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MANIFESTO DEGLI STUDI
A.A. 2018/2019
CORSO DI DOTTORATO IN MATEMATICA
Approvato dal Collegio dei Docenti in Matematica in data 5 dicembre 2018Dipartimento di Matematica
Corso di Dottorato in Matematica
Manifesto degli studi
Il Corso di Dottorato in Matematica ha come finalità il fornire una qualificata preparazione a giovani
ricercatori, avviandoli a svolgere una futura attività che potrà svilupparsi sia come ricerca accademica presso
Università e Enti di ricerca pubblici e privati che nelle applicazioni industriali, economiche e sociali. Il Corso si
configura come naturale completamento della formazione scientifica conseguita con le lauree di primo e
secondo livello, con le quali è coordinata.
Il Corso di Dottorato di ricerca in Matematica è istituito ai sensi del D.M. n. 45/2013.
Il Corso di Dottorato viene proposto dal Dipartimento di Matematica. Alcune borse di studio sono offerte da
enti esterni, sulla base di convenzioni approvate.
Al corso di Dottorato collaborano, mettendo a disposizione degli allievi le proprie competenze e le proprie
strutture, i seguenti enti di ricerca nazionali e locali:
Università di Verona
FBK - Fondazione Bruno Kessler, Trento
COSBI
FEM
INdAM–COFUND-DP-2015
FAIRMAT
Indirizzi di ricerca
Il Corso di Dottorato di ricerca in Matematica, per l’anno accademico 2018/2019 è articolato nei seguenti
indirizzi di ricerca:
a. Indirizzo generale
L’indirizzo riguarda una o più delle seguenti aree scientifiche caratterizzanti:
- Calcolo delle variazioni: analisi in spazi metrici, teoria geometrica della misura, convergenze
variazionali (Gamma-convergenza), trasporto ottimo
- Analisi Geometrica, Geometria Riemanniana, Flussi Geometrici.
- Equazioni alle derivate parziali (EDP) non lineari: problemi a frontiera libera, modelli di isteresi,
comportamento asintotico ed omogeneizzazione di EDP, metodi variazionali e topologici, equazioni
di Ginzburg-Landau e Schrödinger non lineari
- Geometria analitica e geometria algebrica:
Curve algebriche e spazi di moduli.
Superfici di tipo generale e spazi dei moduli
Varietà di dimensione alta: teoria di Mori, varietà di Fano.
Geometria algebrica reale, analisi complessa e ipercomplessa.
- Fisica matematica: aspetti fondazionali, analitici e geometrici delle teorie quantistiche e relativistiche.
Tecniche geometriche in meccanica analitica; applicazioni alla teoria del controllo.
- Sistemi Dinamici e Teoria del Controllo: esistenza, molteplicità, stabilità di soluzioni periodiche di
equazioni differenziali, sistemi lagrangiani e hamiltoniani; giochi differenziali e problemi di controllo
ottimo, soluzioni di viscosità di equazioni di Hamilton-Jacobi; ottimizzazione di sistemi ibridi.
- Processi stocastici: equazioni differenziali stocastiche alle derivate parziali, Integrazione funzionale ed
applicazioni.
- Statistica: Statistica Robusta, Misure statistiche di profondità dei dati, Statistica Computazionale.
2Dipartimento di Matematica
Corso di Dottorato in Matematica
- Logica matematica ed informatica teorica: applicazione di tecniche non standard (alla A. Robinson) in
analisi funzionale, sviluppo di logiche non classiche, teoria dei linguaggi di programmazione, sistemi
di tipi, analisi statica, aspetti generali e filosofici, fondamenti, matematica costruttiva e programma
di Hilbert.
- Teoria dei gruppi, in particolare gruppi di permutazioni e p-gruppi finiti, gruppi e algebre di Lie, metodi
computazionali e applicazioni in fisica teorica. Algebra commutativa e computazionale, algebre
monomiali e strutture combinatorie ad esse associate. Algoritmi per il calcolo di invarianti algebrici e
combinatori. Teoria dei codici e crittografia. Decomposizione tensoriale, varietà secanti, algoritmi e
applicazioni alla teoria della complessità, alla quantum information e all’analisi dei dati.
Rappresentazioni di algebre, algebra omologica.
b. Indirizzo "Modellizzazione Matematica e Calcolo Scientifico (MOMACS)"
L’indirizzo è trasversale alle seguenti aree scientifiche caratterizzanti:
- Processi Stocastici: equazioni integro-differenziali e equazioni alle derivate parziali stocastiche per
modellizzazione di fenomeni fisici, biologici e finanziari.
- Metodi Numerici per Equazioni a Derivate Parziali: modellizzazione di fenomeni elettromagnetici, e
di problemi di fluidodinamica (classica e quantistica), metodi di approssimazione basati su elementi
finiti, elementi di frontiera, differenze o volumi finiti.
- Approssimazione/interpolazione numerica di funzioni multivariate: metodi efficienti ed applicazioni.
- Matematica discreta: modellizzazione in ricerca operativa, teoria dei grafi, ottimizzazione
combinatoria e applicazioni in biologia computazionale.
- Controllo ottimo, ottimizzazione: applicazioni alla scienza delle decisioni, trattamento delle immagini,
patrimonio culturale.
- Modelli matematici e computazionali in medicina: simulazione dei meccanismi fisiologici e patologici
dell'organismo umano, con particolare attenzione ai sistemi circolatorio e linfatico e alle loro
interazioni con il sistema nervoso centrale.
c. Indirizzo "Metodi algebrici e geometrici in crittografia e teoria dei codici"
L’indirizzo si propone di avviare la ricerca nel campo di una varietà di metodi matematici impiegati in
crittografia, e nella teoria dei codici a correzione d'errore, in particolare:
Metodi algebrici: algebra lineare, algebra commutativa, algebra computazionale, basi di Groebner, campi
numerici, teoria dei gruppi;
Metodi geometrici: geometria algebrica, curve ellittiche.
I problemi di ricerca proposti spaziano da classificazioni puramente teoriche a problemi vicini alla ricerca
industriale, questi ultimi complementati da stage presso aziende leader del settore.
d. Indirizzo “Biologia matematica e computazionale”
L’indirizzo si propone di avviare alla ricerca nel vasto campo dell’applicazione di modelli matematici e
computazionali nella biologia.
I metodi che verranno approfonditi possono andare dalla teoria delle equazioni differenziali ordinarie, parziali
e con ritardo o dei processi stocastici, alla statistica computazionale, alla bioinformatica, ai metodi formali per
la descrizione di sistemi complessi.
Anche le aree della biologia coinvolte sono molteplici, dall’epidemiologia ed ecologia, alle reti molecolari con
applicazioni alla farmacologia sistemica e alla nutrigenomica.
3Dipartimento di Matematica
Corso di Dottorato in Matematica
Il lavoro di tesi consisterà nella modellizzazione di un problema biologico, con l’ausilio di metodi matematici,
statistici, computazionali, informatici. Esso potrà coinvolgere altri centri di ricerca in provincia di Trento
(COSBI, FBK, FEM) o altrove.
e. Indirizzo "Mathematical applications to Quantum Science and Technologies"
L'indirizzo si propone di preparare il dottorando per attività di ricerca teorica ed applicativa su argomenti
legati alla fisica quantistica in senso lato (meccanica quantistica, quantum information, quantum field theory)
sia dal punto di vista teorico-fondazionale che applicativo che necessitino e coinvolgano tecniche
matematiche avanzate di algebra, analisi, geometria, fisica matematica, informatica e probabilità.
Lo studente potrà inoltre collaborare con il progetto di dottorato trans-disciplinare "Quantum Science and
Technology" nell'ambito del consorzio Q@TN.
Corsi attivati per l’anno accademico 2018/2019:
I corsi attivati sono contenuti nell’allegato documento “Manifesto dei corsi”. In accordo con l’art. 6 del
regolamento interno del dottorato, uno studente del I anno di corso dovrà scegliere tre corsi del manifesto:
- Algebra e logica matematica
- Analisi Matematica
- Analisi Numerica
- Geometria
- Fisica Matematica
- Probabilità e Statistica/Metodi matematici dell’economia
- Ricerca Operativa/Informatica
4Dipartimento di Matematica
Corso di Dottorato in Matematica
Allegato
Courses of the PhD School in Mathematics, a.y. 2018-19
AREA: MATHEMATICAL LOGIC AND INFORMATION SCIENCE
Set Theory (borrowed from the master degree in mathematics at Trento)
- Lecturer: Stefano Baratella (University of Trento)
- Period: February-May 2019
- Venue: Trento
- Contents: The course is an introduction to the Zermelo-Fraenkel set theory with emphasis on its links
with the mathematical practice. The goal is to become acquainted with the basic notions and techniques
from set theory so to be able to use them in the mathematical practice.
Axioms for the Zermelo-Fraenkel set theory. Well-orderings and ordinals. Equivalents of the axiom of
choice. Cardinal numbers. Cardinal arithmetic. Generalized continuum hypothesis. Consequences of set-
theoretic assumptions on the mathematical practice. The axiom of determinacy. Introduction to relative
consistency results. Constructible sets.
- Kunen K., Set theory -- An introduction to independence proofs, North-Holland.
- Jech T., Set theory -- The 3rd millenium edition, Springer
- Devlin K., The joy of sets, Springer.
Proofs and Computations (borrowed from the master degree in mathematics at Verona)
- Lecturer: Peter Schuster (University of Verona) and Daniel Wessel (University of Verona)
- Period: March-May 2019
- Venue: Verona
- Contents: Gödel's incompleteness theorems and their repercussion on Hilbert's programme, with
elements of computability theory (recursive functions and predicates, etc.). Elements of constructive set
theory.
AREA: ALGEBRA
Advanced Group Theory (borrowed from the master degree in quantitative and computational biology at
Trento)
- Lecturer: Andrea Caranti (University of Trento)
- Period: February-May 2019
- Venue: Trento
- Contents: The course builds upon the introduction to group theory given in the BSc course. Its main aim
is to give an introduction to the theory of representations and characters of finite groups.
Primitive permutation groups. Soluble and nilpotent groups. Representation and character theory of
finite groups.
- Isaacs, I. Martin . Finite group theory. Graduate Studies in Mathematics, 92. American Mathematical
Society, Providence, RI, 2008.
5Dipartimento di Matematica
Corso di Dottorato in Matematica
- Isaacs, I. Martin . Character theory of finite groups. Corrected reprint of the 1976 original [Academic
Press, New York; MR0460423]. AMS Chelsea Publishing, Providence.
Representation Theory (borrowed from the master degree in mathematics at Verona)
- Lecturers: Lidia Angeleri (University of Verona) and Francesca Mantese (University of Verona)
- Period March-May 2019
- Venue: Verona
- Contents: A first introduction to the representation theory of quivers, an important branch of modern
algebra with connections to geometry, topology and theoretical physics
Homological Methods in Representation Theory (borrowed from the master degree in mathematics at
Verona)
- Lecturers: Lidia Angeleri (University of Verona)
- Period: November 2018-February 2019
- Venue: Verona
- Contents: Two fundamental tools in homological algebra: purity and localization. The focus will lie on
their use in the current research in representation theory of algebras.
Categorical Methods in Algebra and Geometry
- Lecturers: Ryo Takahashi (Univ. Nagoya e Univ. Kansas), Peter Arndt (Univ. Düsseldorf), Jorge Vitoria (City,
Univ. of London). Ten hours each
- Period: December 2018-May 2019
- Venue: Verona
- Contents: seminar activities
AREA: GEOMETRY
Algebraic Geometry II (borrowed from the master degree in mathematics at Trento)
- Lecturers: Luis E. Sola Conde (University of Trento)
- Period: February-May 2019
- Venue: Trento
- Contents: Toric geometry studies a special type of algebraic varieties that can be described in terms of
discrete –combinatorial– data. Though small and manageable, the category of toric varieties is rich
enough to be used to illustrate many of the basic concepts of algebraic geometry: affine and projective
varieties, group actions, smoothness and singularity, divisors, line and vector bundles, etc. The course is
meant as an introduction to algebraic geometry for PhD and Master students, for which we assume
familiarity with undergraduate geometry, basic algebraic structures and complex numbers.
- Affine Varieties. Affine toric varieties. Cones and lattices. Abstracts algebraic varieties. Toric varieties.
Fans. The orbit-cone correspondence. Properties of toric varieties. Smoothness. Projective toric
varieties. Divisors and line bundles on toric varieties. Toric Surfaces. Toric Resolutions and Toric
Singularities.
6Dipartimento di Matematica
Corso di Dottorato in Matematica
- D. Cox, J. Little and H. Schenk: Toric Varieties, Graduate Texts in Mathematics 124, AMS 2009,
http://www.cs.amherst.edu/~dac/toric.html
- M. Atiyah and I. G. MacDonald: Introduction to commutative algebra, Addison-Wesley, Reading, MA,
1969.
- G. Barthel, L. Bonavero, M. Brion, D. Cox: Notes of the (first week of the) course “Geometry of Toric
Varieties”, available at:
http://www-fourier.ujf grenoble.fr/~bonavero/articles/ecoledete/ecoledete.html.
- D. Cox, J. Little and D. O’Shea: Ideals, varieties and algorithms: an introduction to computational
algebraic geometry and commutative algebra, 3rd ed., Springer, New York, 2007.
- D. Eisenbud: Commutative algebra with a view toward algebraic geometry, Springer, New York, 2000.
- D. Eisenbud and J. Harris: The geometry of schemes, Springer, New York, 2000.
- W. Fulton: Introduction to toric varieties, Princeton University Press, Princeton, NJ, 1993.
- J. Harris: Algebraic geometry: a first course, Springer, New York, 1992. [8] R. Hartshorne: Algebraic
geometry, Springer, New York, 1977.
- J. Little: Software for Toric Varieties, at:
http://mathcs.holycross.edu/~little/MathInTheMountains/MMTTSoftware.pdf.
- H. Matsumura: Commutative ring theory, Cambridge University Press, Cambridge, 1986.
- E. Miller and B. Sturmfels: Combinatorial commutative algebra, Springer, New York, 2005.
- T. Oda: Convex bodies and algebraic geometry, Springer, New York, 1988. [13] M. Reid: Undergraduate
algebraic geometry, Cambridge University Press, Cambridge, 1989.
- H. Schenck: Computational algebraic geometry, Cambridge University Press, Cambridge, 2003.
- I. Shafarevich: Basic algebraic geometry, v. 1 and 2 , Springer, New York, 1994 and 1996.
- G. Ziegler: Lectures on polytopes, Springer, 1995.
AREA: MATHEMATICAL ANALYSIS
Geometric Analysis (borrowed from the master degree in mathematics at Trento)
- Lecturer: Lorenzo Mazzieri (University of Trento)
- Period: November-December 2018
- Venue: Trento
- Contents: The fundamental equations of Riemannian Geometry. Second fundamental form. Gauss and
Codazzi equations. Variational theory of the geodesics. The length functional and the energy functional.
First and second variation formulas with applications. Introduction to minimal sufarces. First and second
variation of the area functional. Comparison Geometry, Bishop-Gromov volume comparison Theorem.
The Cheeger-Gromoll Splitting Theorem, with applications.
- T. Colding and W. Minicozzi, A course in minimal surfaces
- M. Do Carmo, Riemannian Geometry
- S. Gallot, D. Hulin and J. Lafontaine, Riemannian Geometry
- Q. Han and F. Lin, Elliptic Partial Differential Equations
- P. Li, Geometric Analysis
- P. Petersen, Riemannian Geometry
- R. Wald, General Relativity
7Dipartimento di Matematica
Corso di Dottorato in Matematica
Geometric Measure Theory (borrowed from the master degree in mathematics at Trento)
- Lecturer: Francesco Serra Cassano
- Period: February-May 2019
- Venue: Trento
- Contents: Recalls and complements of measure theory; Differentiation of Radon measures; Introduction
to Hausdorff measures and area and coarea formulas; Rectifiable sets and blow-ups of Radon measures;
Introduction to minimal surfaces and sets of finite perimeter.
Mean Field Games and Optimal Transport
- Lecturers: Fabio Bagagiolo (University of Trento) and Antonio Marigonda (University of Verona)
- Period: January-March 2019
- Venue: Trento and Verona
- Contents: In the Mean Field Games models (MFG), in the view of an optimization criterion, many (even
infinitely many) agents individually take their decisions, being anyway influenced by the behavior of all
other agents. Each single agent has only the perception of the average behavior of the others. This fact
brings to a limit model given by two partial differential equations: one (Hamilton-Jacobi) for the optimal
behavior of the single agent, the other one (transport, Fokker-Planck) for the evolution of the distribution
of the population. MFG is a recent subject of research and may suitably describe:
crowd/opinion/electrical grid dynamics, financial markets, and many others.
Optimal Transport theory (OT) deals with the optimal transportation/allocation of resources, and was
formalized firstly by G. Monge in 1781 and developed by L. Kantorovich in 1942-1948, More recently,
optimal transport methods have earned an increasing importance, both from the point of view of the
applications and from a theoretical point of view. From a modeling point of view, optimal transport
theory is useful to model complex systems, where the number of particles is so large that only a statistical
description is viable (as in statistical mechanics).
- MFG (Bagagiolo): Optimal control problems: Dynamic Programming and Hamilton-Jacobi equations.
The deterministic and the stochastic cases. Equilibrium in non-cooperative games. Mean field games:
from N players to infinitely many players. The mean field games system of PDEs. Switching mean field
games and mean field games on networks.
- M. Bardi & I. Capuzzo Dolcetta: Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellman
Equations, Birkhäuser, 1997
- A. Bressan: Noncooperative Differential Games: a Tutorial (from the website of the author)
- P. Cardaliaguet: Notes on Mean Field Games (from the website of the author)
- L. C. Evans: Partial Differential Equations, AMS, 1998.
- OT (Marigonda): Review on measure theory; The optimal transport problem in the Monge's
formulation; Relaxation of the optimal transport problem: Kantorovich's formulation; Kantorovich
Duality and its consequences; Special costs: |x-y|, h(|x-y|) with h strictly convex.; Benamou-Brenier's
Dynamical formulation of the optimal transport problem; The Wasserstein space and its differential
structure; Applications.
- L. Ambrosio, N. Gigli and G. Savaré, Gradient flows in metric spaces and in the spaces of probability
measures. Lectures in Mathematics, ETH Zurich, Birkhäuser, 2005
- F. Santambrogio. Optimal Transport for Applied Mathematicians, Progress in Nonlinear Differential
Equations and Their Applications, Vol. 87, Birkhäuser Basel, 2015.
- C. Villani. Topics in Optimal Transportation. Graduate Studies in Mathematics, AMS, 2003.
8Dipartimento di Matematica
Corso di Dottorato in Matematica
- C. Villani, Optimal transport: Old and New, Springer Verlag (Grundlehren der mathematischen
Wissenschaften), 2008
AREA: PROBABILITY AND MATHEMATICAL STATISTICS/MATHEMATICAL MODELS FOR ECONOMICS
Bayesian Statistics (borrowed from the master degree in mathematics at Trento)
- Lecturers: Pier Luigi Novi Inverardi (University of Trento) and Claudio Agostinelli (University of Trento)
- Period: February-May 2019
- Venue: Trento
- Contents: Likelihoods, Priors and Bayesian rule; Frequentist and Bayesian Approaches: Differences; How
to specify the priors: conjugate, non informative, improper, Jeffreys priors. Regular Exponential Family:
univariate and multivariate. Point estimation, Credibility intervals. Examples: binomial-beta model,
normal-inversegamma, etc.; Exchangeability. Bayes factor and model comparison. Models Averaging.
Predictive Distributions. Examples: regression model (also with G-priors). Hierarchical models. Laplace
(Normal) Approximations. Pseudo Random number generator for a given distribution.
Acceptance/Rejection and Sampling Resampling Techniques. Markov Chain Monte Carlo. Metropolis-
Hasting. Gibbs sampler. Several Examples.
- A. Gelman, J.B. Carlin, H.S. Stern and D.B. Rubin (2004) Bayesian Data Analysis, Chapman and
Hall/CRC.
- J. Albert (2009) Bayesian Computation with R, Springer
- C.P. Robert and George Casella (2010) Introducing Monte Carlo Methods with R, Springer
- D. Gamerman and H.F. Lopes (2006) Markov Chain Monte Carlo. Stochastic Simulation for Bayesian
Inference. Chapman and Hall/CRC
Empirical Processes and Applications in Statistics
- Lecturers: Claudio Agostinelli (University of Trento) and Anand N. Vidyashankar (George Mason
University, USA)
- Period: 10-23 June 2019
- Venue: Trento
- Contents: The goal of this short course is to introduce students with background in mathematical
statistics to the modern empirical processes theories. Over the past decade, the developments in
empirical process theory have proven to be powerful in working with the flexible models consisting of
both parametric and nonparametric components, e.g., the Cox proportional hazards model or
generalized additive models. However, this increased flexibility makes distribution theory quite
challenging and modern empirical process techniques are usually required.
- Kosorok, M.R (2008), Introduction to Empirical Processes and Semiparametric Inference, Springer,
New York.
- Van der Vaart (1998), Asymptotic Statistics, Cambridge U. Press
- Van de Geer (2000), Empirical Processes in M-estimation, Cambridge U. Press.
Stochastic Processes for Actuarial Sciences
- Lecturers: Stefano Bonaccorsi (University of Trento) and Ermanno Pitacco (University of Trieste)
- Period: 11-22 February 2019
- Venue: Trento
9Dipartimento di Matematica
Corso di Dottorato in Matematica
- Contents: Actuarial mathematics: origin, scope, evolution. The multistate approach. Evolution of a risk:
states, transitions, cash-flows. Applications to insurances of the person. The time-continuous Markov
model. The semi-Markov model. Splitting of states. Finding transition probabilities. Increment-
decrement tables. Premiums, reserves, expected profits. Parametric models for mortality and disability.
Examples: life insurance, life annuities, disability insurance. Representing the disability process.
Distributions of random present values. Time-discrete models vs time-continuous models. The time-
discrete Markov model. Examples
- S. Haberman, E. Pitacco (1999), Actuarial Models for Disability Insurance, Chapman and Hall / CRC
(Chaps. 1, 2, 3)
AREA: MATHEMATICAL PHYSICS
Topics in the Mathematical Physics of Quantum Theories (borrowed from the master degree in
mathematics at Trento)
- Lecturer: Romeo Brunetti (University of Trento)
- Venue: Trento
- Period: February-May 2019
- Contents: The aim of the course is to give explicit examples of constructions of states and their dynamics
in the application of C*-algebras to quantum theories. We shall mainly deal with oscillatory lattice
systems, i.e. non-relativistic many-body theories on lattices. To this end, we shall quickly recall some
basic properties of C*-algebras, mainly representation theory and C*-dynamical systems, inductive limits
and tensor products of C*-algebras, ideals in C*-algebras, AF algebras, type-I C*-algebras (postliminal),
nuclear C*-algebras. For the applications to physics, it will be recalled why Weyl C*-algebras are not well
suited for the implementation of Heisenberg kind of time evolution in presence of non trivial interactions
and that replacing Weyl algebra by the resolvent algebra of Buchholz and Grundling may help to solve
the problem. We then show how such a feeling finds its way through the explicit construction of
interesting examples of dynamics for oscillatory lattice systems in non relativistic many-body theory, and
likewise, the contruction of ground and equilibrium states (KMS states).
- Prerequisites: Knowledge of basic questions in single operator theory in Hilbert space (trace-class,
Hilbert-Schmidt and compact operators, spectral theorem for bounded and unbounded (normal)
operators) and some knowledge of basic questions in C*-algebras theory and its application to quantum
mechanics.
Mathematical Physics (borrowed from the master degree in mathematics at Trento)
- Lecturer: Enrico Pagani (University of Trento)
- Period: February-May 2019
- Venue: Trento
- Contents: The course treats some advanced topics of Mathematical Physics, and some applications to
Analytical Mechanics, Calculus of Variations, Continuum Mechanics, Special and General Relativity
Theory.
Tensor calculus, differential geometry, fiber-bundles, connections, jet-spaces, Lie groups. Analytical
mechanics. Symplectic manifolds. Lagrangian and Hamiltonian formulation of Classical Mechanics. Non-
holonomic constraints. Symmetries and conserved quantities. Geometric theory of first order partial
differential equations. Wave propagation. Calculus of Variation in presence of non-holonomic
constraints. Geometric Optimal Control Theory. Second variation, conjugate points, Maslov theory.
10Dipartimento di Matematica
Corso di Dottorato in Matematica
Continuum Mechanics. Ideal and viscous fluids. Elasticity theory. Basic notions of Special and General
Relativity Theory.
- B. A. Dubrovin, S. P. Novikov, A. T. Fomenko, Geometria Contemporanea, Editori Riuniti/Mir, 1987
- W.M. Boothby, An Introduction to Differentiable Manifolds and Riemannian Geometry, Academic
Press, 1975
- V. I. Arnold, Metodi Matematici della Meccanica Classica, Editori Riuniti, 1979
- R. Abraham, J. Marsden, Foundations of Mechanics, Benjiamin Cummings, Reading, 1978
- R. Abraham, J. Marsden, T. Ratiu, Manifolds, tensor analysis, and applications, Springer, 2003
- R. Courant, D. Hilbert, Methods of Mathematical Physics, Interscience Publ., 1937
- D. McDuff, D. Salamon, Introduction to Symplectic Topology, Clarendon Press, 1988
- S. W. Hawking, G. F. R. Hellis, The large scale structure of space-time, Cambridge U.P., 1973
- P. Olver, Equivalence, Invariants and Symmetries, Cambridge University Press, 1995
AREA: NUMERICAL ANALYSIS
Scientific Computing (borrowed from the master degree in mathematics at Trento)
- Lecturer: Michael Dumbser (University of Trento)
- Period: February-May 2019
- Venue: Trento
- Contents: This course deals with basic and advanced issues in scientific computing concerning the
efficient solution of partial differential equations arising in fluid and solid mechanics. Examples of
possible applications will concern the linear and nonlinear heat conduction equation, the Richards
equation for flow in unsaturated porous media, the Poisson equation and the incompressible Navier-
Stokes equations. If time allows, the governing equations of hemodynamics will be discussed, namely
the Euler equations in systems of compliant blood vessels.
Within this course the following topics are treated: Introduction to to MATLAB. Implementation of basic
algorithms of numerical analysis for linear algebra. Practical implementation of the standard methods
for hyperbolic, parabolic and elliptic partial differential equations. Finite volume, finite difference and
finite element schemes.
- V. Comincioli. Analisi Numerica. McGraw-Hill, 1990.
- A. Quarteroni, F. Saleri. Introduzione al Calcolo Scientifico. Springer, 2001
- E. F. Toro. Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer, 2009.
- R. J. LeVeque. Finite Volume Methods for Hyperbolic Problems. Cambridge University Press 2002.
- D. Kröner. Numerical Schemes for Conservation Laws. Wiley-Teubner 1997.
- E. Godlewski, P.A. Raviart. Numerical Approximation of Hyperbolic Systems of Conservation Laws,
Springer, 1996.
- A. Quarteroni, A. Valli. Numerical Approximation of Partial Differential Equations, Springer 1997.
AREA: OPERATIONAL RESEARCH
An introduction to Dynamic, Linear, and Integer Linear Programming. (With modeling examples and
exercises through TuringArena.)
- Lecturer: Romeo Rizzi (University of Verona)
- Period: February 2019
11Dipartimento di Matematica
Corso di Dottorato in Matematica
- Venue: Trento (if possible and requested we broadcast on Verona).
- Contents: Introduction to Linear Programming (LP). Introduction to Integer Linear Programming (LP).
Introduction to Dynamic Programming (DP). Modeling Combinatorial Optimization (CO) problems as
ILPs.
AREA: MATHEMATICAL MODELING AND SCIENTIFIC COMPUTING (MAMOSC)
Numerical Modeling
- Lecturers: Ana Maria Alonso Rodriguez (University of Trento) and David Mac Taggart (University of
Glasgow)
- Period: March-May 2019
- Venue: Trento
- Contents:
Part I: Modeling and computational electromagnetism (Alonso): Electrostatic, Magnetostatic, Eddy
current models.
Part II: Modeling and computational fluid dynamics (MacTaggart): Fluid dynamics and
magnetohydrodynamics (MHD), Fluid stability theory, Numerical modeling in MHD
- A. Bermudez, D. Gomez, P. Salgado: Mathematical Models and Numerical Simulation in
Electromagnetism, Springer, 2014.
AREA: ALGEBRAIC CODING THEORY AND CRYPTOGRAPHY
Advanced Coding Theory and Cryptography (borrowed from the master degree in mathematics at Trento)
- Lecturer: Edoardo Ballico (University of Trento)
- Period: February-May 2019
- Venue: Trento
- Contents: The course is aiming to Goppa codes, AG codes, evaluation codes, and Elliptic Curve
Cryptography.
Abelian group Cryptography, affine-variety codes, n-th root codes, Order-Domain codes. Abelian groups
Cryptography. Elliptic curves. Elliptic Curve Cryptography and Hyperelliptic Curve Cryptography.
Quantum Elliptic Curve Cryptography (if required by at least one student). Curves over finite fields;
Hasse-Weil. Goppa codes and their generalizations. Quantum Cryptography (if people is interested).
- L. Washington, Elliptic curves: number theory and cryptography, 2nd Ed. Chapman & Hall7/CRC 2006
- S. A. Stepanov, Codes on algebraic curves, Kluwer Academic / Plenum Publishers, Ney York, 1999.
- H. Niederreiter and C. Xing, Algebraic Geometry in Coding Theory and Cryptography, Princeton
University Press, Princeton, NJ, 2009
- J. Walker, Codes and curves,Codes and curves. Student Mathematical Library, 7. IAS/Park City
Mathematical Subseries. American Mathematical Society, Providence, RI; Institute for Advanced
Study (IAS), Princeton, NJ, 2000.
- M. Sala e al.; Groebner bases, coding, and cryptography; Springer; 2009.
12Dipartimento di Matematica
Corso di Dottorato in Matematica
AREA: MATHEMATICAL APPLICATIONS TO QUANTUM SCIENCE AND TECHNOLOGIES
Introduction to Entanglement and Quantum Information
- Lecturer: Sonia Mazzucchi (University of Trento)
- Period: April-May 2019
- Venue: Trento
- Contents:The course will focus on some topics of quantum theory with relevance in both theoretical
developments of quantum mechanics and its technological applications.
The interplay of quantum classical and quantum probability theory, measurement theory, entanglement
(BCHSH inequality, Kochen-Specher’as theorem, local-realism and non-contextuality) and information
theory will be presented and discussed, also in connection with some tecnological applications as
quantum generation of random numbers and quantum cryptography.
1. Knowledge and understanding skills.
Good knowledge of the basic formulation of quantum theories (notion of observable, pure and mixed
quantum state in a Hilbert space) and basic notions of classical probability
2. Ability to apply knowledge and understanding.
Ability to look for relevant literature on the subject and to understand the relevent open research
problems on the subject.
3. Autonomy of judgment.
Ability to formalize and tackle related physical problems with the correct mathematical formalism
4. Communicative Skills.
Ability to expose subjects at the oral level in the possible presentation of a topic taught at a lecture
through a public seminar.
Tensor Decomposition for Big Data Analysis (borrowed from the master degree in mathematics at Trento)
- Lecturer: Alessandra Bernardi (University of Trento)
- Period: September-December 2018
- Venue: Trento
- Contents: An introduction to big data science from the point of view of tensor decomposition.
The course will begin with concrete examples of big data problems. The central part of the course will
be based on geometric structures for modeling the extraction of information from problems of large data
collections. Part of the course will be devoted to computational aspects.
1. Knowledge and understanding skills.
Good knowledge of the basic arguments of tensor decomposition from the geometric point of view
and concrete examples of big data.
2. Ability to apply knowledge and understanding.
Inductive and deductive reasoning ability to deal with issues that are provided individually or in a
group from time to time.
3. Autonomy of judgment.
Ability to develop logical arguments and produce correct demonstrations. Ability to identify the
most appropriate methods for analyzing, interpreting, and modeling information extraction issues
from large data collections.
4. Communicative Skills.
13Dipartimento di Matematica
Corso di Dottorato in Matematica
Ability to expose subjects both at the written / computational level by carrying out exercises
handed out by the instructor both at the oral level in the possible presentation of a topic taught at a
lecture through a public seminar.
AREA: EXTERNAL SCHOLARSHIPS
An Introduction to Network Science
- Lecturer: Alex Arenas (University of Tarragona)
- Period: 15-19 April 2019
- Venue: Trento
- Contents: Network science has emerged as a branch of study focussing interest on the connectivity
interactions between elements of a system. The most central object of study in network science are the
so-called complex networks. Complex weblike structures describe a wide variety of systems of high
technological and intellectual importance. For example, the cell is best described as a complex network
of chemicals connected by chemical reactions; the Internet is a complex network of routers and
computers linked by various physical or wireless links; fads and ideas spread on the social network,
whose nodes are human beings and whose edges represent various social relationships; the World Wide
Web is an enormous virtual network of Web pages connected by hyperlinks. These systems represent
just a few of the many examples that have recently prompted the scientific community to investigate the
relationship between the topology of complex networks and the dynamics that take place on them. A
complex network is just a graph with several non-trivial topological properties, not present in simple
models of networks. Some of them are: scale-free degree distributions, high clustering coefficients (i.e.
more triangles than expected in a random network), assortativity (correlations between connected
nodes’ degrees), and community structure. On the contrary, simple graphs such as random networks or
grids show a homogeneous structure in which all nodes are almost indistinguishable, unlike what is
observed in real networks. In this course we will review the state-of-the-art in network science and put
focus on the applications, and open problems faced so far.
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