## Laboratory AGM: Analyse, Géométrie, Modélisation

Besides, a science made solely in view of applications is impossible; truths are fecund only if bound together. If we devote ourselves solely to those truths whence we expect an immediate result, the intermediary links are wanting and there will no longer be a chain.

H. Poincaré, The value of science

The master is a national diploma that can be obtained in two years after the bachelor degree. The Master of Mathematics of the University of Cergy-Pontoise is a high-level training whose purpose is threefold: to give a solid foundation in mathematics with an introduction to research, to train experts in industry, and to prepare for the Agrégation – a national competition for becoming teachers at high schools. The Master of Mathematics is attached to the Laboratory AGM – an internationally recognised research group with broad scientific interests.

The master has three directions – research, applications and teaching, which determine the main career opportunities. Students following the directions of research or applications can apply for a grant to pursue their PhD studies. However, most of the students of these two directions will head towards the labor market after their master. The teaching orientation aims to prepare for Agrégation (Option B: scientific computing) and is restricted essentially to French nationals.

First year: M1
Second year: M2
Secretariat
Apply for the Master of Mathematics

#### First year: M1

The courses take place from September to May and are organised into two terms. Students must validate at least 30 ECTS per term, that is, 60 ECTS in total.

First term

The schedule will be available in the beginning of the first term. Click on the title of a course to have more detail on it.

• Functional analysis and PDEs (7.5 ECTS)

• Lecturer: Dr. François Fillastre

Contents

1. Hilbert spaces
2. Banach spaces
3. Laplace abd Poisson equations
4. Wave equation
5. Heat equation
6. Introduction to numerical methods for PDEs

References:

• L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 2002.
• K. Yosida, Functional Analysis, Springer-Verlag, Berlin, 1978.

• Advanced differential calculus and applications (7.5 ECTS)

• Lecturer:
Prof. Emmanuel Hebey

Contents

1. Euclidean differential calculus
2. Differential calculus in Banach spaces
3. The big theorems of differential calculus in Banach spaces
4. The concept of a differentiable manifold
5. Tangent space and differentials
6. Elements of tensor calculus and higher-order differentials

References

• H. Cartan, Cours de calcul différentiel, Hermann, 2007.
• M. Berger, B. Gostiaux, Géométrie différentielle. Variétés, courbes et surfaces, PUF, 2013.
• T. Aubin, A course in Differential Geometry, AMS, 2001.

• Dynamical systems (7.5 ECTS)

• Lecturer:
Dr. Irina Ignatiouk

Contents

1. Preliminaries on some techniques for solving ordinary differential equations
2. Cauchy-Lipschitz theorem. Local and global existence, uniqueness, and continuous dependence on problem data
3. The concepts of flow, phase space, orbit, period, equilibrium point, and stability
4. Linear systems: solution, equilibrium point, stability, classification of equilibrium points for linear systems of dimension 2
5. Link between stability of a nonlinear equation and linearised problem. First integral, Lyapunov function. Hamiltonian systems. Examples arising in physics, biology, and dynamics of populations
6. Poincaré-Bendixon theorem (in dimension 2)

References

• M. W. Hirsch, S. Smale, Differential Equations, Dynamical Systems, and Linear Algebra, Academic Press, New York–London, 1974.
• J.-L. Pac, Systèmes dynamiques. Cours et exercices corrigés, Dunod, Paris, 2012.
• C.-M. Marle, Systèmes dynamiques. Une introduction, Ellipses, Paris, 2003.

• Probability theory (7.5 ECTS)

• Lecturer: Dr. Smail Alili

Contents

1. Preliminaries on the measure theory and integration
2. Gaussian vectors
3. The $0-1$ law. Borel-Cantelli lemma and applications to the almost sure convergence for sequences and series of random variables
4. Conditional expectation
5. Discrete-time martingales
6. Markov chains with a countable state space

References

• P. Barbe, M. Ledoux, Probabilité, EDP Sciences, Paris, 1999.
• M. Cottrel, C. Duhamel, V. Genon-Catalot, Exercices de probabilités, Cassini, Berlin-Paris,1980.
• D. Dacunha-Castelle, M. Duflo, Probabilités et statistiques, Masson, Paris, 1982-83.
• L. Mazliak, P. Priouret, P. Baldi, Martingales et chaînes de Markov, Hermann, Paris, 1998.
• J. Neveu, Martingales à temps discret, Masson, Paris, 1972.

All the courses of the first term are mandatory.

Second term

The schedule will be available in the beginning of the second term. Students have to follow the courses Numerical analysis and Statistics and to choose two other courses with at least 12 ECTS (completing them, if necessary, with the short course How to find an Internship of 1 ECTS). In addition, they have to do Memoir/Internship (6 ECTS).

• Numerical analysis (6 ECTS)

• Lecturers: Prof. Françoise Demengel (lectures) and Dr. Christian Daveau (tutorials)

Contents

1. Diagonalisation and trigonalisation for normal matrices
2. Gauss and LU decompositions. Iterative methods for solving linear systems, methods of Jacobi, Gauss-Seidel and relaxation
3. Iterative methods for computing the eigenvalues of a symmetric matrix
4. Partial differential equations in dimension $1$. Approximation by finite differences for linear partial differential equations in dimension $1$. Convergence of a method
5. Nonlinear partial differential equations in dimension $1$. Theoretical study of their solvability
6. The spaces $H^1 (]0,1[)$ and $H_0^1 (]0,1[)$, existence of a solution by using a variational formulation: application to the problem $$-u^{\prime \prime } + j^\prime (u) = f, \quad u(0)= u(1)=0,$$ where $j$ is an appropriate convex function
7. The spaces $W^{1,p} (]0,1[)$ and $W_0^{1,p}(]0,1[)$ with $p\ge 1$. Application to solving the equation for the $p$-Laplacian with $p>1$

References

• P.G. Ciarlet, Introduction à l'analyse numérique matricielle et à l'optimisation: Cours et exercices corrigés, Dunod, 2007.

• Statistics (6 ECTS)

• Lecturer: Dr. Irina Ignatiouk

Contents

1. Statistical models
2. Basic concepts of point estimation: bias, quadratic risk, convergence. Construction of estimators: methods of moments and the maximum likelihood
3. Optimal estimation: exhaustive statistics, Rau-Blackwell theorem, complete statistics, Lehmann-Scheffé theorem, Fisher information,  Cramer-Rao inequality. Exponential models
4. Bayesian estimation, minimax estimator
5. Confidence intervals
6. Statistical tests
7. Linear models

References

• D. Dacunha-Castelle, M. Duflo, Probabilités et statistiques I, Masson, Paris, 1982.
• D. Fourdrinier, Statistique inférentielle : cours et exercices corrigés, Dunod, Paris, 2002.
• X. Milhaud, Statistique, Belin, 2001.
• A. Monfort, Cours de statistique mathématique, Economica, 2001.
• G. Saporta, Probabilités, analyse des données et statistiques, Technip, 2011.

• Calculus of variations, convex analysis, and optimisation (6 ECTS)

• Lecturer: Prof. Elisabeth Logak

Abstract: This course is an introduction to the optimisation, that is, investigation of extrema for real-valued functions. The concept of convexity (of sets and functions) plays a central role, and the convex analysis constitutes a substantial part of this course. The final chapter deals with some basic tools of the calculus of variations.

Contents

1. Convex functions of one or several variables
2. Optimisation under constraints: the Lagrangian, the Kuhn-Tucker conditions. Gradient algorithm with fixed step
3. Topology and geometry of convex sets: projection; separation properties. Cones and duality. Legendre transform
4. Introduction to calculus of variations: the Euler-Lagrange equation; applications to the isoperimetric and Dirichlet problems

References

• R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, NJ, 1970.
• J.B. Hiriart-Urruty, C. Lemaréchal, Fundamentals of Convex Analysis, Springer, New York, 2001.

• Programming in Scilab, R, C, C++ (5 ECTS)

• Lecturer: Christian Daveau

• Algebra (6 ECTS)

• Lecturer: Dr. Michela Varagnolo

Abstract: In the first par of this course, we study (finite) groups. We prove a structure theorem in the Abelian case and study also the general situation, for which we establish the Sylow theorem. We turn then to the theory of representation of gourps over the fild of complex numbers. The second part is devoted to extensions of fileds and algebraic elements. We establish the existence and uniqueness of the closure of a field and study in more detail the case of finite fields.

Contents

1. Action of a group on a set, class equation for a group action on a finite set, Burnside formula. Sylow theorem
2. Abelian groups of finite type: structure theorem
3. Generators and relations for a group. Simple groups
4. Representation theory for a finite group over the field of complex numbers. Table of characters.
5. Extensions of fields, degree formula. Algebraic elements on a field. Rupture and splitting fields. Algebraic closure. Finite fields

References

• N. Jacobson, Basic algebra I, Dover, 2009.
• Y. Kosmann-Schwarzbach, Groupes et symétries : Groupes finis, groupes et algèbres de Lie, représentations, Éditions de l'École Polytechnique, 2006.
• J.-P. Serre, Représentations linéaires des groupes finis, Hermann, 1998.
• D. Perrin, Cours d'algèbre, Ellipses, 1998.
• P. Tauvel, Cours d'algèbre, Dunod, 2000.

• Stochastic analysis (6 ECTS)

• Lecturer: Prof. Eva Löcherbach

Contents

1. Revision: Continuous-time processes, martingale laws, consistency theorem, and Kolmogorov continuity theorem; application: existence of the Brownian motion
2. Construction of the Itô integral and its properties (Itô isometry, integration up to a stopping time, maximal inequality, Lenglart inequality)
3. Itô's formula and applications
4. Stochastic differential equations (SDE), existence and uniqueness under Lipschitz and linear growth hypotheses, Picard's iterations, Gronwall's lemma
5. Diffusions and the strong Markov property. Generator of an SDE and Dynkin's formula. Forward and backward equations, the Feynman-Kac formula
6. Outlook : Large-time behaviour of diffusions. Example: Ornstein-Uhlenbeck processus

References

• D. Nualart, Lecture Notes on Stochastic Processes, Personal web page of D. Nualart
• B. Øksendal, Stochastic Differential Equations. An Introduction with Applications, Berlin: Springer, 1998.

#### Second year: M2

The courses take place from September to May and are organised into two terms. Students must validate 30 ECTS per term, that is, 60 ECTS in total.

First term

The schedule will be available in the beginning of the first term. All courses are optional, and students need to get at least 30 ECTS.
• Quasi-periodic Schrödinger operators: spectral theory and dynamics (10 ECTS)

• Lecturer: Prof. Raphaël Krikorian

Abstract
:
The aim of this course is to study spectral properties of 1D Schrödinger operators with a quasiperiodic potential. An important tool in this approach is the investigations of the dynamics of the associated Schrödinger cocycles. This approach, which has proven its efficiency in the last twenty years, is the basis of the recent spectacular results by A. Avila.

Contents

1. Spectral theory of self-adjoint operators, Schrödinger operators, Berezansky theorem, dynamically defined operators, spectral measures and integrated density of states
2. Schrödinger cocycles, preliminaries on the ergodic theory, rotation number and Lyapunov exponent, $m$ functions, uniform and non-uniform hyperbolicity, Oseledec theorem
3. Links between the spectral and dynamical aspects,spectrum/hyperbolicity, density of states/rotation number. Thouless formula
4. Reducibility of cocycles, KAM theory, Dinaburg-Sinai and Eliasson theorems. Link with the absolutely continuous spectrum
5. Anderson localisation, importance of non-uniform hyperbolicity
6. Aubry duality. Application to studying the almost Mathieu operator

References

• R. Carmona, J. Lacroix, Spectral Theory of Random Schrödinger Operators, Birkhäuser, Boston, 1990.
• T. Ransford, Potential Theory in the Complex Plane, Cambridge University Press, Cambridge, 1995.
• P. Walters, An Introduction to Ergodic Theory, Springer, New York-Berlin, 1982.

• Distributions and introduction to PDEs (10 ECTS)

• Lecturer: Prof. Françoise Demengel

Abstract: This course aims at giving solid bases of functionals spaces necessary for the theory of elliptic partial differential equations (PDEs).

The course begins with an introduction to distributions. We first define the space of ${\cal C}^\infty$ functions with compact support and its topology, then the space of distributions and the corresponding topology, and finally, fast decaying functions and tempered distributions, as well as their Fourier transform. We study next the Sobolev spaces, including Sobolev embedding theorems and compactness of embeddings. In the case of domains with $C^1$ smooth boundary, we establish trace theorems, which require introduction of Sobolev spaces of fractional regularity.

The second part of the course is devoted to studying the problem of existence, uniqueness, and regularity for some PDEs of which the simplest one is the following problem considered in a bounded domain $\Omega$ with smooth boundary $\partial\Omega$:
$$-\Delta u=f \quad\mbox{in \Omega}, \qquad u=0\quad\mbox{on \partial\Omega}.$$
The existence (and in some cases, uniqueness) of solution is proven with the help of a fundamental result of convex analysis, namely, the minimisation of a convex functional on a reflexive Banach space.

We establish regularity results for those equations, as well as some qualitative properties, such as the (strong) maximum principle. We study also the equation for $p$-Laplacian.

Contents

1. The spaces of test functions and distributions. Topology and operations on distributions
2. Sobolev spaces $W^{m,p}(\Omega)$: elementary properties and embeddings
3. Sobolev spaces of fractional regularity. The case of $W^{m-\frac1p,p}(\partial\Omega)$ and generalised Green formula
4. A fundamental theorem of the calculus of variations: existence of a minimum for a coercive convexe functional on a reflexive Banach space. Application to the Dirichlet problem for the Poisson equation
5. Regularity of solutions. Nonlinear equations. Strong maximum principle

References

• F. Demengel, G. Demengel, Functional Spaces for the Theory of Elliptic Partial Differential Equations, Springer, London, 2012.

• Introduction to mathematical physics and spectral theory (10 ECTS)

• Lecturer: Dr.

Abstract: This course is an introduction to the spectral theory for self-adjoint operators, which can be regarded as an extension of the result on diagonalisation for Hermitian matrices to the infinite-dimensional case. The spectral theory plays an important role in mathematical physics and is one of the fundamental objects in quantum mechanics. We begin with a study of bounded operators in a Banach space and introduce the important concepts of spectrum and resolvent. We next turn to self-adjoint operators in a Hilbert space. The main result here is the spectral theorem. In the second part, we extend these results to the case of unbounded operators. We conclude this course by a study of groups of unitary operators in a Hilbert space and of the Stone theorem. The results will be illustrated by a number of examples, such as the discrete and continuous Schrödinger operators.

Contents

1. Bounded operators in Banach spaces. Spectrum and resolvent
2. Bounded self-adjoint operators in Hilbert spaces. Spectral theorem
3. Compact, trace class, and Hilbert–Schmidt operators
4. Unbounded operators in Hilbert spaces
5. Spectral theorem for unbounded self-adjoint operators
6. Unitary groups. Stone theorem

References

• P.D. Hislop, M. Sigal, Introduction to spectral theory: with application to Schrödinger operators, New-York, Springer, 1996.
• M. Reed, B. Simon, Methods of Modern Mathematical Physics I, Functional Analysis, Academic Press, 1980.
• M. Reed, B. Simon, Methods of Modern Mathematical Physics II, Fourier Analysis, Self-Adjointness, Academic Press, 1975.

• Methods of time series analysis (8 ECTS)

• Lecturer: Prof. Paul Doukhan

Abstract: This course provides some essential tools for understanding linear time series introduced in M1. The time series analysis requires various tools, such as Gaussian and moving averages models, or models with memory. Some knowledge of functional analysis will be necessary for this part of the course. Moreover, we shall consider the problem of estimation of time series, using mainly empirical methods and confining ourselves to the most basic techniques for their investigation. Two types of conditions are considered: long-range dependence or weak dependence. In the first case, which is technically rather involved, we study the simplest models, and in the second case, we specify the techniques related to the short-range dependence.

References:

1. P. J. Brockwell, R. A. Davis, Time Series: Theory and Methods, Springer, New York, 1991.
2. P. Doukhan, Tools for Nonlinear Time Series, 2016.
3. M. Rosenblatt, Curve Estimation, 1991.
4. M. Rosenblatt, Stationary sequences and random fields, Birkhäuser, Boston, 1985.

• Continuous time simulation (3 ECTS)

• Lecturer: Dr. Tristan Guillaume

Contents

1. Introduction : basic principles of the Monte Carlo simulation and main models used in evaluation of options in finance
2. Numerical simulation of random variables: 1) Random number generators (pseudo-random generators, low discrepancy sequences) 2) Sampling of laws (inversion, acceptation/rejection, various algorithms)
3. Application to evaluation of options: 3) Brownian bridge and conditioning (application to evaluation of barrier and lookback options) 4) Control variables (application to evaluation of average and basket options) 5) Simulation of diffusion processes with jumps

• Risk management (3 ECTS)

• Lecturer: Prof. Jean-Luc Prigent

Abstract: This course is an introduction to the financial risk management. The first part is devoted to the concept of Value-at-Risk a quantitative risk measure introduced by the Basel II directive. The second part treats the implementation of regulation for operational risks.

Contents

• Chapter 1 : Quanititaive management of financial risks
Preliminaries and complements in probability theory
Quantile of a probability distribution
Concepts of copula for modeling the dependence
Concepts of point processes for modeling risks of loss
Concepts of Value-at-Risk and applications in market risks
Calculation of the law for accumulated losses (application in actuarial sciences and operational risks management)

• Chapter 2 : Operational risks management
The Basel II directive and its extension to Basel III
Typology of operational risks and activity lines
Construction of an internal model for operational risks management

References:

• Basel Committee on Banking Supervision: International Convergence of Capital Measurement and Capital Standards. A Revised Framework - Comprehensive Version, Bank for International Settlements, June 2006.
• Basel Committee on Banking Supervision. Basel III: A global regulatory framework for more resilient banks and banking systems, December 2010 (rev June 2011)
• E. Bouyé, V. Durrleman, A. Nikeghbali, G. Riboulet, T. Roncalli, Copulas for Finance. A Reading Guide and Some Applications.
• D. Duffie, J. Pan, An overview of value at risk, Journal of Derivatives 4 (1997), no. 3, 7-49.
• A. Chapelle, G. Hübner, J.-P. Peters, Le risque opérationnel – Implications de l’Accord de Bâle pour le secteur financier, Larcier, 2005.
• A. P. Frachot, P. Georges, T. Roncalli, Loss distribution approach for operational risk, Groupe de Recherche Opérationnelle, Crédit Lyonnais, France.
• T. Roncalli, La Gestion des Risques Financiers, Editions Economica, 2009.

• Statistical learning (6 ECTS)

• Lecturer: Dr. William Kengne

Abstract: Statistical treatment of multidimensional data (in particular, data of large dimension) is an important issue in a number of domains, including finance, marketing, insurance, and biology. This question is even more important nowadays, when information and communication technologies allow one to stock easily'' large quantities of data. In this course, we implement the statistical theory and machine learning techniques for constructing various algorithms. Applications to real data are performed with the help of R.

Contents

1. General introduction
2. Elements  of the theory of supervised learning: risk, concentration inequality, minimisation principle for the empirical risk, Vapnik--Chervonenkis dimension
3. Regression models: generalities, parametric estimation, selection of variables, ridge regression, LASSO, nonparametric estimation, spline regression, kernel estimators
4. Basic methods for supervised classification: the k closest neighbours, logistic regression, binary decision trees
5. Support vector machines: general principles, linear and nonlinear separators, Mercer conditions

References

• O. Bousquet, S. Boucheron, G. Lugosi, Introduction to statistical learning theory, In O. Bousquet, U.V. Luxburg, G. Rätsch, Advanced Lectures on Machine Learning, Springer (2004), 169-207.
• C.J.C. Burges, A tutorial on support vector machines for pattern recognition, Data mining and knowledge discovery, (1998), 121-167.
• P.-A. Cornillon, E. Matzner-Løber, Régression avec R. Springer-Verlag, 2011.
• L. Devroye, L. Györfi, G. Lugosi, A Probabilistic Theory of Pattern Recognition, Springer, 1996.
• G. Dreyfus, J.-M. Martinez, M. Samuelides, M. B. Gordon, F. Badran, S. Thiria, Apprentissage statistique : Réseaux de neurones. Cartes topologiques. Machines à vecteurs supports, Eyrolles, Collection : Algorithmes, 2008.
• T. Hastie, R. Tibshirani, J. Friedman, The Elements of Statistical Learning : Data Mining, Inference, and Prediction, Springer, 2009.
• G. James, D. Witten, T. Hastie, R. Tibshirani, An Introduction to Statistical Learning, Springer, 2013.
• P. Massart, Concentration Inequalities and Model Selection. Springer, 2007.
• G. Saporta, Probabilités, analyse des données et statistique, Editions Technip, 2011.
• S. Tufféry, Data mining et statistique décisionnelle : l’intelligence des données, Editions Technip, 2010.
• V. N. Vapnik, Statistical Learning Theory, Wiley-Blackwell, 1998.

• Compléments d'algèbre (5 ECTS)

• Lecturer: Dr. Michela Varagnolo

Contents

1. Compléments de théorie des groupes (produit semi-direct, classification de groupes de petit cardianl, presentation par générateurs er relations)
2. Compléments de théorie des représentations de groupes fini sur les nombres complexes (calcul de tables des caractères)
3. Compléments de théorie de l'extension des corps (existence et unicité de la clôture algébrique, corps finis, polynômes cyclotomiques et théorème de Wedderburn)
4. Autres sujets à la demande

References

• D. Perrin, Cours d'algèbre, Ellipses, 1998.
• J. L. Alperin, R. B. Bell, Groups and Representations, Springer, New York, 1995.
• D. Robinson, A Course in the Theory of Groups, Springer, New York, 1996.
• J.-P. Serre, Représentations linéaires des groupes finis, Hermann, 1998.
• J. Rotman, An Introduction to the Theory of Groups, Springer, New York, 1995.
• H. E. Rose, A Course in Finite Groups, Springer, London, 2009.
• D. Hernandez, Y. Laszlo, Introduction à la théorie de Galois, École Polytechnique, 2012.
• J.-P. Escofier, Théorie de Galois, Cours et exercices corrigés, Dunod, 2004.

• Compléments d'analyse (5 ECTS)
Second term

The schedule will be available in the beginning of the term. All the courses are optional, and students must get at least 30 ECTS.
• Hyperbolic equations (10 ECTS)

• Lecturer: Prof. Nikolay Tzvetkov

Abstract: The goal of this course is to give a complete proof of a classical result due to Kruzhkov on the existence and uniqueness of entropy solution for a scalar conservation law.

Contents

1. Linear equations. Transport equation and wave equation in dimensions 1 and 3. Finite speed of propagation, qualitative consequences. Dispersion properties
2. Nonlinear scalar conservation laws. Method of characteristics and nonexistence of global-in-time classical solutions. Concept of a weak solution. Nonuniqueness of weak solutions
3. Concept of an entropy solution. Compactness results. Existence and uniqueness of an entropy solution
4. Riemann problem, shock and rarefaction waves. Link with entropy solutions

References

• C. Dafermos, Hyperbolic Conservation Laws in Continuum Physics, Springer, Berlin, 2000.
• S. N. Kruzkov,  First-order quasilinear equations in several independent variables, Math. USSR Sb. 10 (1970), 217-243.

• Spectral analysis of self-adjoint differential operators (10 ECTS)

Abstract: The aim of this course is to study Schrödinger operators with short- and long-range potentials with the help of abstract techniques of functional analysis, initiated in the works of Putnam, Kato, Lavine and Mourre. We shall also discuss more recent developments in these subjects.

Contents

1. Introduction to unbounded and self-adjoint operators. Rellich-Kato theorem, differential operators, Schrödinger operators
2. Borel functional calculus for arbitrary self-adjoint operators, spectral measure, boundary values for the resolvent, Stone formula, propagators, explicit formulas for the free Schrödinger operators. Calculation of commutators
3. Spectral types, criteria for absolute continuity, Putman, Lavine, and Mourre theorems. Application to short-range scattering

References

• G. K. Pedersen, Analysis Now, Springer-Verlag, New York, 1989.
• H. Brezis, Analyse fonctionnelle: Théorie et applications, Masson, Paris,1983.
• K. Schmüdgen, Unbounded Self-adjoint Operatorson Hilbert Space, Springer, Dordrecht, 2012.
• M. Reed, B. Simon, Methods of Modern Mathematical Physics. IV. Analysis of Operators, Academic Press, New York–London, 1978.

• Reaction-diffusion equations (10 ECTS)

• Lecturer: Prof. Élisabeth Logak

Abstract: This course is devoted to studying qualitative properties of semilinear parabolic partial differential equations, or reaction-diffusion equations. An important tool is the maximum principle, which will be studied in detail in the elliptic and parabolic cases. We next turn to some applications in various models of biomathemaitcs and study, in particular, their long-time dynamics.

Contents

1. Examples of linear elliptic and parabolic PDE’s: Laplace and Poisson equations, heat equation
2. Maximum principle for linear elliptic or parabolic equations; application to semilinear boundary value problems
3. Reaction–diffusion equation: super- and sub-solutions; qualitative properties: asymptotic behaviour in time, examples of blow-up
4. Introduction to nonlinear differential equations: stability of equilibrium points. Travelling waves: the case of the Fisher and bistable equations, long-time dynamics
5. Modelling in population dynamics: epidemic models in compartments (SIS, SIR), structured models

References

• L. C. Evans, Partial Differential Equations, American Mathematical Society, Providence, RI, 1998.
• J. Smoller, Shock Waves and Reaction-Diffusion Equations, Springer-Verlag, New York–Berlin, 1983.
• J. D. Murray, Mathematical Biology. I–II, Springer-Verlag, New York, 2002- 2003.

• Probabilistic evaluation of financial assets (3 ECTS)

• Lecturer: Dr. Tristan Guillaume

Abstract: This course deals with probabilistic evaluation of contingent assets in finance, mainly that of options for the equity or foreign exchange type underlyings. The instantaneous variations of prices of underlying assets are modelled by stochastic differential equations. The no-arbitrage prices for contingent assets are represented by conditional expectation under an equivalent martingale measure.

After introducing the Ito integral and changes of probability measure, we turn to applications of the continuous-time stochastic calculus in a model of complete market:

• standard options
• options depending on the trajectory of underlying; in particular, options whose repayment at maturity depends on the maxima and minima achieved by the underlying (barrier and lookback options)
• written options on several correlated underlyings (best-of and worst-of, spread, and exchange options)

The difficulties related to the introduction of stochastic interest rates will also be addressed. If the time permits, some elements of evaluation in incomplete market will be given at the end of the course (mixed models for diffusion with jumps, models of stochastic volatility)

• Risk measure: theory and applications (4 ECTS)

• Lecturer: Prof. Jean-Luc Prigent

Abstract: This course is an extension of that on financial risks management taught in the first term. The first part is devoted to the axiomatisation of the risks measure in response to flaws arising in Value-at-Risk quantitative risk measure introduced by the Basel II directive. The second part treats the implementation of regulation for credit  risks, in particular, in terms of the securitisation of this type of risk.

Contents

• Chapitre 1 : Theory of risk measures
The Value-at-Risk : principles and limitations. The expected shortfall
Concepts of coherent risks measures: axiomatisation and representation
Concepts of convex risks measures: axiomatisation and representation

• Chapitre 2 : Credit risks management
Credit risk: generalities
The risk of default of bonds: Merton structural approach, the approach in terms of the default intensity, the bond rating
Structured credit products : CDS and CDO

References:

• P. Artzner, F. Delbaen, J. M. Eber, D. Heath, Coherent measures of risk, Mathematical Finance 9 (1999), 203-228.
• A. De Servigny, O. Renault, Measuring and Managing Credit Risk, Standard & Poor's Press, 2004.
• D. Duffie, K. J. Singleton, Modeling term structures of defaultable bonds, Review of Financial Studies 4 (1997), 7-49.
• H. Föllmer, A. Schied, Convex measures of risk and trading constraints, Finance and Stochastics, 6 (2002), 429-447.
• R. Apostolik, C. Donohue, Foundations of Financial Risk: An Overview of Financial Risk and Risk-based Financial Regulation, GARP (Global Association of Risk Professionals), 2015.
• P. Jorion, Financial Risk Manager Handbook: FRM Part I / Part II, + Test Bank, 2011.
• R. C. Merton, On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance 29 (1974), 449-470.

• Models in population genetics (3 ECTS)

• Lecturer: Prof. Thierry Huillet

Contents

• Chapter 1. Markovian models, birth and death chains, branching processes, diffusion processes on an interval: definitions, properties, and examples

• Chapter 2. Models in population genetics (discrete space-time and diffusion approximations)
- Wright neutral model (Poisson branching that preserves the total number of individuals in successive generations)
- Wright-Fisher diffusion (scalante limit)
- Feller model
- Moran model
- Wright model with mutations
- Wright model with selection, haploid and diploid cases (dominance)
- Models with frequency-dependent selection
- Wright model with fluctuating selection
- Multiparallel case and sufficient conditions for polymorphisme
- Dynamics with recombination

• Chapter 3. Diffusion models on the interval $[0, 1]$
- Classification of Feller boundaries
- Evaluation of additive functionals, Dirichlet problem, Green function
- Application to problems with extinction and fixation

- Modification of processes (Doob transform), diffusion with killing: application to conditionning and Yaglom limit laws, determining the age of a mutant; evaluation of additive functionals for the modified process

• Chapter 4. Introduction to questions related to genealogy (coalescence). Models in population genetics backwards in time. Duality between Markov chains and between Wright-Fisher diffusion and Kingman coalescence

References:

• J. F. Crow, M. Kimura, An Introduction to Population Genetics Theory, Harper & Row, New York-London, 1970.
• S. N. Ethier, T. G.Kurtz, Markov Processes. Characterization and Convergence, John Wiley & Sons, New York, 1986.
• W. J. Ewens, Mathematical Population Genetics. I. Theoretical Introduction, Springer-Verlag, New York, 2004.
• J. H. Gillespie, The Causes of Molecular Evolution, Oxford University Press, 1991.
• S. Karlin, H. M. Taylor, A Second Course in Stochastic Processes, Academic Press, New York-London, 1981.
• P. Mandl, Analytical Treatment of One-dimensional Markov Processes, Springer-Verlag, New York, 1968.
• T. Maruyama, Stochastic Problems in Population Genetics, Springer-Verlag, Berlin-New York, 1977.
• T. Nagylaki, Anecdotal, historical and critical commentaries on Genetics. Gustave Malécot and the transition from classical to modern population genetics, Genetics 122 (1989), 253-268.
• S. Tavaré, Ancestral inference in population genetics, Lecture Notes in Math. 1837, Springer, 2004, pp. 1-188.

• Advanced probability theory (10 ECTS)

• Lecturer: Prof. Eva Löcherbach

Abstract: In this course, we will establish the basic tools to study Markov jump processes and piecewise deterministic Markov processes (PDMP’s) aris- ing in applications such as insurance, fiability, biology, in particular population genetics, neuroscience, and many other fields. The basic tools such as the gen- erator and the transition semigroup, the invariant measure and the Ito formula with jumps will be introduced and explained. We will then turn to the study of the longtime behavior of such processes and the speed of convergence to equilibrium, relying on the total variation and/or the Wasserstein coupling..

Contents

1. Introduction : Theory of stochastic processes in continuous time. Example : Poisson process and compound Poisson process
2. Piecewise deterministic Markov processes : existance and uniqueness of solutions, moments, etc.
3. Study of the embedded Markov chain and of its invariant measure. Link with the continuous time process
4. Generator and transition semigroup. Ito formula with jumps. Lyapunov type functions and Meyn and Tweedie approach to stability and ergodicity of the process
5. Some examples : Growth/fragmentation processes. Interacting particle systems
6. Poisson random measures (PRM), integration with respect to  PRM, SDE's driven by PRM. Acceptance-rejection method to simulate the solution of a SDE driven by PRM.

References:

• P.A. Ferrari, A. Galves, Coupling and Regeneration for Stochastic Processes, notes for a minicourse presented in XIII escuela venezolana de matematicas, 2000.
• D. A. Levin, Y. Peres, L. W. Elizabeth, Markov Chains and Mixing Times, With a chapter on “Coupling from the past” by J. G. Propp and D. B. Wilson, Providence, RI: American Mathematical Society, 2009.
• E. Löcherbach, Ergodicity and Speed of Convergence to Equilibrium for Diffusion Processes, Personal web page of E. Löcherbach

• Modélisation, option B : Calcul Scientifique (10 ECTS)

• Lecturer: Prof. Philippe Gravejat

Abstract: Ce cours propose une préparation à l’oral de modélisation de l’agrégation externe (Option B : calcul scientifique). Cet oral consiste à détailler un texte scientifique proposé au candidat et à illustrer certains des points de ce texte à l’aide de l’outil informatique. Le cours a pour but de décrire les principales notions d’analyse numérique au programme officiel de cette épreuve suivant le plan ci-dessous. Il se complète d’une initiation au langage Scilab à travers l’implémentation pratique des diverses méthodes numériques introduites, ainsi que de leçons sur un échantillon des textes rendus publics par le jury de l’agrégation.

Contents

1. Résolution des systèmes linéaires
2. Optimisation
3. Résolution des systèmes non linéaires
4. Interpolation polynomiale
5. Intégration numérique
6. Résolution approchée des équations différentielles ordinaires
7. Résolution approchée des équations aux dérivées partielles

References:

• P.G. Ciarlet, Introduction à l'analyse numérique matricielle et à l'optimisation, Dunod, 2007.
• A.-L. Crouzeix, M. Mignot, Analyse numérique des équations différentielles,  Masson, 1992.
• J.-P. Demailly, Analyse numérique et équations différentielles,  EDP Sciences, 2006.
• L. Dumas, Modélisation à l'oral de l'agrégation - Calcul scientifique, Ellipses, 1999.

• Préparation à l'oral d'agrégation (10 ECTS)

• Lecturer:  Prof. Philippe Gravejat

Abstract: Ce cours propose une préparation aux oraux d’algèbre et géométrie, et d’analyse et probabilités de l’agrégation externe. Ces oraux consistent à rédiger le plan d’un cours sur l’un des sujets parmi une liste établie par le jury de l’agrégation, puis à développer l’un des points de ce plan. Chaque séance du cours prépare à cette épreuve à travers deux leçons sur un éventail caractéristique des sujets proposés en algèbre et géométrie, ainsi qu’en analyse et probabilités.

Contents

1. Groupe opérant sur un ensemble ; Séries de Fourier
2. Exemples de sous-groupes distingués et de groupes quotients ; Suites et séries de fonctions
3. Utilisation des groupes en géométrie ; Continuité et dérivabilité des fonctions réelles d’une variable réelle
4. Exemples et représentations de groupes finis de petit cardinal ; Espaces de Hilbert
5. Groupe linéaire d’un espace vectoriel de dimension finie ; Espace complets
6. Exemples d’actions de groupes sur les espaces de matrices ; Exemples de parties denses et applications
7. Déterminant ; Espaces de Schwartz et distributions tempérées
8. Polynômes d’endomorphisme et réduction d’un endomorphisme en dimension finie ; Équations différentielles
9. Endomorphismes remarquables d’un espace vectoriel euclidien (de dimension finie) ; Utilisation de la notion de compacité
10. Sous-espaces stables par un endomorphisme ou une famille d’endomorphismes d’un espace vectoriel de dimension finie ; Théorème d’inversion locale, théorème des fonctions implicites
11. Corps finis ; Fonctions holomorphes.
12. Nombres premiers ; Suites vectorielles ou réelles définies par une relation de récurrence
13. Formes quadratiques réelles ; Méthodes d’approximation des solutions d’une équation
14. Barycentres dans un espace affine réel de dimension finie, convexité ; Utilisation de la notion de convexité en analyse
15. Méthodes combinatoires, problèmes de dénombrement ; Extremums : existence, caractérisation, recherche
16. Fonctions caractéristiques et transformée de Laplace d’une variable aléatoire ; Modes de convergence d’une suite de variables aléatoires

References:

• E. Amar, E. Matheron, Analyse complexe, Cassini, 2003.
• M. Audin, Géométrie, EDP Sciences, 2006.
• V. Beck, J. Malick, G. Peyré, Objectif agrégation, H & K, 2005.
• J.-P. Demailly, Analyse numérique et équations différentielles, EDP Sciences, 2006.
• F. Demengel, G. Demengel, Mesures et distributions, théorie et illustrations par les exemples,  Ellipses, 2001.
• S. Francinou, H. Gianella, S. Nicolas, Exercices corrigés d’oraux X-ENS - Algèbre 1, 2, et 3, et analyse 1, 2, 3 et 4, Cassini, 2009 à 2014.
• X. Gourdon, Les maths en tête - Algèbre et analyse, Ellipses, 2009 et 2008.
• P. Ortiz, Exercices d’algèbre,  Ellipses, 2004.
• J.-Y. Ouvrard, Probabilités 1 et 2, Cassini, 2008 et 2009.
• D. Perrin, Cours d’algèbre, Ellipses, 1998.
• A. Pommelet, Agrégation de mathématiques - Cours d’analyse, Ellipses, 1997.
• H. Queffélec, C. Zuily, Analyse pour l’agrégation, Dunod, 2013.
• F. Rouvière, Petit guide de calcul différentiel, Cassini, 2009.

The courses are completed by a Research initiation memoir or an Internship (10 ECTS).

#### Secretariat

Department of Mathematics
University of Cergy-Pontoise
95302 Cergy-Pontoise Cedex
France

#### Apply for the master of mathematics

A portal for applying for the master programme online will be available in May of 2017.