mathematical theory and a wide range of applications.  More recently, their theory has been put in correspondence with the theory of non-necessarily Markovian stochastic proceses.  This correspondence has exhibit common aspects but also important differences that could be exploited in the analysis of random sequences (or signals). The course will start with the precise definition of Gibbs measures and the description of their main properties.  The limitations of Gibbsian theory will be subsequently explored, exposing a number of mechanisms leading to non-Gibbsian measures.  The second part of the course will focus on one-dimensional lattices and compare Gibbs measures with discrete-time stochastic processes and other measures introduced in the theory of processes and symbolic dynamical systems (Bowen measures, $g$-measures).   

DIOPHANTINE EQUATIONS

In this course, I will teach the students how to solve exponential diophantine equations using Baker's method and the Baker-Davenport reduction method.
 

• [13-3-2019] Chapter Five Dedekind rings: Noetherian ring, characterize of Noetherian ring, Kurll Dimension, Fractional ideal. (By: Florian Luca)
• [14-3-2019] Chapter Five proof In Dedekind ring The set Id(R) is an abelian group and every fractional ideal can be written as a finite product of prime ideals.(By: Florian Luca)
• [20-3-2019] Chapter Six &The Dedekind ζ-function: ????Fis a Dedekind ring, proof of the norm of two non zero ideals of ????F is multiplicative, prime decomposition of the ideal generated by p in ????fand classification to totally split, totally ramified and remains prime(inert). (By: Florian Luca)
• [21-3-2019] Fibonacci Diophantine Triples.(By: Florian Luca)

Introduction to Dynamical Systems (Mini-course)

Tentative plan (will be adapted to the audience)

1. Continuous Dynamical Systems: Weak contraction fixed point Theorem,
Gronwall’s Lemma, push forwards of measure, ODE (existence,
uniqueness, global existence, flows, Poincar´e map, stability, Linear systems,
Floquet Theory, Hartman–Grobman Theorem, Flow Box theorem,
invariant sets and measures, Poincar´e recurrence Theorem),...
2. Discrete Dynamical Systems: Definition, periodic and fixed points
(example of interval maps, Sharkovskii’s Theorem), ! ( )–limit sets, recurrent
points, transitivity (characterization of transitivity in Polish spaces),
minimality, non-wandering points.
Tentative schedule
• Their will be six(6) lectures of 2h each, holding once a week
• Starting: week of 6–10 May, 2019.
References
[1] Coddington, E., and N. Levinson. ”Theory of ordinary differential equations,
New York, 1955.” Translated under the title Teoriya obyknovennykh
differentsial’nykh uravnenii, Moscow: Inostrannaya Literatura (1958).
[2] Guckenheimer, John, and Philip Holmes. ”Nonlinear oscillations, dynamical
systems and bifurcations of vector fields.” J. Appl. Mech 51.4 (1984): 947.
[3] Katok, Anatole, and Boris Hasselblatt. ”Introduction to the modern theory
of dynamical systems.” Vol. 54. Cambridge university press, 1995.
[4] Hasselblatt, Boris, and Anatole Katok. ”A first course in dynamics: with a
panorama of recent developments.” Cambridge University Press, 2003.
[5] Zehnder, Eduard. ”Lectures on dynamical systems: Hamiltonian vector
fields and symplectic capacities.” Vol. 11. European Mathematical Society,
2010.
1

Advanced course in Noetherian and Homological Commutative Algebra

The course, composed by 8 two-hours lectures, will deal with advanced topics
of Noetherian commutative algebra, with the help of homological techniques,
following on the subjects treated in the course of AL410 (Algebra commutativa)
during the II semester of 2018/2019 a.a.
In the following are listed some of the topics that will be treated during the
course. The teacher is also willing to arrange with the students some variations
in the program in the range of topics indicated by the title.
1. Noether Normalization Lemma
2. Completions
(a) Hensel Lemma
(b) Cohen Structure Theorem
(c) Completion of 1-dimensional Noetherian domains
3. Cohen-Macaulay rings
4. Principal Ideal Theorem
5. Regular rings
6. Homological algebra: projective and injective modules, invertible modules,
projective and injective resolutions, Tor and Ext, Kaplansky's theorem.
Lectures will start after Easter. A complete schedule of lessons will be
available in the next two weeks.

For further information, please contact:
Dario Spirito (spirito@mat.uniroma3.it)
Francesca Tartarone (tfrance@mat.uniroma3.it)

Advanced Graphics for Scientific Data

Abstract:
We will introduce some useful tools for Scientific Data visualization, starting from GNUplot for drawing 1D and 2D objects, then moving to Paraview for exploring more complex features of 3D data sets.
 

HANDS ON CONTINUUM MECHANICS:FROM THEORY TO EXPERIMENTS

HANDS ON CONTINUUM MECHANICS: from THEORY to EXPERIMENTS
Continuum mechanics has been the central pillar of mathematical-physics until the mid of XIX century, attracting the most talented mathematicians. After decades of fading interest, it is now regaining an important role in both mathematics and physics, with the main boost coming from diverse fields such as bio-mechanics, nano-mechanics, electro-mechanics, and even physics-based visual effects in featured movies.

This short course, consisting of 24 hours of lectures, aims at introducing the key notions of continuum mechanics, and at giving the know-how to transfer these notions in working examples.

In particular, the students will learn the fundamentals of continuum mechanics, and will then be able to implement and solve a selection of problems by using the Finite Element Method. The distinctive feature of this course lies in its twofold nature: a rigorous mathematical treatment of the key notions of continuum mechanics, coupled with the possibility of exploring, through numerical experiments, the wide range of phenomena that can be described within this framework.

Contents.
The pillars of mechanics:
1) The Principle of Virtual power and the Balance of forces;
2) The Principle of invariance;
3) The Dissipation Principle;

Kinematics and strain measures;
Constitutive relations: from strain to stress.
The Integral and differential form of the balance equations.

Worked example: Gradus ad Parnassum
Topic 1) Linear Anisotropic Solids
Topic 2) Hyper elastic Solids
Topic 3) Linearly-viscous Fluids

References
Morton E. Gurtin, An Introduction to Continuum Mechanics, Academic Press, 1981
 

Syzygies of Algebraic Varieties (mini-course)

Alex Küronya
(Goethe-Universität Frankfurt am Main)

The topic of the course is the study of embeddings of varieties into projective
spaces. Given a very ample line bundle L on a projective variety X, the Kodaira
map associated to L gives rise to an embedding of X into some projective
space, thus realizing X as a common zero set of equations in a polynomial ring.
Syzygies of the pair (X;L) are algebraic invariants of this embedding which
describe the higher order relations among the arising system of equations.
Following a quick introduction to the basics of the subject, in particular
reviewing the necessary material from commutative algebra, we will start fo-
cusing on pairs (X;L) where the associated syzygy modules are as simple as
possible. This is made more precise by the so-called 'property (Np)', which was
first considered by Green and Lazarsfeld, and which means that the first p + 1
syzygies are linear. We will study how verifying property (Np) can be reduced
to checking the vanishing of higher cohomology of vector and line bundles, and
look at the case of abelian varieties where one obtains a surprisingly uniform
answer.
In the last part of the course we consider the case of surfaces in more detail
and see how one can characterize property (Np) in terms of forbidden subvari-
eties. The necessary prerequisites are the basics of graded rings and a working
knowledge of positivity, cohomology, and vanishing theorems for projective va-
rieties.

Topological Quantum Matter

The discovery of the quantum Hall effect and of topological insulators stimulated the interest of condensed matter physicists for the toolbox of topology and geometry, initiating a fruitful interplay between the two communities. This course will present a selection of topics from both mathematics (vector bundles, their invariants, K-theory) and physics (periodic tables of topological insulators, topological transport) in order to illustrate some basic aspects of the thriving research field on topological quantum matter.
The specifics of the program of the lectures will depend on the inclination, background and interests of the audience, and could include more advanced topics like K-theory for C*-algebras and applications to disordered topological insulators, or obstruction theory and constructive algorithms for Wannier functions.