Mathematics - Computer Science - Economics

• Programs of the course units Mathematics 1A, Computer Science 1A and Economics-Business 1A from Ecole Centrale Méditerranée Engineering programme (see syllabus 1A)
• Basics of the Python language

• Apply a field of applied mathematics (Probability-Statistics, Finite Elements, Optimal Transport) to applications
• Design a computer program by implementing the necessary steps with the adequate tools: modeling, algorithms, programming environment in Python
• Understand advanced concepts in economics about markets or strategic behavior or process data related to an economics problem.

The major course unit Mathematics-Computer Science-Economics (MIE) is divided into 3 periods, each period is equal to 4 weeks (18h  et 6h d’autonomie). For each period, the student must choose one subcourse offered in one of the following fields: Mathematics, Computer Science or Economics. There is one constraint for the student's choices (except for international credit mobility students such as Erasmus+ students): at the end of the semester the student must have studied subcourses in at least two different fields among Mathematics, Computer Science and Economics.

Choice of a course among Mathematics, Computer Science or Economics at each period

Mathématiques

1. M1 : Probability and statistics
   
   1. Conditional probability and conditional expectation: definition, conditional probability distribution, properties, Bayes formula, martingales
   2. Inferential statistics:  parametric estimation (maximum likelihood, moment method, regular model and Fisher information, confidence interval, parametric tests (likelihood ratio test) and nonparametric tests (chi-square)
2. M2 : Variational methods and finite elements
   
   1. Distributions : definition, convergence, derivative
   2. Hilbert space, Sobolev spaces, inequalities (Cauchy-Schwarz, Minkowski), Green formula, semi-norm
   3. Variational methods: Lax-Milgram theorem, Galerkin methode (definition, convergence and order).
      
   4. Finite elements: definition, approximation space, convergences for the local approximation and for the global approximation, convergence theorem for the Lagrange finite-element method
3. M3 : Introduction to the theory of optimal transport
   
   1. Monge and Kantorovitc formulations,
   2. Kantorovitch duality, c-concave functions, applications in economics: matching equilibrium
   3. Wasserstein distance, generalized Wassertsein distance (unbalanced optimal transport)
   4. Computational methods: Sinkhorn algorithm, entropic regularization, Benamou-Brenier  formulation

Computer Science

1. I1 : Algorithm design
   1. Algorithm divide & conquer
   2. Sequence alignment problem: sequence alignement problem, sequence alignement algorithm
      
   3. Dynamic programming and NP-completeness
   4. Greedy algorithms: principles and  implementation
   5. Enumeration problem: branch-and-bound strategy and backtracking strategy
2. I2 : Data driven programming
   1. Event-driven programming and persistent objects. Python object. CRUD principle. MVC design template.
   2. Persistence managers: ORM, DAO. web servers. HTML/CSS. Django, pony ORM, Django ORM.
3. I3 : Scientific Python
   1. Data manipulation and data analysis with Python: libraries Numpy and Scipy
   2. Graphs in Python: library Matplotlib
   3. Dataframe manipulation and representation: libraries Pandas and Seaborn
   4. Image processing: library Scikit-image

Economics

1. E1 : Innovation and market power: monopoly and rents
   1. Market, market structures, competitive case
   2. Monopoly: simple monopoly, production of a sustainable good, price discrimination, product selection
   3. Strategic interactions: oligopoly, Cournot model, Bertrand model, industrial economic strategies
      
2. E2 : Strategic behaviours: game theory
   
   1. Dominated strategy and Iterated elimination of strictly dominated strategies (IESDS)
   2. Nash equilibrium: definition, best answers, connection between IESDS-Nash
   3. Mixed strategy: definition, investigation of mixed equilibrium, Nash theorem, interpretation of mixed Nash equilibrium
   4. Games with continuous action space
   5. Sequential games
   6. Matching
3. E3 : Economics : data and public policies
   
   1. inequalities in economics: definition, measure, inequality factors
   2. Public policies addressing inequalities
   3. Public policy evaluation: experimental and quasi-experimental evaluation methods

• M1: Model a statistical experiment for an i.i.d. sample and implement standard pointwise  and interval estimation methods and testing procedures
• M2: Write and analyze a week formulation for a PDE. Implementation in Finite Element software.
• M3: Formulate the optimal transport problem and calculate Wasserstein distances
• I1: Understand the main categories of algorithms and how to implement them
• I2: Implement event-driven programming in Python and understand the notion of persistence
• I3: Program in Python with the Numpy, Scipy, Matplotlib, Pandas, Seaborn and Scikit-image libraries
• E1: Identify the different types of markets, understand the notions of monopoly and oligopoly
• E2: Classify strategies and understand Nash equilibrium
• E3: Understand inequalities in economics and implement tools to evaluate public policies addressing them

Continuous assessment

A bibliography will be proposed at the beginning of each subcourse offered in the major course unit Mathematics-Computer Science-Economics.

1. Mathematics
   
   1. M1 : Christophe Pouet, Mitra Fouladirad, Frédéric Schwander
   2. M2 : Guillaume Chiavassa
   3. M3 : Magali Tournus
2. Computer Science
   
   1. I1 : François Brucker, Pascal Préa
   2. I2 : Emmanuel Daucé, Manon Philibert
   3. I3 : Muriel Roche, Manon Philibert
3. Economics
   1. E1 : Nicolas Clootens, Santiago Lopez-Cantor
   2. E2 : Nicolas Fournier (Aix-Marseille Université), Hajare El Hadri, Santiago Lopez-Cantor, Ayoub Salih
   3. E3 : Hajare El Hadri