Dal 27 aprile p.v. saranno attivati in modalità online i seguenti insegnamenti nell'ambito dell'attività formativa ad hoc per il dottorato in "Matematica e Applicazioni":

Mathematics of the Finite Element Method: Essentials for the numerical treatment of elliptic pdes - Prof. Francesco Calabrò (April 27-May 28, 2020)

Advanced Approximation Algorithms for Hard Combinatorial Optimization Problems - Prof. Paola Festa (May 4-25, 2020)

A differential geometric approach to gauge theory - Prof. A. Zampini (May-June 2020)

Fundamentals and Selected Topics in the Geometry of Tensors- prof. A De Paris (June 22-July 7)

Convective Motions Porous Media - Prof. F. Capone (July-September 2020)

Introduction to Homogenization Theory - Proff. U. De Maio; A. Gaudiello (September 14-26, 2020)

Gli interessati possono richiedere, ai rispettivi docenti, i codici Teams dei corsi.

I contenuti di ciascun corso sono riportati negli allegati di seguito, eccetto per gli insegnamenti tenuti da professori stranieri che probabilmente saranno rinviati al prossimo anno, in considerazione dell'emergenza COVID.

Mathematics of the Finite Element Method

(La Matematica del metodo degli elementi finiti: fondamenti per il trattamento numerico di problemi ellittici)

Prof. Francesco Calabrò

calabro@unina.it

List of topics (draft):

i. Essential notions on Sobolev spaces (traces in H^1, dual spaces); variational formulation of the Poisson problem (in dimension > 1) and good position (Lax-Milgram).

ii. Galerkin method, Cea lemma in the general and symmetric case; various examples of elliptic problems.

iii. Estimation of the interpolation error: definition of the interpolator; Deny-Lions theorem; related finite elements and reference element, scaling argument; error estimate for the Galerkin method in the Poisson case both in norm H^1 and L^2 (Aubin-Nitsche).

iv. First Strang lemma and quadrature error analysis for linear elements; Second Strang lemma and analysis of the error of approximation of the domain for linear elements.

v. Some implementation issues: the structure of a finite element code. vi. The diffusion-transport problem with dominant transport: exact solution and

numerical difficulties. Description of the "non-conforming artificial-diffusion" (NCAD) and "streamline-upwind Petrov-Galerkin" (SUPG) methods; error analysis for SUPG.

vii. Stokes Equation: inf-sup condition for the Babuška–Brezzi theorem. Mixed finite element methods.

viii. Darcy problem: implementation of mixed finite elements RT0-P0. ix. Isogeometric method for elliptic problems. x. Extensions and open questions.

Schedule of Lectures (To Be Confirmed):

Lecture 1: April 27, from 10 to 12; Lecture 2: April 30, from 9 to 11; Lecture 3: May 4, from 11 to 13; Lecture 4: May 7, from 9 to 11; Lecture 5: May 11, from 11 to 13;

Lecture 6: May 14, from 9 to 11; Lecture 7: May 18, from 11 to 13; Lecture 8: May 21, from 9 to 11; Lecture 9: May 25, from 11 to 13; Lecture 10: May 28, from 9 to 11.

How to Register to the Course

Students interested in attending the Course should

ü download MICROSOFT TEAMS (if they have not already done) from the web site http://softwaresso.unina.it/teams/

ü join the Team “Mathematics of the Finite Element Method - PhD course” through the following code

17e9mj9

Advanced Approximation Algorithms for

Hard Combinatorial Optimization Problems (Algoritmi Avanzati di Approssimazione per Problemi Difficili di Ottimizzazione Combinatoria)

Prof. Paola Festa

Tentative list of topics:

1. Introduction to Hard Combinatorial Optimization Problems: P vs NP, NP Optimization problems, Approximation Ratio.

2. Techniques: Duality theory; Linear programming rounding methods (randomized, primal-dual, dual-fitting, iterated rounding); Semi-definite program based rounding; Greedy and combinatorial methods; Local search; Dynamic programming and approximation schemes; Metric methods

3. Problems: Tour problems: Metric-TSP, Asymmetric TSP, TSP Path Number Problems: knapsack, bin packing Scheduling: multiprocessor scheduling, precedence constraints, generalized assignment Connectivity and network design: Steiner trees, Steiner forests, Survivable Network Design Covering problems: vertex cover, set cover Constraint satisfaction: max k-Sat Clustering: k-center, k-median, facility location Cut problems: max cut, k-cut, multicut, sparsest cut, bisection

4. Open Problems.

Schedule of Lectures:

Lecture 1: May 4, from 9 to 11; Lecture 2: May 6, from 9 to 11; Lecture 3: May 8, from 11 to 13; Lecture 4: May 11, from 9 to 11; Lecture 5: May 13, from 9 to 11; Lecture 6: May 15, from 11 to 13; Lecture 7: May 18, from 9 to 11; Lecture 8: May 20, from 9 to 11; Lecture 9: May 22, from 11 to 13; Lecture 10: May 25, from 9 to 11.

How to Register to the Course

Students interested in attending the Course must

ü download MICROSOFT TEAMS (if they have not already done) from the web site http://softwaresso.unina.it/teams/

ü join the Team “Prof.ssa Festa - Advanced Approximation Algorithms for Hard Combinatorial Optimization Problems” through the following code

5v4jl1n

ü send an email to Prof. Paola Festa (paola.festa@unina.it) informing her about the successful adhesion to the Team of the Course.

Course Program

Fundamentals and Selected Topics in the Geometry of Tensors

1 Motivations

Historical landmarks in the discovery of tensors, in particular, the stress tensorand the curvature tensor. Distinction between tensor fields and tensors as lin-ear algebraic objects. Discussion on the several formal implementations of theconcept of a tensors. Basic examples and first theoretic elements.

2 Fundamentals

Fundamental results and definitions, in particular, about coordinate represen-tation, duality, notational convention in Physics, tensor algebra, symmetric al-gebra, exterior algebra. Examples.

3 Applications

Outline of classical applicative results in Physics or Engineering, mainly regard-ing di↵erentiable tensor fields. Outline of recently found applications of tensorsas linear algebraic objects, taken from [5].

4 Tensor rank

Deeper description of applications of tensor theory, particularly in Computa-tional Complexity and ‘soft sciences’. Tensor decompositions. Symmetric tensordecompositions. Tensor rank, symmetric rank and Waring rank. Examples.

5 Intermediate results

Presentation and discussion of open problems about rank. Further technicaltools, in particular about contraction, insertion, derivation and polarization.Geometric viewpoints. Secant varieties.

6 Tensor rank and algebraic geometry

Outline of the Alexander-Hirschowitz theorem and developments about relatedproblems.

7 Apolarity

Detailed introduction to apolarity. Apolarity lemma. First applications andexamples.

8 Lower bounds on Waring rank

Detailed presentation of recent results on lower bounds on the Waring rank ofspecific forms, in particular for the set of all forms of given degree and numberof variables. Examples.

9 Upper bounds on Waring rank

Detailed presentation of recent results on upper bounds on the Waring rank offorms of given degree and number of variables. Examples.

10 Additional insights, advances and research perspectives

Deeper analysis of some aspects that has been touched upon in the precedinglectures, depending on the interest of the audience. Presentation of advancedtopics and research perspectives.

Main Sources

[1] M.C. Brambilla, G. Ottaviani, On the Alexander-Hirschowitz theorem. Jour.Pure Applied Algebra. 212:1229–1251, 2008http://dx.doi.org/10.1016/j.jpaa.2007.09.014

[2] A. De Paris, Seeking for the maximum symmetric rank. Mathematics,6(11):247, 2018http://dx.doi.org/10.3390/math6110247

[3] A. De Paris, The asymptotic leading term for maximum rank of ternaryforms of a given degree. Linear Algebra Appl. 500:15–29, 2016.http://dx.doi.org/10.1016/j.laa.2016.03.012

[4] Geramita, A.: Expose I A: Inverse systems of fat points: Waring’s problem,secant varieties of Veronese varieties and parameter spaces for Gorensteinideals. In: The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995),pp. 2–114. Kingston: Queen’s University, 1996.

[5] J.M. Landsberg, Tensors: Geometry and applications, American Mathe-matical Society (AMS), Providence, RI, 2012.http://dx.doi.org/10.1090/gsm/128

Programma

Fondamenti ed Argomenti Scelti di Geometria dei Tensori

1 Motivazioni

Passaggi storici fondamentali nella scoperta dei tensori, in particolare il tensoredegli sforzi e il tensore di curvatura. Distinzione tra campi tensoriali e tenso-ri come oggetti di algebra lineare. Discussione sulle varie formalizzazioni delconcetto di tensore. Elementi teorici ed esempi elementari.

2 Fondamenti

Definizioni e risultati fondamentali, in particolare sulla rappresentazione in coor-dinate, la dualita, le convenzioni in uso in Fisica e sulle algebre tensoriali,simmetriche ed esterne. Esempi.

3 Applicazioni

Presentazione di risultati classici in Fisica ed Ingegneria, principalmente riguar-danti campi tensoriali di↵erenziabili. Presentazione di applicazioni dei tensoricome oggetti di algebra lineare, che sono emerse di recente (tratte da [5]).

4 Rango di tensori

Approfondimenti sulle applicazioni della teoria dei tensori, in particolare allacomplessita computazionale e alle ‘soft sciences’. Decomposizione di tensori edi tensori simmetrici. Rango tensoriale, rango simmetrico e rango di Waring.Esempi.

5 Risultati intermedi

Presentazione e discussione di problemi aperti sul rango. Ulteriori strumen-ti tecnici, in particolare contrazione, inserzione, derivazione e polarizzazione.Descrizioni geometriche. Varieta delle secanti.

6 Rango tensoriale e geometria algebrica

Presentazione del teorema di Alexander-Hirschowitz e di sviluppi in ambiticorrelati.

7 Apolarita

Introduzione dettagliata all’apolarita. Lema di apolarita. Prime applicazionied esempi.

8 Limitazioni dal basso per il rango di Waring

Presentazione dettagliata di risultati recenti sulle limitazioni dal basso per ilrango di Waring di forme assegnate, in particolare per l’insieme di tutte leforme di assegnato grado e numero di variabili. Esempi.

9 Limitazioni dall’alto per il rango di Waring

Presentazione dettagliata di risultati recenti sulle limitazioni dall’alto per ilrango di Waring delle forme di assegnato grado e numero di variabili. Esempi.

10 Approfondimenti, sviluppi e prospettive di ricerca

Approfondimenti su alcuni degli argomenti a↵rontati nelle lezioni precedenti,sulla base degli interessi dei partecipanti. Presentazione di sviluppi avanzati eprospettive di ricerca.

Principali fonti di riferimento

[1] M.C. Brambilla, G. Ottaviani, On the Alexander-Hirschowitz theorem. Jour.Pure Applied Algebra. 212:1229–1251, 2008http://dx.doi.org/10.1016/j.jpaa.2007.09.014

[2] A. De Paris, Seeking for the maximum symmetric rank. Mathematics,6(11):247, 2018http://dx.doi.org/10.3390/math6110247

[3] A. De Paris, The asymptotic leading term for maximum rank of ternaryforms of a given degree. Linear Algebra Appl. 500:15–29, 2016.http://dx.doi.org/10.1016/j.laa.2016.03.012

[4] Geramita, A.: Expose I A: Inverse systems of fat points: Waring’s problem,secant varieties of Veronese varieties and parameter spaces for Gorensteinideals. In: The Curves Seminar at Queen’s, Vol. X (Kingston, ON, 1995),pp. 2–114. Kingston: Queen’s University, 1996.

[5] J.M. Landsberg, Tensors: Geometry and applications, American Mathema-tical Society (AMS), Providence, RI, 2012.http://dx.doi.org/10.1090/gsm/128

Corso di Dottorato

Geometria di↵erenziale in teorie di gauge

Lo scopo del corso e di illustrare i fondamenti di geometria di↵erenziale in teoria di gauge. I temi

delle lezioni sono:

1. varieta di↵erenziali

2. campi vettoriali, forme di↵erenziali, il calcolo esterno di Cartan

3. gruppi ed algebre di Lie

4. strutture metriche su una varieta

5. fibrati vetotriali

6. fibrati principali

7. connessioni, curvature

8. le equazioni di Yang-Mills

9. l’elettromagnetismo come una teoria di gauge

10. fibrati di Hopf e connessioni di monopolo

Graduate course

A di↵erential geometric approach to gauge theory

The aim of this course is to describe the di↵erential geometrical foundations of a gauge theory.

A plan for it is as follows

1. Di↵erentiable and smooth manifolds

2. Vector fields and di↵erential forms on a manifold, the exterior Cartan calculus

3. Lie groups and Lie algebras

4. Metric structures on a manifold

5. Vector bundles

6. Principal bundles

7. Connections and curvatures

8. The Yang-Mills equations

9. Electromagnetism as a gauge theory

10. Hopf bundles and monopole connections

alessandro zampini

1

Convective Motions in Porous Media

(Moti Convettivi in Mezzi Porosi)

Prof.ssa Florinda Capone

Tentative list of topics:

1. Mechanics of fluid flow through a Porous Medium: Porosity; Seepage velocity and equation of continuity; Momentuum equations (Darcy law, Forchheimer’s equation; Brinkman’s equation); Acceleration and other inertial effects; Boundary conditions.

2. Heat Transfer through a Porous Medium: Energy equation; Oberbeque-Boussinesq approximation; Thermal boundary conditions; Local Thermal non-equilibrium

3. Mass Transfer in Porous Media: Multicomponent flow; Mass Conservation in a mixture; Combined heat and Mass Transfer.

4. Internal Natural Convection: Horton-Rogers-Lapwood problem: Heating from below; Linear and Nonlinear Stability Analysis of the Conduction Solution; Double Diffusive Convection; Influence of Rotation on the Onset of Convection; Influence of Inertia term on the Onset of Convection; Influence of Magnetic Field on the Onset Of Convection.

5. Open Problems.

Course Period: July-September, 2020

Phd course

Umberto De Maio, Antonio Gaudiello

Time period of phd course: September 14-26, 2020

Title: Introduction to Homogenization Theory

Description: Homogenization is a mathematical theory that aims to establish the macroscopic behavior of a "microscopically" heterogeneous system, in order to describe some relevant characteristics of the heterogeneous medium (for example its electrical or thermal conductivity). Which means that a heterogeneous material is replaced by a homogeneous fictitious material (the "homogenized" material) whose overall characteristics are a good approximation of those of the original material. From a mathematical point of view this essentially means that the solutions of a family of problems that depend on a "small" parameter (boundary value problem, minimum of suitable functional), converge in some sense, with the parameter tending to zero , to a "homogenized" problem independent of the parameter (boundary value problem, minimum of a suitable functional). The course is devoted to some basic problems and methods of the homogenization theory.