| AM8000 – Graduate Seminar |
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This course extends over four academic terms (Fall and Winter of the first year and Fall and Winter of the second year.
It is graded as a Pass or Fail course. All students are required to attend regularly as well as to participate.
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| Group A: Foundation Courses |
| AM8001 – Analysis and Probability |
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A selection from the following topics in Applied Analysis and Probability:
curvilinear coordinates; contour integration; conformal mapping; Fourier
series; eigenfunction expansions; measure spaces, integration, random variables
and conditional expectation; random variables, modes of convergence,
discrete time martingales and filtrations; Brownian motion, continuous time
stochastic processes and martingales; stochastic calculus; stochastic and ordinary
differential equations; introduction to partial differential equations.
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| AM8002 – Discrete Mathematics and its Applications |
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Selected topics from discrete mathematics: graph isomorphisms and homomorphisms; Ramsey theory, random graphs; infinite graphs; automorphism
groups; graph searching games (such as Cops and Robbers); Steiner triple
systems; graph decompositions; Latin squares; finite fields; polynomial rings;
finite projective and affine planes.
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| Group B: Core Courses |
| AM8101 – Principles and Techniques in Applied Mathematics, Part I |
| Integral Transform; Discrete Fourier Transforms; Asymptotic Expansions; Perturbation Methods; Calculus of Variations. |
| AM8102 – Principles and Techniques in Applied Mathematics, Part II |
| Numerical Methods; Numerical Linear Algebra; Numerical methods for ODEs; Numerical Methods for PDEs; Numerical Simulations. |
| Group C: Elective Courses |
| AM8201 – Financial Mathematics |
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This course covers the fundamentals of mathematical methods in finance.
After providing a background in Stochastic Calculus, it considers the study
of financial derivatives. Fixed income instruments, derivative pricing in discrete and continuous time,
including Black-Scholes formulation, American, Exotic and Futures derivatives. Elements of Portfolio Management and
Capital Asset Pricing Model are taken into account.
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| AM8202 – Digital Signals and Wavelets |
| Digital Signals; Wavelets; Adaptive Methods; Two and Three Dimensional Transforms; Applications |
| AM8203 – Topics in Functional Analysis |
| Normed Spaces; Fundamental Results; Hilbert Spaces; Calculus in Banach Spaces; Additional Topics |
| AM8204 – Topics in Discrete Mathematics |
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Selected advanced topics from discrete mathematics: random graphs; models of complex networks; homomorphisms and constraint satisfaction; adjacency properties; Ramsey theory; graph searching games; Latin squares;
designs, coverings, arrays, and their applications.
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| AM8205 – Applied Statistical Methods |
| This course covers a wide variety of Statistical methods with
application in Medicine, Engineering, Economics. Exploratory data analysis. Parametric probability distributions.
Sampling and experimental designs. Estimation, confidence intervals and tests of hypothesis.
Analysis of variance. Multiple regression analysis, tests for normality. Nonparametric statistics.
Statistical analysis of time series; ARMA and GARCH processes. Practical techniques for the analysis
of multivariate data; principal components, factor analysis. |
| AM8206 – Partial Differential Equations |
| Hyperbolic equations, weak solutions, shock formation, nonlinear
waves, reaction-diffusion equations, traveling wave solutions,
elliptic equations, numerical methods, applications. |
| AM8207 – Topics in Biomathematics |
| Discrete and Continuous time processes applied to biology and chemistry. Deterministic and stochastic descriptions for birth/death processes in chemical kinetics. Numerical methods for spatially distributed systems including multi-species reaction-diffusion equations. Applications will include some or all of: chemical waves, traveling wave fronts in excitable media, spiral waves, pattern formation, blood flow and flow in chemical reactors. |
| AM8208 – Topics in Mathematics |
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The Banach fixed point theorem and iterative methods, the Schauder
fixed point theorem and compactness, ordinary differential equations
in Banach spaces, predator-prey systems, epidemic disease models,
differential calculus and the implicit function theorem, continuation
with respect to a parameter, positive operators, nonexpansive
operators and iterative methods, applications to ordinary differential
equations, fractional differential equations, partial differential
equations and integral equations.
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| AM8209 – Directed Studies in Math |
