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PhD program in
Mathematics in the Natural, Social and Life Sciences
Courses Organization and Content
Academic Year 2019–20
Contents
1 Pillar courses 2
2 Colloquia 6
3 Short courses 7
The area of Mathematics is characterized by four large macro research sectors:
1. Applied Partial Differential Equations
2. Continuum Mechanics Modeling
3. Numerical Methods
4. Probability and Statistical Mechanics
For each of the four areas we offer one “pillar” course covering a range of graduate-level topics.
First year students are required to attend all the four pillar courses and they are asked to pass
the final qualifying examination for all of them. Each pillar course consists of 60 hours and
starts the first week of November. All pillar courses are over by February.
After a one-month break in February, we offer a number of short courses covering a broad range
of different advanced topics. Examination methods for these course are typically in the form
of at-home assignments which will be given at the discretion of the lecturers. Participation to
these courses is mandatory while passing the exams is not considered compulsory.
Each student is also asked to attend the Mathematics Colloquia which take place on Thursdays
across the span of the whole academic year. The purpose of the Colloquia is to expose the
students to a range of active research topics within each macro sector and at their intersection,
that are not covered by the other courses offered at GSSI.
Examination methods for the pillar courses may vary (details are provided in the descriptions
below) and are decided by the course’s convenor. Instead, the grading system is uniform
across them and is based on the grading system defined in the European Credit Transfer and
Accumulation System (ECTS) framework by the European Commission.1
1
The ECTS grading scale is based on the class percentile of each student and each examination and looks at
each student’s performance relative to other students in the class. Following this logic, ECTS system classifies
students into the following broad groups: A (10%): outstanding performance without errors; B (25%): above
the average standard but with minor errors; C (30%): generally sound work with some errors; D (25%): fair but
with significant shortcomings; E (10%): performance meets the minimum criteria; FX: Fail – some more work
(i.e. re-sit) required before the credit can be awarded; F: Fail – considerable further work is required.
1
GSSI Gran Sasso Science Institute
Viale Francesco Crispi, 7 - 67100 L'Aquila, Italy
Tel. +39 0862 4280262 email: info@gssi.it C.F. 01984560662
At the end of each year, students are asked to take an admission exam in the form of a one-
hour presentation, which is used to asses students learning and research development progress.
Admission to the second year depends on the above requirements (positive evaluation at all
four pillar courses, active participation to all colloquia and short courses).
1. Pillar courses
Title Hours
Applied Partial Differential Equations Sara Daneri 60 hours
Paolo Antonelli
Introduction to Continuum Mechanics Roberto Verzicco 60 hours
Francesco Viola
Advanced topics in Numerical Analysis Nicola Guglielmi 60+20 hours
Francesco Tudisco
Probability and Statistical Mechanics Stefano Olla 60 hours
Enrico Presutti
Stefano Marchesani
Applied Partial Differential Equations
Lecturers
Paolo Antonelli, paolo.antonelli@gssi.it,
Sara Daneri, sara.daneri@gssi.it
Timetable and workload
Total number of hours: 60 (30 hours for each part)
Within those hours TA sessions will be organized.
Course description
This course presents the main techniques and tools developed for the study of applied PDEs,
by reviewing some results and problems in fluid dynamics and dispersive equations. In the first
part the focus will be on PDEs such as the Euler and Navier-Stokes equations and in particular
on the questions of existence, uniqueness and regularity of solutions in different settings. The
second part, focused on the general theory for nonlinear Schrödinger equations, discusses the
existence of solutions and their asymptotic behavior or possible formation of singularities. In
this way the student will get acquainted with the fundamental tools exploited in this field, such
as semi-group theory, fixed point arguments, a priori estimates and compactness arguments.
Course requirements
Basic knowledge of functional analysis, notions of Lp spaces, measure theory and Fourier spaces.
Also the knowledge of Sobolev spaces is strongly advised, eventually to be covered in a parallel
series of tutorial lectures.
2
GSSI Gran Sasso Science Institute
Viale Francesco Crispi, 7 - 67100 L'Aquila, Italy
Tel. +39 0862 4280262 email: info@gssi.it C.F. 01984560662Course content
The course will be divided in two parts, the first one focused on incompressible fluid dynamics
and the second related to the analysis on nonlinear Schrödinger equations.
• Part 1: Incompressible fluid flows. Derivation of the Euler and Navier-Stokes equa-
tions from conservation principles in the continuum hypothesis; Conserved quantities and
special solutions; Local existence of solutions for regular initial data via energy meth-
ods; Yudovich theorem on existence and uniqueness of two-dimensional solutions with
bounded vorticity; Leray-Hopf solutions of the Navier-Stokes equations; Strong solutions
and weak-strong uniqueness; Serrin’s regularity result.
• Part 2: Nonlinear Schrödinger equations. Review of basic tools from harmonic
analysis: real and complex interpolation. Derivation of effective equations for nonlinear
dispersive waves. Invariances and conserved quantities: the Noether’s theorem. Existence
of local regular solutions: the energy method. Local and global smoothing estimates asso-
ciated to the linear propagator: dispersive estimates, Strichartz estimates, Kato smoothing
estimates. The local Cauchy problem for the nonlinear Schrödinger equation in H 1 and
L2 . Global existence and asymptotic behavior for repulsive nonlinearities; scattering the-
ory. Formation of singularities at finite times: blow-up results based on virial arguments.
Stability of solitary waves: concentration-compactness. Instability of solitary waves in the
mass-critical case, universality of the blow-up profile with minimal mass.
Examination and grading
The students will be evaluated on the basis (a) a reading seminar on a research paper related to
modern developments of the topics handled during the course and (b) a written exam to assess
the skills developed during the course. The evaluation grid is: Excellent, Very Good, Good,
Sufficient, Fail.
Books of reference
• T. Cazenave, Semilinear Schrödinger equations, Courant Lecture Notes.
• A.J. Majda, A.L. Bertozzi, Vorticity and incompressible flow, Cambridge University Press.
• C. Marchioro, M. Pulvirenti, Mathematical theory of incompressible nonviscous fluids,
Springer.
• J.C. Robinson, J.L. Rodrigo, W. Sadowski The three-dimensional Navier-Stokes equations,
Cambridge University Press.
• C. Sulem, P.L. Sulem, The nonlinear Schrödinger equation. Self-focusing and wave col-
lapse, Springer.
An Introduction to Continuum Mechanics
Lecturers
Roberto Verzicco, roberto.verzicco@gssi.it
Francesco Viola, francesco.viola@gssi.it
Timetable and workload
Lectures: 60 hours
Homework assignments: 4 at 4 hours each
Final project and exam: 20 hours
3
GSSI Gran Sasso Science Institute
Viale Francesco Crispi, 7 - 67100 L'Aquila, Italy
Tel. +39 0862 4280262 email: info@gssi.it C.F. 01984560662Course description and outcomes
The students attending this course are expected to become familiar with vector spaces relevant
to continuum mechanics and to perform vector and tensor manipulations. They will be able
to describe motion, deformation and forces in a continuum, derive equations of motion and
conservation laws and use constitutive models for fluids and solids. With these tools at hand
students will be able to solve simple boundary value problems for fluids and solids. As an
application of a challenging problem of continuum mechanics, the final part of the course will
be devoted to introduce the basics of turbulence and some related computational method.
Topics
1. Reminders on Linear Algebra and Tensor Calculus
2. The Continuum hypothesis: from microscopic to macroscopic
3. Kinematics of deformable bodies
4. Eulerian and Lagrangian descriptions of motion
5. The balance laws of continuum mechanics: Conservation of Mass and Energy, Momentum
Balance
6. Constitutive Relations
7. Solid mechanics: nonlinear and linearized elasticity
8. Fluid dynamics: the Navier Stokes equations
9. An introduction to the physics of fluid turbulence
10. Energy cascade
11. Kolmogorov theory and wall turbulence
12. Basic concepts on computational methods for fluid dynamics and turbulence simulation
Examination and grading
Each student, after having delivered a written report on the final project, will be evaluated and
ranked according to the grades Excellent, Very Good, Good, Sufficient, Fail.
Suggested references
M. Gurtin, Introduction to Continuum Mechanics, Academic Press 1981
S. Pope, Turbulent Flows, Cambridge University Press 2000
Advanced topics in numerical analysis
Lecturers
Nicola Guglielmi, nicola.guglielmi@gssi.it
Francesco Tudisco, francesco.tudisco@gssi.it
Timetable and workload
Lectures: 60 hours
Labs: 20 hours
Final project assignment: 24 hours
Course description and outcomes
This course is an introduction to modern numerical analysis. The primary objective of the
course is to develop graduate-level understanding of computational mathematics and skills to
solve a range real-world mathematical problems on a computer by implementing advanced
4
GSSI Gran Sasso Science Institute
Viale Francesco Crispi, 7 - 67100 L'Aquila, Italy
Tel. +39 0862 4280262 email: info@gssi.it C.F. 01984560662numerical algorithms using a scientific computing language (such as MATLAB or Julia).
Course requirements
Calculus and basic linear algebra and numerical analysis. Previous programming experience in
any language may help.
Course content
The course will cover the following topics
• Boundary value problems (BVP): Finite differences; Variational methods; Rayleigh
Ritz Galerkin methods
• Numerical optimization: Unconstrained optimization; Gradient descent methods; Con-
jugate directions method; Constrained optimization; Penalization methods
• Iterative methods for eigenvalue problems: Power method; Subspace iteration;
Krylov subspace methods; Application to spectral clustering
• Methods for sparse linear systems: Sparse direct solvers; General projection meth-
ods; CG and GMRES; Preconditioning;
• Numerical quadrature: Order conditions; Error analysis; Superconvergence; Orthogo-
nal polynomials; Gaussian quadrature
• Linear multistep methods for ODEs: Explicit and implicit Adams’ methods; Local
error and stability; Convergence; Variable step size multistep methods; General linear
multistep methods
• Runge Kutta methods for ODEs: General form; Convergence theory; Order condi-
tions; Stability theory; A stability; B stability; Stiff problems; Von Neumann theorem;
Evolution PDEs
Books of reference
• E. Hairer, G. Wanner, S. P. Nørsett; Solving Ordinary Differential Equations I
• E. Hairer, G. Wanner; Solving Ordinary Differential Equations II
• Y. Saad; Iterative methods for Sparse Linear Systems (Free Online Version)
• Y. Saad; Numerical methods for Large Eigenvalue Problems (Free Online Version)
Examination and grading
Students will be evaluated on the basis of a written exam and computational assessment to be
taken at the end of the course. The grade scale for both the tests is: Excellent, Very Good,
Good, Sufficient, Fail.
Probability and Statistical Mechanics
Lecturers
Stefano Olla, stefano.olla@gssi.it
Errico Presutti, errico.presutti@gssi.it
Stefano Marchesani, stefano.marchesani@gssi.it
Timetable and workload
Lectures: 60 hours
5
GSSI Gran Sasso Science Institute
Viale Francesco Crispi, 7 - 67100 L'Aquila, Italy
Tel. +39 0862 4280262 email: info@gssi.it C.F. 01984560662Course description and outcomes
The course consists of two main parts:
1. Dynamics and thermodynamics through statistical mechanics. (Olla, Marchesani)
An introduction to statistical mechanics as a tool to understand thermodynamics as
emerging from the dynamics of a very large system in contact with exterior forces and heat
bath. In very simple one dimensional examples, like hard core particles or chain of oscilla-
tors, we will study their equilibrium states (Gibbs distributions) and the thermodynamic
processes than connect them.
2. Mass transport, Markov chains and Gibbs measures. (Presutti)
An introduction to phase transitions in equilibrium states, in particular on the Ising
model, with tools coming from transport theory and Markov chains.
Course requirements
Elementary probability.
Examination and grading
Written exercises during the course which will be evaluated and ranked according to the grades
Excellent, Very Good, Good, Sufficient, Fail.
2. Colloquia
We organize regular colloquia (which all the students are required to attend) and seminars
across the four research areas. We list below details of some of the colloquia planned so far,
more details and more updated informations on date, hours and rooms can be found at the
addresses:
www.gssi.it/seminars/seminars-and-events-2020/itemlist/category/201-seminars-maths-2020
www.gssi.it/seminars/seminars-and-events-2019/itemlist/category/193-seminars-maths-2019
March 19, 2020
Denis Serre (ENS Lyon)
The role of the Hilbert metric in the Lipschitz estimate for a minimal surface equation
January 21, 2020
Ricardo Grande Izquierdo (MIT, USA)
Discrete NLS-type equations and their continuum limit
January 15, 2020
Alfio Quarteroni (Politecnico di Milano ed EPFL, Lausanne)
Numerical models for multiphysics: theory, algorithms, applications
December 20, 2019
Gigliola Staffilani (MIT, USA)
Some results on the almost everywhere convergence of the Schroedinger flow
November 28, 2019
Giovanni Gallavotti (La Sapienza, Roma)
Non-equilibrium ensembles: Navier-Stokes example
6
GSSI Gran Sasso Science Institute
Viale Francesco Crispi, 7 - 67100 L'Aquila, Italy
Tel. +39 0862 4280262 email: info@gssi.it C.F. 01984560662November 14, 2019
Michele Benzi (Scuola Normale Superiori, Pisa)
Some uses of the field of values in numerical analysis
November 13, 2019
Barbara Mazzolai (IIT, Genova)
Towards a new generation of self-growing plant-inspired robots
3. Short courses
Short courses take place between March and September. Participation to the courses is manda-
tory for all first-year students.
Applied Partial Differential Equations
• M. Palladino (GSSI) and P. Cannarsa (Università di Roma “Tor Vergata”)
Optimal Control of Finite and Infinite Dimensional Systems
• P. Marcati (GSSI)
Topics in Fluid Dynamics
• Luca Alasio (GSSI)
Foundations of functional spaces
Numerical Methods
• A. Quarteroni (Politecnico Milano and EPFL Lousanne)
Numerical models for multiphysics: theory, algorithms, applications
• D. Boffi (Pavia) - Finite elements
• V. Simoncini (Bologna) - Cancelled due to covid-19 emergency
• D. Kressner (EPFL Lausanne) - Cancelled due to covid-19 emergency
• V. Noferini (Aalto, Finland)
Combinatorial Network Analysis
• L. Pareschi (Ferrara) - Cancelled due to covid-19 emergency
• P. Antonietti (Politecnico Milano)
Theory and application of discontinuous Galerkin methods for PDEs
Probability and statistical mechanics
• D. Ioffe (Technion)
Topics in the Ventsel-Freidlin theory
• S. Shlosman (University of Moscow)
Modern topics in percolation
Statistical Physiscs: Models and Applications
• G. Gradenigo (GSSI)
Condensation phenomena in quantum and classical statistical mechanics
• A. Vulpiani (Sapienza, Roma)
Transport, diffusion and front propagation
Mathematical problems in quantum mechanics
• S. Cenatiempo (GSSI)
7
GSSI Gran Sasso Science Institute
Viale Francesco Crispi, 7 - 67100 L'Aquila, Italy
Tel. +39 0862 4280262 email: info@gssi.it C.F. 01984560662Introduction to Quantum Mechanics
• S. Cenatiempo (GSSI)
Interacting bosons in the Gross-Pitaevskii regime: statics and dynamics
• A. Hannani (University Paris-Dauphine, PSL)
Hydrodynamic Limit for a Disordered Quantum Harmonic Chain
8
GSSI Gran Sasso Science Institute
Viale Francesco Crispi, 7 - 67100 L'Aquila, Italy
Tel. +39 0862 4280262 email: info@gssi.it C.F. 01984560662You can also read
