Program in Applied Mathematics




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НазваниеProgram in Applied Mathematics
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TOPICS IN FINANCIAL MATHEMATICS



  • Prerequisites: Basic Calculus, Probability




  • Books:
    1. J. V. Allen, Lectures Notes on Actuarial Mathematics, manuscript, 2005.
    2. J.-P. Aubin, Analyse Non Lineaire et ses Motivations Economiques, Masson, 1984.




  • Commitment: 3 hours/week, 3 credits




  • Contents:


1. Financial markets, financial derivatives, payoff functions
2. Asset price model
3. Black-Scholes analysis; American and European options; Black-Scholes formula
4. Variations on Black-Scholes models; Future options
5. Numerical methods: Monte Carlo method, binomial method, finite difference method, fast algorithms for solving linear systems
6. Exotic options
7. Path-dependent options: General method, average strike options, look-back options
8. Bonds and interest rate derivatives: Bond models, interest models


APPROXIMATION THEORY



  • Prerequisites: Real Analysis and Complex Function Theory




  • Textbooks:
    1. G.G. Lorentz: Approximation of Functions, Holt, Rinehart and Winston, 1966
    2. R. DeVore and G. G. Lorentz, Constructive Approximation, Springer, 1993




  • Commitment: 3 hours/week, 3 credits




  • Contents:


Best polynomial and rational approximation, moduli of smoothness, Jackson-Timan type quantitative direct estimates for polynomial approximation, converse theorems, positive linear operators, Korovkin theorems, interpolation (Lagrange, Newton, Hermite), Fourier series.


APPLIED NUMERICAL ANALYSIS



  • Prerequisites: Undergraduate Calculus, Linear Algebra




  • Books:
    1. A. Quarteroni, A. Valli,  Numerical Approximation of Partial Differential Equations, Series: Springer Series in Computational Mathematics, Vol. 23, 1997.
    2. L. J. Segerlind, Applied Finite Element Analysis, J. Wiley & Sons, New York 1987.
    3. M.N.O. Sadiku, Numerical Techniques in Electromagnetics, 2nd edition, CRC Press 2001.
    4. K. Kunz, R. Lubbers, The Finite Difference Time Domain Method for Electromagnetics, CRC Press 1993.




  • Commitment: 3 hours/week, 3 credits




  • Contents:


1. Approximations of differential equations (finite difference, finite element, Galerkin and collocation methods)
2. Applications (Heat transfer, Maxwell equations, air-pollution transport model, torsion of noncircular sections, irrotational flows, Black-Scholes' equation)
3. Operator splitting techniques, matrix exponentials and their applications to air-pollution transport models and to Maxwell equations
4. Qualitative properties of mathematical and numerical models
5. Computer examples and implementations


MATHEMATICAL MODELS IN BIOLOGY AND ECOLOGY


  • Prerequisites: Basic Calculus, Ordinary Differential Equations




  • Books:
    1. L. Edelstein-Keshet, Mathematical Models in Biology, SIAM Classics in Applied Mathematics 46, 2004
    2. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2001




  • Commitment: 3 hours/week, 3 credits




  • Contents:


1. Discrete and continuous single species models. Exponential and logistic growth. The delayed logistic equation
2. Multi-species communities: competition, comensualism, coexistence.
3. Predator-prey models. The Lotka-Volterra model and more complicated models (Gause, Kolmogorov). Prey-dependent and ratio-dependent predation.
4. Chemical reaction kynetics: Michaelis-Menten theory
5. Simple oscillatory reactions. Nerve impulses and Hodgkin-Huxley theory. FitzHugh-Nagumo model.
6. Reaction-diffusion equations. Convection, advection. Chemotaxis.
7. Ecological epidemiology: integrated pest management strategies.
8. Numerical simulations and visualisations by means of XPP, Phaser, Maple (or equivalent)


THE MATHEMATICAL THEORY OF INFECTIOUS DISEASE PROPAGATION


  • Prerequisites: Basic Calculus, Ordinary Differential Equations




  • Books:
    1. F. Brauer and C. Castillo-Chavez, Mathematical Models in Population Biology and Epidemiology, Springer, 2001
    2. V. Capasso, Mathematical structures of Epidemic Systems, Lecture Notes in Biomathematics, Springer Verlag, Berlin 1993.




  • Commitment: 3 hours/week, 3 credits




  • Contents:


1. Basic concepts of mathematical epidemiology. Deterministic models. Compartmental models.
2. Single population models with constant population size. Models with no immunity.
3. Models with nonconstant population size and immunity effects. Basic reproduction number of a disease. Stability and persistence.
4. Infective periods of fixed length. Models with delay. Arbitrarily distributed infective periods.
5. Seasonality and periodicity. Orbital stability of periodic solutions.
6. Models with pulse vaccination.
7. Multigroup models (models with patchy structure).
8. Numerical simulations and visualisations by means of XPP, Phaser, Maple (or equivalent).


EVOLUTIONARY GAME THEORY AND POPULATION DYNAMICS



  • Prerequisites: Basic Calculus, Ordinary Differential Equations, Mathematical Models in Population Dynamics




  • Book: J. Hofbauer and K. Zigmund, Evolutionary Games and Population Dynamics, Cambridge University Press, 1998.




  • Commitment: 3 hours/week, 3 credits




  • Contents:


1. Evolutionary stability. Normal form games. Evolutionarily stable strategies. Population games
2. Replicator dynamics. The equivalence of the replicator equation to the Lotka-Volterra equation. The rock-scissors-paper game. Partnetship games and gradients
3. Other game dynamics. Imitation dynamics. Monotone selection dynamics. Best-response dynamics. Adjustment dynamics. A universally cyclic game.
4. Adaptive dynamics. The repeated Prisoner's Dilemma. Adaptive dynamics and gradients.
5. Asymmetric games and replicator dynamics for them.
6. Population dynamics and game dynamics
7. Game dynamics for Mendelian populations
8. Numerical simulations and visualisations


BIOINFORMATICS


  • No. of Credits: 3 and no. of ECTS credits: 6




  • Course Level: introductory




  • Brief introduction to the course:


Stochastic models: HMMs, SCFGs and time-continuous Markov models and their algorithmic aspects.



  • The goals of the course:


 To learn the stochastic transformational grammars, especially HMMs and SCFGs
 To learn time-continuous Markov models describing sequence evolution
 To learn the algorithmic background of these models
  To learn the statistical background and tools, like Maximum Likelihood and Expectation Maximization


  • The learning outcomes of the course:


The students will be able to read and understand scientific papers related to the topic.


  • More detailed display of contents:


Lecture 1.
Theory: Score based dynamic programming algorithms. Linear, concave and affine gap penalties.

Lecture 2.
Theory: Conditional probability, Bayes theorem. Unbiased, consistent estimations. Statistical testing. Local alignment, extreme value distributions for local alignments, p and E value estimations.

Lecture 3.
Theory: Hidden Markov Models. Parsing algorithms: Forward, Backward and Viterbi. Posterior probabilities. Expectation Maximization. The Baum-Welch algorithm.

Lecture 4.
Theory: Profile HMMs. Aligning sequences via profile-HMMs. Pair-HMMs.
Practice: HMM topology design.

Lecture 5.
Theory: Substitution models. Felsenstein’s algorithm for fast likelihood calculation of a tree.


Lecture 6.
Theory: Predicting protein secondary structures with profile HMMs and evolutionary models. Gene prediction with HMMs.

Lecture 7.
Theory: Modeling insertions and deletions with time-continuous Markov models: The Thorne-Kishino-Felsenstein models.

Lecture 8.
Theory: Describing the TKF models as pair-HMMs. Extension to many sequences: multiple-HMMs. The transducer theory for evolving sequences on an evolutionary tree.

Lecture 9.
Theory: Stochastic transformational grammars. Stochastic regular grammars are HMMs. Stochastic Context-Free Grammars. Parsing algorithms for SCFGs: Inside, Outside and CYK.

Lecture 10.
Theory: Posterior decoding of SCFGs. Expectation Maximization. Combining SCFGs with evolutionary models: the Knudsen-Hein algorithm.

Lecture 11.
Theory: Covarion Models as ‘profile-SCFGs’. The RFam database. Predicting tRNAs in the human genome.

Lecture 12.
Theory: The Zuker-Tinoco model for RNA secondary structures. Calculating the partition function of the Boltzmann distribution and other moments of the Boltzmann distribution.


COMPUTATIONAL NEUROSCIENCE


  • Prerequisites: Undergraduate Calculus, Elementary Linear Algebra, Basic knowledge of Differential Equations,
    Further useful skills: programming in Matlab/Scilab/Octave, NEURON, XPPaut




  • Books:
    1.Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, Dayan P, L.F. Abbott, MIT press, 2001.
    2.Spikes: Exploring the Neural Code, Rieke F, Warland D, van Steveninck RR, Bialek W, MIT press,1997.
    3. Spiking Neuron Models, Wulfram Gerstner and Werner M. Kistler Cambridge University Press 2002.
    4. Neural Organization: Structure, Function and Dynamics. Arbib MA, Érdi P, Szentágothai J, MIT Press,1997.




  • Commitment: 3 hours/week, 3 credits




  • Contents :


1. General introduction to Computational Neuroscience
2. General introduction to the anatomy, evolution and cellular basis of the nervous system
3. Basics of nerve cell electrochemistry and electrophysiology. Conductance based models of neurons
4. Parallel conductance model. Mechanism of action potential generation. The Hodgkin-Huxley model. Ionic currents, ionchannels, gate kinetics
5. Simplified neuron models. Simplifications of the Hodgkin-Huxley model: the FitzHugh-Nagumo-Rinzel model, phase-space analysis. Explanation of bursting by bifurcation analysis. Abstract models: phase model, rate model, McCulloch-Pitts neuron, integrate and Fire neuron model
6. Beyond the Hodgkin_Huxley model. Diverse voltage- and ligand gated kinetics in single-compartment models. Role of cellular morphology, dendritic effects. The cable-equation and multi compartmental models. What is detailed modeling good for? Taxonomy of neuron models. Synapses and synaptic plasticity. Detailed, simplified and phenomenological models od the synaptic function
7. Cellular bases of learning: synaptic plasticity. The Hebbian rule of learning. variations for the Hebbian rule. Long term synaptic potentiation and depression. Synaptic plasticity on different time scales. Metaplasticity. Basics of modeling neural networks. The two (three) levels of neural dynamics. Learning rules: reinforcement, supervised and unsupervised learning. Basic neural architectures: feedforward and feedback structures, lateral connections, attractor networks
8. Windows to the World: traditional and modern measuring and data processing techniques: EEG, PET, fMRI, electrodes, intra- and extracellular measurements, patch-clamp. Fourier- and wavelet transformations, EEG/MEG imaging, spike-sorting
9. Neural oscillations: generation of oscillations an the cellular and network level. Oscillation based neural computations: timing and dynamic linking. Oscillations in memory models
10. The hyppocampus: modeling memory and spatial navigation. Place cells and place fields. Phase and rate coding. Dynamic modes of the hyppocampus
11. Modeling neurological and psychiatric disorders. Epilepsy, Parkinson's disease, Alzheimer's disease, schisophrenia.


PROBABILISTIC MODELS OF THE BRAIN AND THE MIND


  • Prerequisites: Undergraduate Calculus, Elementary Linear Algebra, Probability and Statistics




  • Books:
    1. Dayan & Abbott. Theoretical Neuroscience: Computational and Mathematical Modeling of Neural Systems, MIT press, 2001.
    2. MacKay. Information Theory, Inference & Learning Algorithms, Cambridge University Press, 2002.
    3. Sutton & Barto. Reinforcement Learning: An Introduction, MIT Press, 1998.
    4. Doya et al. Bayesian Brain: Probabilistic Approaches to Neural Coding, MIT Press, 2007.
    5. Rao et al. Probabilistic Models of the Brain: Perception and Neural Function, MIT Press, 2002.




  • Commitment: 3 hours/week, 3 credits




  • Contents:


Machine learning, unsupervised learning, Bayesian networks, reinforcement learning, sampling algorithms, variational methods, computer vision, Cognitive science,
inductive reasoning, statistical learning, semantic memory, vision as analysis by synthesis,
sensorimotor control, classical and instrumental conditioning, behavioural economics,
Neuroscience, neural representations of uncertainty, probabilistic neural networks,
probabilistic population codes, natural scene statistics and efficient coding,
neuroeconomics, neuromodulation


COMPUTATIONAL NUMBER THEORY



  • Prerequisities: Basic Algebra 1, Real Analysis




  • Book: Richard Crandall, and Carl Pomerance, Prime Numbers - A Computational Perspective, Springer,2000




  • Commitment: 3 hours/week, 3 credits




  • Contents:


  Primes, primes of special form, the prime number theorem
  Sieving
  Arithmetic on large numbers
  Primality test: Fermat and Frobenius test
  Proving primality: the polynomial algorithm
  Factoring primes: Pollard rho method, Baby-step Giant-step method
  Solving the discrete logarithm problem
  Subexponential pactoring algorithm, quadratic sieve
  Elliptic curves, using elliptic curves in factoring


PROTOCOLS

Prerequisites: Basic Algebra 1, Introduction to Computer Science


  • Book: Colin Boyd and Anish Maturia, Protocols for Authentication and Key Establishment, Springer, 2003.




  • Commitment: 3 hours/week, 3 credits




  • Contents:


What protocols are; properties, attacks agains protocols
Turing Machines, Oracle, computational and decisional Diffie-Hellman
problems
Protocols involving two or more parties, security notions, simulability
ZK protocols, concurrent ZK, resettable ZK protocols, AM protocols
Public key and symmetric key protocols
Protocols for key establishment
Identity based protocols, signatures
Famous attacks against famous protocols

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