A professional skills course, focusing on the development of technical proficiencies that are essential to succeed as practicing mathematicians in academia, government, or industry. The emphasis is on delivering professional presentations and using modern mathematical research tools. A high level of active student participation is required. Course Hours:1.5 units; (3S-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. MAY BE REPEATED FOR CREDITNOT INCLUDED IN GPA

Abstract measure theory, basic integration theorems, Fubini's theorem, Radon-Nikodym theorem, Lp spaces, Riesz representation theorem. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 601 and either Mathematics 501 or Pure Mathematics 501 will not be allowed.

Sequences and series of functions; Lebesgue integration on the line, Fourier series and the Fourier transform, pointwise convergence theorems, distributions and generalized functions. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 603 and either Mathematics 545 or Pure Mathematics 545 will not be allowed.

Systems of ordinary differential equations. Existence and uniqueness. Introduction to partial differential equations. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 605 and Applied Mathematics 605 will not be allowed.

A sophisticated introduction to modules over rings, especially commutative rings with identity. Major topics include: snake lemma; free modules; tensor product; hom-tensor duality; finitely presented modules; invariant factors; free resolutions; and the classification of finitely generated modules over principal ideal domains. Adjoint functors play a large role. The course includes applications to linear algebra, including rational canonical form and Jordan canonical form. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 607 and any of Mathematics 511, Pure Mathematics 511, 607 or 611 will not be allowed.

Introduction to Hilbert and Banach spaces, linear operators, weak topologies, and the operator spectrum. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 617 and Applied Mathematics 617 will not be allowed.

A rigorous study of function of a single complex variable. Holomorphic function, Cauchy integral formula and its applications. Conformal mappings. Fractional linear transformations. Argument principle. Schwarz lemma. Conformal self-maps of the unit disk. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 621 and Mathematics 521 will not be allowed.

Introduction to the algebraic invariants that distinguish topological spaces. Focus on the fundamental group and its applications, and homology. Introduction to the basics of homological algebra. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 625 and either Mathematics 525 or Pure Mathematics 607 will not be allowed.

Introduction to modern algebraic geometry sufficient to allow students to read research papers in their fields which use the language of schemes. Topics will include Spectra of rings; the Zariski topology; affine schemes; sheaves; ringed spaces; schemes; morphisms of finite type; arithmetic schemes; varieties; projective varieties; finite morphisms, unramified morphisms; etale morphisms. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

The interplay of the group-theoretic notion of lattice and the geometric concept of convex set, the lattices representing periodicity, the convex sets geometry. Topics include convex bodies and lattice points, the critical determinant, the covering constant and the inhomogeneous determinant of a set, Star bodies, methods related to the above, and homogeneous and inhomogeneous forms. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

An excursion into the infinite world, from Ramsey Theory on the natural numbers, to applications in Number Theory and Banach Spaces, introduction to tools in Model Theory and Logic, fascinating homogeneous structures such as the rationals and the Rado graph, and possibly further explorations into the larger infinite world. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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

Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

An investigation of major problems in computational number theory, with emphasis on practical techniques and their computational complexity. Topics include basic integer arithmetic algorithms, finite fields, primality proving, factoring methods, algorithms in algebraic number fields. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 643 and any of Mathematics 527, Pure Mathematics 527, or Pure Mathematics 627 will not be allowed.

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

Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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 modern scientific computations, with a natural emphasis on numerical linear algebra, and including modern computing architectures and their implications for 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

Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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

Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

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

Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department.

Martingales in discrete and continuous time, risk-neutral valuations, discrete- and continuous-time (B,S)-security markets, the Cox-Ross-Rubinstein formula, Wiener and Poisson processes, Itô’s formula, stochastic differential equations, Girsanov’s theorem, the Black-Scholes and Merton formulas, stopping times and American options, stochastic interest rates and their derivatives, energy and commodity models and derivatives, value-at-risk and risk management. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 681 and any one of Mathematics 581, Applied Mathematics 681, or Applied Mathematics 581 will not be allowed. Also known as:(formerly Applied Mathematics 681)

Basic computational techniques required for expertise quantitative finance. Topics include basic econometric techniques (model calibration), tree-based methods, finite-difference methods, Fourier methods, Monte Carlo simulation and quasi-Monte Carlo methods. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 683 and any one of Mathematics 583, Applied Mathematics 683, or Applied Mathematics 583 will not be allowed.
Also known as:(formerly Applied Mathematics 683)

Stochastic processes are fundamental to the study of mathematical finance, but are also of vital importance in many other areas, from neuroscience to electrical engineering. Topics to be covered: Elements of stochastic processes, Markov chains and processes, Renewal processes, Martingales (discrete and continuous times), Brownian motion, Branching processes, Stationary processes, Diffusion processes, The Feynman-Kac formula, Kolmogorov backward/forward equations, Dynkin’s formula. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics or consent of the Department. Antirequisite(s):Credit for Mathematics 685 and Statistics 761 will not be allowed.

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

Course Hours:3 units; (3-0) Prerequisite(s):Mathematics 681 and admission to a graduate program in Mathematics and Statistics or consent of the Department.

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

Course Hours:3 units; (3-0) Prerequisite(s):Mathematics 681 and 683 and admission to a graduate program in Mathematics and Statistics or consent of the Department.

Topics will be chosen according to the interest of the instructors and students. Course Hours:3 units; (3-0) Prerequisite(s):Admission to a graduate program in Mathematics and Statistics. MAY BE REPEATED FOR CREDIT