• Recap on Machine Learning approaches and principal metrics to assess the model’s performance.
• The bias-variance trade-off.
• Classifiers, SVM, Kernel methods.
• Regularization and the Curse of Dimensionality.
• Tree models.
• Recap on Deep Learning: gradient descent, backpropagation, initialization, optimization, layers, activations.
• The Gradient vanishing issue.
• Principal Architectures: Convolutional Neural Networks, ResNets, Transformers and more recent approaches.
• Recurrent Neural Networks.
• Approaches to make Deep Neural Networks more efficient: pruning, quantization, batch norm fusion, knowledge distillation.
• Application and specific designs: semantic segmentation, self-supervision, generative models.

• Calculus of variation, the basic problem: fixed endpoints conditions, no constraints, scalar case. The Euler equation. Comments and classical examples. The hamiltonian formalism.
• Extensions of the classical problem: several variables, variable endpoints (transversality conditions), holonomic and nonholonomic constraints (Lagrange multipliers).
• Introduction to control systems (represented by ordinary differential equations). Reachable set and the controllability problem.
• The optimal control problem, general formulation. An example: vertical landing. Approach to the optimal control problem by classical calculus of variation methods (under severe restrictive assumptions). Approach by dynamic programming principle.
• General approach to the optimal control problem: statement of the Pontrjagin maximum principle. Applications: vertical landing revisited and Zermelo navigation problem.
• Linear control systems and the quadratic regulation problem (on the infinite horizon).
• Minimal time control between any two states for a normal time-invariant linear system.
• Optimal control of the motion of a satellite's attitude.

References 

• A. Bressan, B. Piccoli; Introduction to the Mathematical Theory of Control, AIMS books.

Part I - Cosmology (S. Camera)

• Introduction to cosmology and the study of the cosmic large-scale structure in the context of the present observational effort. Case study for the module: the European Space Agency's Euclid satellite mission.
• Heuristic derivation of the equations governing the evolution of the universe: Friedmann's equations and the cosmological continuity equation. A bird's eye view on Friedmann models and the current best-fit to data: the concordance cosmological model.
• Linear theory of cosmological perturbations. Evolution of the density contrast of matter fluctuations.
• Introduction to theory of random fields and N-point correlators.  The matter power spectrum and its significance.

References:

• Daniel Baumann, Cosmology, Cambridge University Press, 2022, ISBN 9781108838078
• Dragan Huterer, A Course in Cosmology - From Theory to Practice, Cambridge University Press, 2023, ISBN 9781316513590
• Scott Dodelson & Fabian Schmidt, Modern Cosmology, Elsevier, 2020, ISBN 9780128159484

Part II - Exoplanets (D. Gandolfi)

•  A brief introduction to exoplanets.
• How to find an exoplanet: direct and indirect detection methods.
• Transit photometry & Doppler spectroscopy: the royal road to exoplanet discovery and characterization.
• The exoplanet zoo: populations, demographics, and architectures of exoplanetary systems.
• Exoplanets from space: CoRoT, Kepler, K2, TESS, CHEOPS, Gaia, JWST.
• The future of exoplanet research from space: PLATO and ARIEL

References:

• "Handbook of Exoplanets". Editors: Deeg, Hans J., Belmonte, Juan Antonio. ISBN 978-3-319-55332-0. Springer International Publishing AG, part of Springer Nature, 2018.
• The Exoplanet Handbook". Michael Perryman. 2018. Cambridge University Press; Second Edition, 952 p., ISBN: 9781108419772.

Part I - Orbit determination

• Introduction to the "orbit determination" problem.
• The classical methods of Laplace and Gauss.
• Charlier's theory on the occurrence of alternative solutions in Laplace's method.
• The least squares method, the linear least squares.
• Confidence ellipsoids: conditional and marginal ellipsoids.
• Propagation of the covariance.
• The method of the admissible region.
• Recent methods for the linkage problem.

References 

• A. Milani, G. F. Gronchi, "Theory of Orbit Determination", Cambridge University Press, 2010.
• G. F. Gronchi, G. Baù, S. Marò. Orbit determination with the two-body integrals. III. Celestial Mechanics and Dynamical Astronomy, 123:105–122, 2015.
• G. F. Gronchi, G. Baù, A. Milani. Keplerian integrals, elimination theory and identification of very short arcs in a large database of optical observations. Celestial Mechanics and Dynamical Astronomy, 127:211–232, 2017.

Part II - Orbits perturbations

• Dynamical models: the restricted 2-body problem,the perturbed 2-body problem.
• Numerical integrations: numerical methods, semi-analytical propagations.

References 

• A. Vallado, Fundamentals of Astrodynamics and Applications (Space Technology Library). 4th Edition, Springer, 2007, ISBN-13 978-0387718316 (Chapters 8 and 9).
• R. H. Battin, An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition, AIAA Educational Series, Reston, 1999 (Chapter 10).
• H. Curtis, Orbital Mechanics for Engineering Students, Second Edition (Aerospace Engineering). 2nd Edition, Butterworth-Heinemann, 2009, ISBN-13 978-0123747785 (Chapter 12).
• C. Colombo, F. Letizia, E.M. Alessi, M. Landgraf, “End-of-life Earth re-entry for highly elliptical orbits: the INTEGRAL mission”, The 24th AAS/AIAA Space Flight Mechanics Meeting, Jan. 26-30, 2014, Santa Fe, New Mexico.
• D. Finkleman D., C. Colombo, P. Cefola, “Analysis of the suitability of analytical, semi/analytical and numerical approaches for important orbital dynamics tasks”, 65th IAC, 2014, Toronto, Canada, IAC14.C1.2.6.

Part III - Optimization in space engineering

• Some brief considerations on optimal control (direct method point of view): optimal control paradigm, indirect/direct methods, and collocation.
• Basic concepts: reminders of linear algebra/calculus, convexity, quasi-convexity, and non-convexity.
• Fundamentals of (finite-dim) optimization: constrained optimization, convex problems, Lagrange multipliers (with economic interpretations), KKT conditions (with interpretations).
• Complements and practical exercises 1: feasibility and non-convexity, gyroids, McCormick’s envelope, recursion, Fermat's (minimum distance) problem, and an optimization problem (sphere packing) very easy to state, but very difficult to solve.
• Basics on mathematical programming: linear, quadratic, combinatorial, and mixed-integer problems, hints on the complexity theory.
• Complements and practical exercises 2: heuristics and mixed-integer applications.

References

• Floudas and Pardalos (eds.): Encyclopedia of Optimization. Kluwer Academic Publishers, Dordrecht, The Netherlands (2009)
• Hillier and Lieberman: Introduction to Operations Research. McGraw-Hill (2014)
• Minoux and Vaida: Mathematical programming: theory and algorithms. Wiley (1986)
• Nemhauser and Wolsey: Integer and Combinatorial Optimization. Wiley (1988)
• Pintér: Global Optimization in Action. Kluwer (1996)
• Williams, H.P.: Model Building in Mathematical Programming. Wiley (2013)

• Introduction to cryptography
• Symmetric cryptography: basic schemes, One Time Pad, AES
• Asymmetric cryptography: Diffie-Hellman, RSA, elliptic curves, digital signatures, cryptanalysis of these schemes
• Post-quantum cryptography: quantum computing, cryptographic schemes based on lattices and coding theory
• Hash functions
• Introduction to blockchain
• The example of Bitcoin
• Introduction to blockchain 2.0
• Examples of blockchain applications: decentralized finance, Internet of Things

References

• A. M. Antonopoulos, Mastering Bitcoin, O'Reilly, 2017
• D. Boneh, Twenty Years of Attacks on the RSA Cryptosystem, https://crypto.stanford.edu/~dabo/pubs/papers/RSA-survey.pdf
• D. Boneh, V. Shoup, A graduate course in applied cryptography, http://toc.cryptobook.us/, 2020.
• T. Duong, J. Rizzo, The CRIME Attack, https://docs.google.com/presentation/d/11eBmGiHbYcHR9gL5nDyZChu_-lCa2GizeuOfaLU2HOU/edit.
• J. H. Silverman, The arithmetic of elliptic curves, Springer-Verlag, 1985.

Part I - Mathematical methods

The course gives an introduction to a statistical analysis of scientific data, with a focus on astronomy.
Working on a real dataset, the following topics will be discussed:

• regression analysis
• time series analysis,

using the R software. Some prerequisites are required (basics of probability and statistics)

References

• Priestley: “Spectral Analysis and Time Series”, Academic Press 1981
• Shumway, Stoffer: “Time Series Analysis and Its Applications With R Examples”,
  Springer 2016
• Cryer, Chan: "Time Series Analysis with applications in R", Springer 2008
• Feigelson, Babu: “Modern Statistical Methods for Astronomy (with R applications)”,
  Cambridge University Press, 2012

Part II - Physical methods

This module will illustrate the basic elements of the data analysis of space based scientific instruments.
The different aspects of the data analysis, from the on board preprocessing to the ground reconstruction and high level analysis, will be discussed. During the classes some practical examples will be given. 

In particular we will propose the following contents:

• description of onboard data processing and ground reconstruction for few
  space based instruments
• tools for high level analysis (instrument response function generation,
  spectrum unfolding)

Slides and material presented during the classes will be shared.

The objectives of this course is to provide basic knowledge of detectors and instruments developed for scientific payloads of space missions. The course will illustrate design elements and operational constraints of specific example observatories, as determined by the required scientific performance.

The course will focus on the following subjects:

• Instruments and detection principles (12h) which include:
  • a) interaction between matter and radiation;
  • b) detection technologies;
  • c) space equipments and instrumentation.
• Cosmic radiation: experiments and results. This module will include also multi messenger studies with networks of observatories. (14h)
• Space environment and space debris. (6h)

Useful websites

• https://ams.nasa.gov
• https://fermi.gsfc.nasa.gov
• https://ixpe.msfc.nasa.gov/

• Reference Frames for Space Sciences, construction and materialization of celestial reference systems and coordinate frames: the ICRS/F, BCRS, and GCRS
• General Relativity (GR) and Space Sciences: fundamentals of the GR theory of measurements in weakly curved space-time, Solar System metrics, Space-Time coordinate transformations in GR
• The measurement/observation protocol in GR. Photon paths, relativistic observables from within the gravitational field of the Solar System, inverse ray-tracing methods, astrometric lensing and relativistic effects. The relativistic observers in space and relativistic satellite attitudes
• Global and differential astrometry. Relativistic observation equations: the O-C equations. Local focal plane coordinates. S/N of the instrumentation aboard satellites. Instrumental calibration. Mathematical methods for the solution of large least squares problems
• Examples in testing GR and Local Cosmology

G. Sartor

• Understanding Geospatial Data and Remote Sensing
• Applications of Remote Sensing
• Deep Learning Applications for Remote Sensing (Python Programming Session)

R. Renzulli

• Intro: What is generative AI?
• Autoregressive Models and Variational Autoencoders
• Generative Adversarial Networks
• Normalizing Flows
• Diffusion Models
• Hands-on training session

R. Fioresi

Introduction to Deep Learning, basic steps of the algorithm analogies with the human visual systems and its mathematical models.

The geometry of the space of data and the space of parameters; KL divergence and its information geometry interpretation.

Geometric Deep Learning: the algorithm of Deep Learning on Graphs.

Message passing and GATs: a geometrical modelling via heat equation and laplacian on graphs.

This course will be self-contained as much as possible. The necessary differential geometric concepts (manifolds, Frobenius theorem, Cartan formalism) will be introduced and explained. The necessary programming skills will NOT be assumed, but a part of the course will be "hands on" illustrating key examples on colab.

• Introduction to Machine Learning. Error measures. Bias - variance trade off. 
• Classification and regression. The theoretical learning framework. Empirical risk and generalization error.
• Linear models. Regularization methods. Spline regression. Logistic regression.
• Linear discriminant analysis.
• Neural Networks

Part II

• Spherical approximation methods, algorithms and codes in Matlab and C/C++ languages, applications to satellite data.
• Machine Learning and radial kernels.

• The deep learning approach. 
• The deep learning general framework to object classification. 
• Convolutional neural networks. 
• Object detection and applications.

References 

• G. Fasshauer, Meshfree Approximation Methods with MATLAB, Interdisciplinary Mathematical Sciences - Vol. 6, World Scientific Publishers, Singapore, 2007
• G. Fasshauer, M. McCourt, Kernel-based Approximation Methods using MATLAB, Interdisciplinary Mathematical Sciences - Vol. 19, World Scientific Publishers, Singapore, 2015.

• Introduction to space engineering and space mission design
• The space environment (solar system and interstellar medium): hazards and eects
• Human factors in Astronautics, with focus on long duration space missions: physiological/physical,
  psychological, habitability
• Thermal control: fundamentals and technology options
• Environmental Control & Life Support System (ECLSS): fundamentals and examples of systems
  to provide a safe & comfortable place for humans in deep space missions
• Elements of astrodynamics and space trajectories, space structures & mechanisms, attitude
  determination and control
• Elements of special relativity and astronautical applications: relativistic rocket equation, relativity
  and mechanics, relativity and thermodynamics, relativistic spaceflight profiles, effects of space
  environment at relativistic speeds, etc
• Space propulsion systems: cold gas, chemical (solid, liquid, hybrid), electric (electrothermal,
  electrostatic, electromagnetic), and focus on advanced (sails, energetic beams, radiations, nuclear,
  antimatter, ramjet, exotics solutions)
• Electrical power system: power source, energy storage, power management & distribution
• Characteristics and main elements of the ground segment: support to space mission communications and operations
• Data handling and communications architecture, with focus on solutions for deep space communications: gravitational lenses, data compression with KLT, etc
• Payload and experiments: elements of design and examples 13. Risk management and reliability for space missions

• Numerical analysis recap: interpolation of functions; approximation of integrals and derivatives; solution of linear systems; approximation of ordinary differential equations.
• Partial differential equations: classification.
• The Poisson equation: analytical methods (existence & uniqueness; separation of variables); numerical methods (weak formulation; Sobolev spaces; the finite element method).
• The heat equation (numerical methods: hints).
• The advection-diffusion equation, with dominant transport (numerical methods: hints).

References 

• A. Quarteroni. Numerical Models for Differential Problems. Springer-Verlag Italia; second edition (2014).

• Schwarzschild spacetime and its geodesics
• Orbits around a black hole
• Propagation of light signals around a black hole
• Synchronization in General Relativity
• Spherically Symmetric Stars in General Relativty

References

• Relativity: Special, General, and Cosmological , W. Rindelr
• The Mathematical Theory of Black Holes, S. Chandrasekhar
• Gravitation, C.W. Misner, K.S. Thorne, J.A. Wheeler