Complex Networks

Large Scale Image Analysis for Natural and Life Sciences

Principles of imaging modalities (optical microscopy, spectroscopy, CT, MRI, PET, SPECT) and their applications in natural and life sciences (Dharmakumar); Basics of image analysis (filtering, segmentation, detection) and basics of statistical mining; Designing robust image analysis methods; Large-scale analysis; Integration with databases and knowledge sharing platforms; Error testing and precision bound repetition studies for longitudonal and group studies (phenotyping); High performance computing for imaging (computer vision); Scientific and data visualization; Prerequisites: Probability an

Convex Optimization

The course aims at giving a modern and thorough treatment of algorithms for solving convex, large-scale and nonsmooth optimization problems. Applications of convex optimization. Convex sets, functions and optimization problems. Optimality conditions. Basic algorithms for unconstrained optimization (gradient, fast gradient and Newton methods). Basic algorithms for constrained optimization (Interior point and active set methods). Subdifferential and conjugate of convex functions. Duality. Proximal mappings. Proximal minimization algorithm. Augmented Lagrangian Method.

Algorithmics

This course covers basic and advanced foundations, problems and solutions of algorithmic computation. A first part offer an overview of the fundamental notions of algorithm analysis and recalls algorithmic solutions (and their complexity) for some basic problems like sorting and searching. The second part of the course will focus on advanced algorithms which are essential in some of the research fields relevant to the different curriculum of the Computer Decision and System Science track.

Network Theory

Course description: Basic of Graph Theory: degree, clustering, connectivity, assortativity, communities. Analysis of Complex Networks, datasets and software. Community Detection, Modularity, Spectral Properties. Fractals, Self-Organised Criticality, Scale Invariance. Random Graph, Barabasi Albert Model, Fitness model, Small world. HITS Algorithm and PageRank. Real instances of Complex Networks in Biology and Social Sciences. Board of Directors, Ownership Networks, measures of Centrality and Control. World Trade Web, Minimal Spanning Trees, Competition and Products spaces.

Management of Complex Systems: Approaches to Problem Solving

Methods and approach to problem solving. Problem analysis; analysis of complex systems (related to cultural heritage, such as a city of art organization, promotion, etc.). The course will include practical simulations. The course will be linked to a seminar on specific Case studies.

Advanced Topics of Complex Networks

This course will be organized as series of reading groups or specialized seminars by members or collaborators of the research unit on Natural Networks (Networks).

Theory and Numerics of Ordinary and Partial Differential Equations

The first lesson of the course will provide a primer on complex variables. Using this mathematical formalism, the focus of the remaining first part of the course will be to introduce linear ordinary and linear partial differential equations, and the "cheap" methods to solve them using Fourier and Laplace transforms. The ordinary and partial differential equations will be placed into a context of applied mathematics (e.g. classic deterministic and stochastic systems) saving the theoretical approach for advanced lectures.

Stochastic Processes and Stochastic Calculus

This course aims at introducing some important stochastic processes (Markov chains, martingales,
Poisson process, Wiener process) and Ito calculus.
Some proofs are sketched or omitted in order to have more time for examples, applications and
exercises.
In particular, the course deals with the following topics:

Statistics Lab.

- Brief introduction to R (http://www.r-project.org/)
- Creating random variables.
- Applications to the central limit theorem and the law of large numbers
- Descriptive statistics: (i) Representing probability and cumulative distribution functions in discrete and continuous cases; (ii) calculating mean, variance, concentration indexes, covariance and correlation coeff.
- Statistical inference: (i) Point estimation and properties; (ii) interval estimation and properties; (iii) hypothesis testing and properties.