No texbook is required.

Recommended Texts:
• Mathematics Methods of Classical Mechanics, by V. I. Arnold, Springer Ver-
lag, 2nd edition.
• Classical Mechanics, by H. Goldstein, Addison Wesley, 2nd edition.
These books are recommended but purchase of them is not required as lecture notes will be comprehensive.

This course is an introduction to applications of mathematics (in the guise of differential equations theory) to the natural sciences, in particular classical mechanics. Topics covered include Newtonian mechanics (energy, momentum, planetary motion), Lagrangian and Hamiltonian mechanics and Hamilton-Jacobi theory. Applications will include celestial mechanics, fluid dynamics and rigid body motion. Along the way topics such as differential forms, manifolds, Lie groups and Lie algebras, symplectic geometry, dynamical systems and ergodic theory will be naturally introduced.

Assessment: There will be one midterm (worth 30 points), a final exam (50 points) as well as 2 to 4 take-home problem sheets (20 points in total).

Recommended text:
Positivity in Algebraic Geometry I, II, by Lazarsfeld
Principles of Algebraic Geometry, by Griffiths-Harris
Diophantine Geometry--An Introduction, by Hindry-Silverman

Text Book: None

Reference Books : Reading assignments will be a  small set of specific chapters extracted by from the following  reference texts, as well as a few published articles.

- The Elements of Statistical Learning, Data Mining :   Freedman, Hastie, Tibshirani
- Kernel Methods in Computational Biology  :   B. Schölkopf, K. Tsuda, J.-P. Vert
- Introduction to Support Vector Machines:    N. Cristianini ,  J. Shawe-Taylor

Automatic Learning of unknown functional relationships Y = F(X) between an output  Y and high-dimensional inputs X , involves algorithms dedicated  to the intensive analysis of large "training sets"  of N "examples" of inputs/outputs pairs (Xn,Yn ), with n= 1…N to discover efficient "blackboxes" approximating the unknown function X->F(X). Automatic learning was first applied  to emulate intelligent tasks involving complex patterns identification,   in artificial vision, face recognition, sounds identification, speech understanding, handwriting recognition, texts classification and retrieval, etc. Automatic learning has now been widely extended to the analysis of high dimensional biological data sets  in  proteomics and  genes interactions networks, as well as to smart mining of massive data sets gathered on the Internet.  

The course will study  major  machine learning algorithms derived from Positive Definite Kernels  and their associated Self-Reproducing Hilbert spaces. We will study the implementation, performances, and drawbacks of  Support Vector Machines classifiers, Kernel based Non Linear Clustering, Kernel based Non Linear Regression, Kernel PCA. We will explore connections between kernel based learning  and Dictionary Learning as well as Artificial Neural Nets with emphasis on  key conceptual features  such as generalisation capacity. We will present classes of Positive Definite Kernels designed to handle the  long "string descriptions" of  proteins involved in genomics and proteomics.

Required Textbook:
Raymond H. Myers, Douglas C. Montgomery, G. Geoffrey Vining, Timothy J. Robinson
Generalized Linear Models: with Applications in Engineering and the Sciences, 2nd ed.
Wiley, 2010.
ISBN: 978-0-470-45463-3.

Recommended Textbooks:
P. MuCullagh and J.A. Nelder: Generealized Linear Models, 2nd ed. 1999 Chapman  Hall/CRC
C. MuCulloch, and Searle: Generalized, Linear, and Mixed Models. 2000 Wiley.
Faraway J. Linear Models with R. 2004 Chapman Hall/CRC

This course is designed for graduate students who have been exposed to basic probability and statistics and would like to learn more advanced statistical theory and techniques in modelling data. The selected topics will include basic regression models for continuous and categorical response variables, model validity checking, likelihood function and parameter estimation methods, variable selection methods, model selection, large sample theory, shrinkage models, ANOVA and some recent advances. Applications will include many examples in epidemiology, health science, medical studies, demography, economics, and social science.

Grading:
Final grades will be based on homework assignment (30%), two midterm exams (20% each), the final research project (presentation and written report,30%).