Teaching 2017-2018 Curriculum in Mathematics for Engineering
November 8th 2017 - Seminar Prof. Patera/Pontrelli
Basic Course
Reference for timetable Prof. Fabrizio Frezza (fabrizio.frezza@uniroma1.it)
Corso di scrittura tecnico-scientifica (3 CFU) Emilio Matricciani (Politecnico di Milano), gennaio-febbraio 2018
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MATHEMATICS FOR ENGINEERING
Course M1
Giovanni Cerulli Irelli, Andrea Vietri
30 ore secondo semestre (15+15 ore)
TITLE: Graphs: from combinatorics to representation theory
First part: Graceful labellings and edge-colourings of graphs.
After a general introduction to graphs, with no specific background required, the first part of the course proceeds with two distinct topics, Graceful Labellings and Edge-Critical Graphs, which have attracted much interest from decades and provide numerous open questions. Classical constructions are shown along the course. Some of the current research issues are presented together with the state of the art and the known techniques. The students are hopefully expected to give their personal contribution to the development of the themes.
Programme
Basic definitions on graphs. Topic 1) Graceful labellings. Decomposition of a complete graph using a graceful labelling. The Ringel conjecture on trees. Rosa's necessary condition. Graceful collages. Graceful polynomials.
Topic 2) Edge colouring and critical graphs. Vizing's theorem and the Classification problem. Colouring of bipartite graphs. Planar graphs.
Critical graphs. Geometrical interpretation of criticality. Construction of critical graphs.
Texts:
V.Bryant, Aspects of Combinatorics, A Wide-ranging Introduction, Cambridge University Press, 1993.
J.A.Gallian, A Dynamic Survey of Graph Labeling, Electr.J.Comb. 16, DS6 (on-line source).
Second part: Representations.
The second part is more algebraic. We will develop the theory of representations of oriented graphs (which
in this context are called quivers). This theory has been developed since the late 60s, and it is now a central
topic of research in algebra and representation theory. We will provide an introduction to the theme, starting
from basic notions of homological algebra. The goal of the course is the proof of the famous Gabriel's
theorem: "a quiver has only finitely many isoclasses of indecomposable representations if and only if it is an
orientation of a simply-laced Dynkin graph of type A, D or E". We will mainly follow the book "Quiver
Representations" by R. Schiffler. Time permitting we will also develop some basics of the theory of quiver
Grassmannians.
Prerequisites: Linear algebra.
Objectives: the category of quiver representations is a perfect object to start working with functors and
derived functors. The student will acquire familiarity with those concepts by several examples and
applications.
Standard facts of linear algebra will be applied in unexpected ways and hence rediscovered.
Exams (for both parts):
The exam will consist in the solution of weekly exercises and a short talk on a theme compatible with the
interest for the student.
Course M2
Boundary value problems in domains with irregular boundaries.
Professors Maria Rosaria Lancia e Maria Agostina Vivaldi (Sapienza Università di Roma)
20/24 hours.
Starting day: February 27; Final day: April 05; Tuesday and Thursday 14:00-16:00 room 1B1 Pal.002
Program: A list of the topics
- Variational solutions to boundary value problems
- Regularity results
- Homogenization and asymptotic analysis
- numerical approximation with the finite element method
- problems with dynamic boundary conditions
- parabolic boundary value problems
Course M3
“Nonlinear diffusion in inhomogeneous environments''
Professors
Anatoli Tedeev
(South Mathematical Inst. of VSC Russian Acad. Sci.Vladikavkaz Russia) and
Daniele Andreucci
(Sapienza Università di Roma)
May- June 2018 Tuesday-Thursday room 1B1 11:00-13:00.
Nonlinear diffusion in inhomogeneous environments
In many problems of diffusion the spatially inhomogeneous character of the medium is important and affects the qualitative behavior of the solutions.
The course will cover problems in unbounded domains of the Euclidean space or in non-compact Riemannian manifolds, providing a general introduction to the basic theory and then focusing on the asymptotic behavior of solutions for large times.
The prerequisites are standard knowledge of Sobolev spaces and basic theory of Riemannian manifolds.
Many results in this field will be recalled in the course.
1) Linear and non-linear diffusion equations; the concept of solutions and the variety of possible behaviors. The energy method.
2) Sobolev spaces on manifolds. Interplay between geometry and embedding results.
3) Criteria of stabilization for inhomogeneous linear parabolic equations.
4) Specific properties of nonlinear, possibly degenerate, diffusion.
5) The case of space-dependent coefficients; blow up of interfaces.
6) Asymptotics for large times: classical results in the Euclidean space. The asymptotic profile in linear and nonlinear diffusion.
7) The case of the Neumann problem in subdomains of the Euclidean space.
8) Asymptotic behavior in manifolds.
Course M4
An introduction to the functions of bounded variation and existence results for elliptic equations with plaplacian principal part (p greater or equal to 1) and singular lower order terms
Professors Virginia De Cicco (Sapienza Università di Roma) and Daniela Giachetti (Sapienza Università di Roma) From January to March 2018
Aula 1B1 ore 10,00-12,00 nei giorni 10, 12, 16, 19, 23, 26 di Gennaio 2018
March 27, 30 April 17 , 20, 24 May 2 - 10:00/12:00 room 1B1 Pal. RM002
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Program
The first part of the course deals with an introduction to the functions of bounded variation.
We first consider only functions of one variable, we give some definitions, some examples, we present some
characterizations, properties, and we recall a brief history. Then we consider the functions of bounded
variation of several variables: we present the definition, we point out some properties, the main theorems,
and some applications.
The second part of the course deals with some existence results for elliptic equations with p-laplacian
principal part (p greater or equal to 1) and singular lower order terms, with the following program:
1) the case p=2, mild singularity and strong singularity, definition of solutions, existence, stability and uniqueness.
2) results for the case1<p, p different from 2.
3) the case p=1.