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Discrete Geometry: Euclidean, spherical and hyperbolic n-spaces, trigonometry, isometries, convex sets, convex polytopes, (mixed) volume(s), classical discrete groups, tilings, isoperimetric inequalities, packings, coverings. Graph Theory: connectivity; trees; Euler trails and tours; Hamilton cycles and paths; matchings; edge colourings; vertex colourings; homomorphisms; plane and planar graphs; extremal graph theory and Ramsey theory.

631.01. Discrete Geometry

631.03. Graph Theory

Algebraic Number Theory: an introduction to number fields, rings of integers, ideals, unique factorization, the different and the discriminant. The main objective to the course will be to prove the finiteness of the class number and Dirichlet's Unit Theorem.
Analytic Number Theory: students will learn tools to aid in the study of the average behaviour of arithmetic functions, including the use of zeta functions, to prove results about the distribution of prime numbers.

641.01. Algebraic Number Theory

641.03. Analytic Number Theory

Modular forms and automorphic representations and their L-functions. Modularity Theorem from two perspectives.

Classical Perspective on Modular Forms: introduction to modular curves as moduli spaces for elliptic curves and as differential forms on modular curves. A study of L-functions attached to modular forms and the modularity theorem.

An Introduction to Automorphic Representations: introduction to the Langlands Programme. A study of partial L-functions attached to automorphic representations and known instances of the Langlands Correspondence.

647.01. Classical Perspective on Modular Forms

647.03. An Introduction to Automorphic Representations

The Convex Optimization: an introduction to modern convex optimization, including basics of convex analysis and duality, linear conic programming, robust optimization, and applications.

Scientific Computation: an introduction to both the methodological and the implementation components underlying the modern scientific computations with the natural emphasis on linear algebra, including modern computing architecture and its implications for the numerical algorithms.

Numerical Differential Equations: fundamentals of solving DEs numerically addressing the existence, stability and efficiency of such methods.

661.01. Convex Optimization

661.03. Scientific Computation

661.05. Numerical Differential Equations

Interior Point Methods: exposes students to the modern IPM theory with some applications, to the extent that at the end of the course a student should be able to implement a basic IPM algorithm.

Theoretical Numerical Analysis: provides the theoretical underpinnings for the analysis of modern numerical methods, covering topics such as linear operators on normed spaces, approximation theory, nonlinear equations in Banach spaces, Fourier analysis, Sobolev spaces and weak formulations of elliptic boundary value problems, with applications to finite difference, finite element and wavelet methods.

Differential Equations: essential ideas relating to the analysis of differential equations from a functional analysis point of view. General topics include Hilbert spaces and the Lax-Milgram’s theorem, variational formulation of boundary value problems, finite element methods, Sobolev spaces, distributions, and pseudo-differential operators.

663.01. Interior Point Methods

663.03. Theoretical Numerical Analysis

663.05. Differential Equations

Focus on the mathematical treatment of a broad range of topics in quantum Shannon theory. Topics include quantum states, quantum channels, quantum measurements, completely positive maps, Neumarkís theorem, Stinespring dilation theorem, Choi-Jamiolkowski isomorphism, the theory of majorization and entanglement, the Peres-Horodecki criterion for separability, Shannon’s noiseless and noisy channel coding theorems, Lieb’s theorem and the strong subadditivity of the von Neumann entropy, Schumacher’s quantum noiseless channel coding theorem, and the Holevo-Schumacher-Westmoreland theorem.

Wavelet Analysis: covers the design and implementation of wavelet methods for modern signal processing, particularly for one- and two-dimensional signals (audio and images).

Mathematical Biology: introduction to discrete models of mathematical biology, including difference equations, models of population dynamics and the like. Topics include stability of models describe by difference equations, continuous spatially homogeneous processes and spatially distributed models.

669.01. Wavelet Analysis

669.03. Mathematical Biology

Topics include specific areas of mathematical finance and build on Mathematics 681.

Lévy Processes (LP): fundamental concepts associated with LP such as infinite divisibility, the Lévy-Khintchine formula, the Lévy-Itô decomposition, subordinators, LP as time-changed Brownian motions, and also dealing with semi-groups and generators of LP, the Itô formula for LP, the Girsanov theorem, stochastic differential equations driven by LP, the Feynman-Kac formula, applications of LP and numerical simulation of LP.

Credit Risk: corporate bond markets, modelling the bankruptcy risk of a firm, and understanding how corporate bonds are priced.

Stochastic Optimal Control and Applications in Finance: An introduction to the theory of stochastic optimal control and applications in finance and economics. Dynamic programming approach to optimal controls, solutions to several classes of typical optimal control problems, and application of the general theory to some classical models in finance and economics.

691.01. Lévy Processes in Finance

691.03. Credit Risk

691.05 Stochastic Optimal Control and Applications in Finance

Topics include specific areas of mathematical finance and build on Mathematics 681 and 683.

Monte Carlo Methods for Quantitative Finance: random number generation, simulation of stochastic differential equations, option valuation, variance reduction techniques, quasi-Monte Carlo methods, computing ‘greeks', valuation of path-dependent and early-exercise options; applications to risk management; Markov Chain Monte Carlo methods.

Energy, Commodity and Environmental Finance: energy and commodity markets; spot, futures, forwards and swap contracts; the theory of storage; stochastic models for energy prices; model calibration; emissions market modelling; weather derivatives; energy risk management; energy option valuation.

Advanced Topics in Mathematical Finance: An introduction to some of the main ideas in mathematical and computational finance through a guided reading of some seminal papers from the last 100 years, starting with Bachelier's 1900 thesis, and including papers by Samuelson, Markowitz, Black & Scholes, Merton, Hull & White, Schwartz, Glasserman, and others.

693.01 Monte Carlo Methods for Quantitative Finance

693.03 Energy, Commodity and Environmental Finance

693.05 Advanced Topics in Mathematical Finance