# Bachelor in Applied Mathematics Course Descriptions

## Bachelor in Applied Mathematics

www.ie.edu/university STUDY PROGRAM

## Year 1

### Calculus I

• Limits and Continuity. Intuitive and rigorous definition. Main properties and computation techniques.

• Differential calculus in one variable. Definition of derivative and interpretation. Main computation techniques. Higher order derivatives. Main related theorems.

• Sequences and series. Definitions and convergence. Main properties and convergence results: divergence test, ratio test, root test. Power series and Taylor expansions.

• Integral calculus in one variable. Definite and indefinite integral. Fundamental Theorem of calculus. Main integration techniques.

### Linear Algebra

• Vector spaces and subspaces: properties, operations, and concepts such as linear independence, basis, and dimension.

• Linear transformations and matrices: properties, operations, and concepts such as eigenvalues, eigenvectors, and diagonalization.

• Determinants and inverses of matrices: properties, operations and concepts such as Cramer’s rule and LU decomposition.

• Systems of linear equations and their solutions, including Gaussian elimination, Gauss-Jordan elimination, and matrix inversion.

### Discrete Mathematics

• Foundations of logic. Basic concepts of propositional logic and introduction to proofs: predicates, quantifiers…

• Basic techniques of combinatorics and counting. Pigeonhole Principle, permutations, combinations...

• Recursivity. Sets and functions defined recursively and their main properties. Recursive algorithms.

• Relations and their properties. Representing relations, equivalence relations, partial ordering. Applications.

• Graph Theory. Graphs and graph models. Connectivity, Euler and Hamilton Paths. Introduction to algorithms over graphs.

### Geometry

• Bilinear and quadratic forms. Analytic and matricial expression of a bilinear form. Quadratic form and its matricial expression. Classification of quadratic forms. Positive, negative and semi-definite matrices. Spectral Theorem.

• Euclidean space Concept of scalar product. Matricial expression of a scalar product. Euclidean space. Norm induced by a scalar product. Properties. Distances and angles between vectors. Orthogonality. Gram Schmidt. Orthogonal linear variety.

• Affine spaces. Isometries. Affine mappings. Fixed points. Euclidean Group and matrix representations in 2D and 3D.

Calculus I 6 ECTS Calculus II 6 ECTS Linear Algebra 6 ECTS Computer Programming II 6 ECTS Discrete Mathematics 6 ECTS Probability and Statistics 6 ECTS Geometry 6 ECTS Databases 3 ECTS Computer Programming I 6 ECTS Data Visualization 3 ECTS IE Impact: Humanities 6 ECTS Total 30 ECTS 30 ECTS FALL (SEMESTER 1) SPRING (SEMESTER 2)

### Computer Programming I

• Fundamentals of programming: variables, flow control (branching, loops), strings, print and input functions.

• Functions: types of arguments, return, notion of scope, the function as an object.

• Basic data structures: lists, tuples, dictionaries, nested structures. Comprehensions.

• The Python ecosystem: installation, management and import of libraries; Anaconda system. Programming modes: console, REPL console, IDEs, notebooks.

• Introduction to vector programming (array programming). NumPy basics: creating and accessing tensors (basic, fancy indexing). Operations with tensors: calculation of descriptive statistics, element-wise functions.

FALL (SEMESTER 1)

### Calculus II

• Differential multivariable Calculus. Limits and continuity. Partial derivatives and differentiability. Chain rule. Gradients and directional derivatives.

• Extremals of multi-variable real valued functions. Higher order derivatives. Taylor expansions. Restricted maxima and minima: Lagrange multipliers.

• Double and triple integrals over different domains. Change of variables and Jacobian matrices.

• Vector valued functions. Trajectories and vector fields. Divergence and curl. Vector differential calculus.

• Integration over curves and surfaces. Arc length. Parametrized surfaces. Areas and volumes.

• Integral theorems in vector calculus (Green, Stokes, Gauss).

### Computer Programming II

• Input/output: reading and writing files in Python. Binary files vs. text files. Special formats: .csv, .npz, .xlsx. Channels stdin, stdout, stderr. Passing arguments to a program. Advanced string formatting.

• Introduction to pandas. Notion of dataframe: index, columns. Load and export dataframes. Compute descriptive statistics in pandas. Advanced access to dataframes: by integer, by name, by filter.

• Classes in Python: declaration and creation of classes. Class methods vs. instance

methods. Inheritance. constructors/ destructors. How to define class operators.

• Advanced Python environments: How to import modules. How to create packages. Version control via git.

• Other advanced concepts in Python: exceptions, regular expressions, generators, decorators, profiling.

### Probability and Statistics

• Probability theory and distributions: basic principles of probability, probability distributions.

• Descriptive statistics: measures of central tendency, variability, and skewness, graphical tools.

• Inferential statistics: estimation, hypothesis testing, and confidence intervals.

• Analysis of variance: one-way and twoway ANOVA.

### Databases

• Introduction to relational algebra. Table concept, primary key, foreign key. Selection and join operations. Normalization vs. denormalization.

• Fundamental SQL statements (select, join, sort, delete, etc.). Grouping and arithmetic and aggregation operations. Installation and work with a real SQL database.

• Key-value databases. Document-based databases. Columnar databases. Graph databases.

SPRING (SEMESTER 2)

### Data Visualization

• Introduction to visualization: visual variables and attributes; desirable properties of a diagram; how to choose the right visual channel for the right task.

• Data wrangling for exploratory analysis in Python. Elementary charts using matplotlib: scatterplots; sequential vs. diverging vs. qualitative colormaps; barplots; line plots; filled area plots. Basic customization.

• Advanced visualization: smooth density lines; dot, box and violin plots; grid density plots; long vs. wide data. Introduction to seaborn. Interactive visualization with Plotly or Bokeh.

• The “curse of dimensionality”: why dimensionality reduction is often needed. Plot ensembles; feature selection; principal component analysis (PCA); t-stochastic neighbor embedding (t-SNE).

• Fundamentals of Tableau: reading data files; dimensions vs. measures; bar plots; line plots; area plots; scatter plots; handling date formats; filtering variables.

• Advanced Tableau: overlaying plots on top of each other; stacked bars; density plots; paths; calculated fields; pages and animations; geovisualization; merging data sources; packed bubbles; treemaps; boxplots; pie and doughnut charts.

### IE Impact: Humanities

IE Impact is a transversal academic program for all IEU students whose mission is to prepare students to be agents of positive change with a dynamic, hands-on learning journey process. By studying a course in the Humanities, the IE Impact program’s aim is to help you to develop critical thinking skills and to help you begin to be aware of an array of societal challenges affecting the world.

In the Humanities pack of courses, students are introduced to some of the most complex issues and challenges faced by humanity, from a humanistic, intercultural and global standpoint. Topics covered in the course might be as diverse as Archaeology; Culture, history, and commerce in pre-modern China; Introduction to critical management thinking; Cross-cultural communication and its impact; The digital human; Empires and the rhetoric of power; Philosophy of happiness; Image, art and power; markets and society; Social movements: Past, present and future of collective politics; Sociology and cultural studies; Women leaders in art and history: Free speech and dangerous ideas, etc.

SPRING (SEMESTER 2)

## Year 2

### Ordinary Differential Equations

• Fundamentals of Ordinary Differential Equations: Initial and Boundary value problems. Existence and uniqueness of solutions. Geometric interpretation. Regularity with respect to initial data.

• Linear systems of ODEs: Homogeneous systems. Structure of solution space. Linear systems with constant and periodic coefficients. Matrix exponentials. Stability of linear systems. Classification of critical points. Inhomogeneous systems.

• Sequences and series. Definitions and Qualitative theory of ODEs: Fundamentals. Phase diagram. Invariant manifolds. Linearization. Fixed points in 2D. Structural stability (Hartman-Grobman). Limit cycles. Poincaré-Bendixson theorem.

• Applications: Population dynamics (logistic equation, Lotka-Volterra), damped and driven oscillators, Chemical kinetics, etc.

### Numerical Methods

• Numerical representation in Python, computational error and operational cost.

• Root-finding methods for nonlinear equations: regula falsi, secant, fixed point and Newton’s method. Convergence analysis.

• Interpolation of functions: linear and polynomial interpolation. Lagrange polynomial. Splines.

• Numerical differentiation and integration. Finite difference method. Quadrature rules and Monte Carlo integration.

• Finite calculus and numerical methods for differential equations. Euler and RungeKutta methods. Shooting method for boundary value problems.

### Optimization I

• Introduction to Mathematical Programming. Type of problems (optimization and decision-making), type of variables (continuous, discrete and integer) and types of objectives (linear, nonlinear and multiobjective).

• Linear and integer programming (basics of linear and integer programming, including the formulation of linear and integer programs, the simplex method, and branch and bound algorithms).

• Nonlinear programming (basics of nonlinear programming, including the formulation of nonlinear programs, gradient and Newton methods.

• Constrained optimization (formulation of constrained optimization problems, the use of penalty and barrier functions).

• Queuing theory (fundamentals of queuing theory, formulation of queuing models, use of queueing theory for optimization and decision making).

Ordinary Differential Equations 6 ECTS Mathematical Modeling 6 ECTS Numerical Methods 6 ECTS Approximation 6 ECTS Optimization I 6 ECTS Statistical Modeling 6 ECTS Machine Learning 6 ECTS Optimization II 6 ECTS Communication Skills 3 ECTS IE Impact: Entrepreneurship 6 ECTS IE Impact: Technology 3 ECTS Total 30 ECTS 30 ECTS FALL (SEMESTER 3) SPRING (SEMESTER 4)

### Machine Learning

• Introduction to machine learning: supervised vs unsupervised learning, regression vs classification problems.

• Linear Regression: simple and multiple linear regression, cross-validation and regularization.

• Classification: Logistic regression and linear discriminant analysis.

• Ensemble methods: decision trees and random forests, bagging and boosting.

• Unsupervised learning: principal component analysis and clustering methods.

### Communication Skills

• Creating and defending arguments supported by scientific evidence and data.

• How to write high quality research papers, proposals, grant submissions, etc.

• Building the message: The Audience Map. Persuasion. The psychological barriers. Simple is better. Visual Thinking Tools to organize the speech. Storytelling. Metaphors, Analogies, and other visual figures. Structure. Types of Narrative. Understanding the differences and applying them in our presentations.

• Design of materials for visual support. Main principles of effective visual communication. Quick overview of existing design tools. Debate and practice.

• Delivering the message: Logos, Pathos, Ethos. Facing fear on the scene. Correct use of our voice, language and body expression on the scene. Interacting with partners.

### IE Impact: Technology

• Analyze how technology is changing the way customers and enterprises interact.

• Study how technology solves the main challenges that customers/individuals face in terms of providing a response to human behavior, motivations and context and new ways of solving daily problems (e.g. smart chatbots) but also to implement the dynamically changing technology enhancing the overall experience for everyone.

• Course will cover the latest technology trends, how these technologies are implemented, how technology varies across generations and cultures, how technology is applied by society, enterprises and individual people.

• Technology trends are studied through a human-centered approach that will help students frame, develop and discover how technology will aid them in future projects, entrepreneurial initiatives, and even in identifying or generating new business model opportunities.and transmission, audio and image compression, computer graphics…

FALL (SEMESTER 3)
FALL (SEMESTER 4)

### Mathematical Modeling

• Main principles of mathematical modeling. Dimensional analysis. Modeling scales and representations. Types of models. Critical analysis and model validation. Fitting parameter values to empirical data. All models are wrong but some are useful.

• Modeling dynamics with differential equations.

• Modeling uncertain outcomes with statistical models.

• Modeling complex interactions with agent based models.

### Approximation

• Fundamentals of approximation. Functional spaces. Basis and orthogonality. Sequences. Convergence. Main results from Functional Analysis (Mercer theorem, Karhunen-Loeve theorem).

• Fourier analysis. Fourier series and main properties. Fast Fourier transform. Discrete Cosine Transform.

• Wavelets. Haar and Daubechies wavelets. Properties.

• Applications: signal processing and transmission, audio and image compression and computer graphics.

### Statistical Modeling

• Mathematical Foundations of Inference: definition of statistics and statistical model, their properties and maximum likelihood estimation.

• The Exponential Family and Generalized Linear Models: notable distributions, ordinal, multinomial and poisson regression.

• Flexible Regression: Polynomial regression, splines, GAM models and robust estimation.

• Density Estimation: mixture models and the EM algorithm, kernels and their use.

• Survival Analysis: survival and hazard functions, Kaplan-Meier tables, Cox models.

### Optimization II

• Introduction. Convex functions, optimality conditions, differentiability.

• Direct and iterative methods for continuous optimization. First and second order methods (gradient descent, and derivatives, Newton’s methods, etc).

• Optimization with constraints. Lagrange multiplier theorem. Barrier and penalty methods.

• Introduction to graph theory and optimization in networks: basic concepts, support tree problem, minimum path problems, flow problems.

• Basic software in continuous and graph optimization.

### IE Impact: Entrepreneurship

• This course is aimed at inviting students to step out of their comfort zones, question both themselves and existing systems, and work in conditions of uncertainty.

• Students will develop entrepreneurial mindsets by creating, validating, and developing innovative business models, which can be applied to societal challenges, providing sustainable solutions.

• As the last of the three IE IMPACT foundational pillars (Humanities, Technology, and Entrepreneurship), it will help students to develop value propositions and sustainable models for society.

SPRING (SEMESTER 4)

## Year 3

### Partial Differential Equations

• Introduction and classification of Partial Differential equations (PDEs): diffusion, elliptic and hyperbolic.

• Linear PDEs: the wave, heat and Laplace equations. Fourier Series. Separation of variables.

• Heat equation, weak maximum principle and global Cauchy problem.

• Poisson equation, fundamental solution and Green’s functions.

• The wave equation, the method of spherical means.

• Numerical methods for PDEs: stability, convergence and consistency. Lax equivalence theorem. Finite differences methods. Spectral methods.

### Algorithms

• Algorithmic complexity: Notion of complexity in time and space, big-O notation and the like. Classes: P, NP, NPcomplete, NP-hard.

• Important types: exact vs. approximate, random vs. deterministic, exhaustive (brute force) and backtracking.

• Greedy algorithms: examples and applications (TSP, minimum spanning tree) of cases in which they are optimal and in which they are not.

• Dynamic programming: examples and applications (knapsack, TSP). Study of its cost in space in addition to time.

• Recursion and divide and conquer: examples and applications (Fibonacci, traversing trees). Master theorem of recursion.

### Optimization III

• Introduction to combinatorial optimization. Exact solutions to integer problems. Metaheuristics of simulated annealing. Examples of the Traveling Salesman Problem and Sudoku solving.

• Genetic algorithms. Population-based methods, fitness function. Crossover and mutation operators. Examples of the knapsack problem and the N-queens problem.

• Evolutionary strategies. Particle Swarm Optimization, variants (e.g. ant colony).

• Optimization algorithms for games. Optimization on decision trees. Minimax algorithm. Alpha-beta pruning.

• Basic software in combinatorial optimization.

Partial Differential Equations 6 ECTS Bayesian Statistics and Stochastic Processes 6 ECTS Algorithms 6 ECTS Nonlinear Dynamics 6 ECTS Optimization III 6 ECTS Deep Learning 6 ECTS Numerical Linear Algebra 6 ECTS Elective Course 1 6 ECTS IE Impact: Challenge 6 ECTS Elective Course 2 6 ECTS Total 30 ECTS 30 ECTS FALL (SEMESTER 5) SPRING (SEMESTER 6)

### Numerical Linear Algebra

• Matrix decompositions and numerical solutions of Linear systems. LU, QR, Cholesky factorizations and pivoting. Singular Value Decomposition (SVD). Higher order systems, sparse and dense linear systems, parallelization, least squares problems, Tikhonov regularization.

• Iterative methods for linear systems. Krylov subspaces methods: Lanczos iteration, Generalized minimal residuals method (GMRES), Conjugate gradients (CG) and similar versions (e.g., Bi-CG). Preconditioning.

• Eigenvalue and eigenvector problems. Gerschgorin theorem. Power method, Rayleigh quotient, Arnoldi iteration, modified Gram-Schmidt method, Householder reflector and reduction to Hessenberg.

### IE Impact: IE Challenge

• The IE Challenge is the hands-on culmination of IE IMPACT and its three foundational pillars (Humanities, Technology and Entrepreneurship).

• Students in this course learn how to work in diverse teams and apply the skills they have acquired to provide consultancy to businesses to help them amplify or scale their impact on one or more of the Sustainable Development Goals (SDGs)

• They will learn methodologies such as Design Thinking and Lean Startup to channel learning and progress during the project, in which they will have to work on 4 main stages: (1) Exploration, (2) Definition, (3) Ideation / Prototyping and (4) Business Model and Go-to-market strategy.

FALL (SEMESTER 5)
SPRING (SEMESTER 6)

### Bayesian Statistics and Stochastic Processes

• Discrete-time stochastic processes: discrete-time Markov chains and their properties, limiting probabilities and ergodicity.

• Continuous-time Markov chains: the Poisson process and properties of the exponential distribution, introduction to queuing theory.

• Introduction to Bayesian reasoning: the Bayesian approach to inference, Bayes theorem, prior and posterior distributions, highest-density intervals.

• MCMC algorithms: Gibbs and MetropolisHastings algorithms and their use in practice, related software.

• Applied Bayesian Modeling: multilevel and hierarchical models, case-studies.

### Nonlinear Dynamics

• Dynamical systems in 1D and 2D. Geometric thinking à la Poincaré. Fixed points and stability. Limit cycles.

• Bifurcations: saddle-node, pitchfork, transcritical, Hopf. Oscillating systems.

• Fast-slow dynamics: decoupling different time scales. Excitable systems.

• Deterministic chaos in continuous time: Lorenz system, Lyapunov exponents, strange attractors. Mixing and ergodicity.

• Nonlinear maps: logistic map and flip bifurcations. Transient chaos. Renormalization and universality.

• Fractal geometry: Mandelbrot and Julia sets. Fractal dimension and its computation. Numerical experiments and applications in finance.

### Deep Learning

• Introduction of neural networks and deep learning. Revisiting the mathematical foundations underpinning them and their applications: linear algebra, optimization (in particular, gradient descent and variations), multivariate calculus (in particular, chain rule and backpropagation), and probability.

• Learn the principles of deep neural network design, training and evaluation, building on the Machine Learning course and including concepts such as

types of layers and activation functions, feature encoding and embeddings, types of gradient descent optimizers, hyperparameter search, and differentiable metrics for various tasks (classification, regression, synthesis…).

• Develop the programming skills necessary to implement these methods using a popular general-purpose deep learning framework (e.g. PyTorch, TensorFlow/ Keras, JAX).

• Understand the most relevant architectures (such as convolutional and recurrent neural networks, autoencoders, and transformers) and their applications to real-world problems, as well as the limitations and risks of deep learning solutions.

SPRING (SEMESTER 6)

## Year 4

Elective Courses can be chosen among all the courses offered at IE University.

Students will be able to do internships in companies or research centers, which will be equivalent to 12 ECTS.

In order to receive a specialization diploma for one of the three concentrations offered (Industrial Mathematics, Artificial Intelligence and Financial Mathematics), students must choose the 6 courses offered below.

Elective Course 3 6 ECTS Elective Course 8 6 ECTS Elective Course 4 6 ECTS Elective Course 9 6 ECTS Elective Course 5 6 ECTS Elective Course 10 6 ECTS Elective Course 6 6 ECTS Capstone Project 12 ECTS Elective Course 7 6 ECTS Total 30 ECTS 30 ECTS FALL (SEMESTER
SPRING (SEMESTER 8)
7)

Professional Coding and Project Management

Complex Systems

Complex Network Theory

Mathematical Models in Biology and Medicine

Mathematical Models in Transport and Logistics

Cryptography

Computer Vision

Natural Language Processing

Reinforcement Learning

Information Theory

Robotics and Automation

Financial Mathematics I

Behavioral Economics and Decision Theory

Time Series

Economics and Computation

Financial Mathematics II

Industrial Mathematics Artificial Intelligence Financial Mathematics

Professional Coding and Project Management

• Code organization and collaboration. Version control systems. Git and main workflows. Testing (unit tests, application tests). CI/CD.

• Architectural software patterns. Software engineering best practices.

• Work organization. People and roles, tasks (issues, ticket systems, project management tools).

• Agile principles, SCRUM and current trends.

### Complex Systems

• Entropy, complexity and information. Emergent phenomena.

• Deterministic chaos (mixing, ergodicity, Lyapunov exponents, strange attractors).

• Critical phenomena and phase transitions (Ising model, Mean field theories).

• Complex adaptive systems and selforganization.

• Cellular automata (Wolfram complexity classes, Game of Life).

• Flocking and herding models (Vicsek, Cucker-Smale).

• Turing instability and pattern formation.

ELECTIVE COURSES

### Complex Networks

• Social, biological, technological and information networks

• Mathematical foundations (representation, metrics, degree distribution, modularity and community structure)

• Complex network models: random graphs, growth models (Barábasi-Albert, WattsStrogatz), small world effect, scale-free networks.

• Processes on complex networks (percolation and resilience, propagation of information and epidemiological models, synchronization).

### Mathematical Models in Biology and Medicine

• Epidemiology and public health: classic compartment models (SIR, SIS, SEIR). Vaccination strategies. Learning parameters from observed data. Models with geographic structure. Epidemic spread on networks. Lessons learnt from COVID.

• Theoretical Neuroscience: Membrane channels and action potentials. HodgkinHuxley model and reduced neuron models (Morris-Lecar, Integrate & Fire, etc.).

• Statistical models in genomics.

• Tumour growth and Immune response models in cancer research.

• Mathematics of evolution: genotypes and phenotypes. Evolutionary dynamics. Stochastic models for evolution. Quasispecies theory.

• Cardiac and vascular dynamics. Mathematical models in sports science.

### Mathematical Models in Transport and Logistics

• Transportation planning and optimization (basics of transportation planning and optimization, including the formulation of transportation problems, the use of linear and integer programming, and the use of network analysis for optimization and decision making).

• Logistics and supply chain management (basics of logistics and supply chain management, including the formulation of logistics problems, the use of inventory

models, and the use of simulation for optimization and decision making).

• Vehicle routing and scheduling (basics of vehicle routing and scheduling, including the formulation of vehicle routing problems, the use of metaheuristics and optimization algorithms).

• Applications (use of mathematical models in various fields of transport and logistics such as fleet management, cargo handling, and logistics services. It includes realworld problems and case studies for different domains).

### Cryptography

• Encryption, Shannon’s lower bound and pseudo-random number generators.

• Two-party protocols and symmetric key encryption. Merkle’s key exchange protocol. Public-key encryption.

• Diffie-Hellman key exchange. Trapdoor functions and RSA.

• Digital signatures. Lamport’s One-time Signature Scheme. Many-time, stateful, signature schemes. Direct construction of digital signatures from RSA.

• Zero knowledge. Proofs for all of NP. Kilian’s Protocol.

• Lattices, Learning with Errors (LWE). Fully Homomorphic Encryption.

• Quantum cryptography protocol and postquantum cryptography.

### Computer Vision

• Introduction. Image formation. Colour.

• Image processing. Basic operators. Linear filter. Non-linear filter. Fourier transform. Pyramids and wavelets. Geometric transformations. Classic filter detectors.

• Deep learning for Computer Vision. Review of deep learning. Convolutional networks. Residual networks and advanced models.

• Main tasks. Image classification. Object detection. Segmentation. Depth estimation. Introduction to video and 3D processing.

ELECTIVE COURSES

### Natural Language Processing

• Basic concepts: distributed representations of words, word vectors, word window classification and introduction to language models. Dependency parsing.

• Recap of deep learning. Recurrent neural networks. Seq2seq models. Applications in machine translation.

• The Transformer architecture. Transfer learning. BERT and GPT (natural language generation)

• Advanced topics: question answering, zero-shot learning, prompting, reinforcement learning from human feedback, contrastive learning (CLIP).

### Reinforcement Learning

• Basic concepts: reinforcement learning, deterministic and stochastic environments, Markov Decision Processes (MDPs). Exploration vs exploitation.

• Value function, Q-function and Bellman equation. Dynamic Programming.

• Classic algorithms in reinforcement learning: value iteration, policy iteration… Use of Monte Carlo techniques.

• Deep Reinforcement Learning.

• Applications in games, robotics, advertising, etc.

### Information Theory

• Introduction to the basic information measures: entropy, divergence and mutual information.

• Basic principles of lossless data compression and error-correcting codes.

• Applications in cryptography and data storage.

• Introduction to information geometry.

### Robotics and Automation

• Spatial description and kinematics: Rigid body motion. Forward Kinematics, Inverse Kinematics, Jacobian Matrix, Homogeneous Transformations.

• Dynamics and Control: Newton-Euler Formulation, Lagrangian Formulation, Dynamics of Serial Robots, Dynamics of Parallel Robots. Control systems. Proportional-Integral-Derivative (PID) Control, State-Space Control, Trajectory Planning.

• Sensors and Actuators: Types of Sensors, Sensor Measurements, Sensor Fusion, Sensor-based Control. Types of Actuators, Actuator Dynamics, Actuator Control.

• Robotics Projects: Implementing Reinforcement Learning algorithms that enable robots to learn simple tasks.

ELECTIVE COURSES
ELECTIVE COURSES

### Financial Mathematics I

• Introduction to Financial Markets and Instruments: derivatives, stocks, bonds, options, futures, swaps and option pricing using the binomial model. Replication and martingale probability measures.

• Forward and Futures Contracts: price discovery and arbitrage, hedging strategies using futures.

• Capital Asset Pricing Model (CAPM) and other factor models. Markovich Theory of portfolio optimization. Sharpe ratios and efficient frontiers. Practical applications.

• Risk Management: Value-at-Risk, Expected Shortfall and applications of derivatives in portfolio management.

### Behavioral Economics and Decision Theory

• Introduction and Foundations of Behavioral Economics: cognitive biases, heuristics, the psychology of risk aversion, emotions.

• Social and Group Behavior in Decision Making: social influence, the economics of happiness, context and framing, nudges and choice architecture.

• Foundations of Utility Theory: preferences and their mathematical representation, multi-criteria decisions and independence concepts.

• Solving Decision Problems: decision trees and influence diagram, expected utility theory.

• Prospect Theory.

### Time Series

• Introduction to times series (stationarity, seasonality, autocorrelation, and periodograms).

• Times series decomposition (smoothing, additive and multiplicative decompositions).

• ARIMA models.

• Dynamic regression models.

• Forecasting methods. Counterfactual techniques and Intervention analysis.

### Economics and Computation

• Introduction to Game Theory. Equilibria and solutions of games. Best responses. Zero-sum two-player games and linear programming formulations. Selfish routing games and the Price of Anarchy. Network Cost-Sharing Games. The Price of Stability.

• Introduction to Mechanism Design. Auctions. Computational aspects. Applications to Ad markets.

• Voting rules. Definitions, Condorcet’s Paradox and May’s Theorem. Impossibility results (Arrow’s theorem).

• Forecasting: Prediction Markets. Proper Scoring Rules, Shared Scoring Rules, Forecasting contests.

• Decentralized markets and blockchains. Bitcoin.

### Financial Mathematics II

• Stochastic processes in finance: brownian motion, diffusion, Lévy flights, fat-tailed distributions.

• Stochastic integrals and stochastic differential equations, Itô interpretation, Stratonovich interpretation, relation between both interpretations.

• Binomial option pricing model and derivation of the Black-Scholes PDE. Application to pricing of European options, option trading strategies and Greeks. Implied volatility and volatility smile.

• Interest Rates and Swaps: bonds, FRAs, swaps and hedging in interest rate derivatives.

ELECTIVE COURSES

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The information in this brochure is subject to revisions or changes. You will find the most up-to-date information on the IE Universityʼs website.

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