V63.0121 Calculus I
Derivatives, antiderivatives, and integrals of functions of one real variable. Trigonometric, inverse trigonometric, logarithmic and exponential functions. Applications, including graphing, maximizing and minimizing functions. Areas and volumes.
V63.0122 Calculus II
Techniques of integration. Further applications. Plane analytic geometry. Polar coordinates and parametric equations. Infinite series, including power series.
V63.0123 Calculus III
Functions of several variables. Vectors in the plane and space. Partial derivatives with applications, especially Lagrange multipliers. Double and triple integrals. Spherical and cylindrical coordinates. Surface and line integrals. Divergence, gradient, and curl. Theorem of Gauss and Stokes.
V63.0140 Linear Algebra
Systems of linear equations, Gaussian elimination, matrices, determinants, Cramer's rule. Vectors, vector spaces, basis and dimension, linear transformations. Eigenvalues, eigenvectors, and quadratic forms.
If you want to go to graduate school, it is essential that you take a course in real analysis. After that, courses in probability theory, Markov chains, and differential equations, probably in that order will be very useful.
V63.0141 Honors Linear Algebra I - identical to G63.2110
Linear spaces, subspaces, and quotient spaces; linear dependence and independence; basis and dimensions. Linear transformation and matrices; dual spaces and transposition. Solving linear equations. Determinants. Quadratic forms and their relation to local extrema of multivariable functions.
V63.0142 Honors Linear Algebra II - identical to G63.2120
V63.0233 Theory of Probability
An introduction to the mathematical treatment of random phenomena occurring in the natural, physical, and social sciences. Axioms of mathematical probability, combinatorial analysis, binomial distribution, Poisson and normal approximation, random variables and probability distributions, generating functions, Markov chains applications.
V63.0234 Mathematical Statistics
An introduction to the mathematical foundations and techniques of modern statistical analysis for the interpretation of data in the quantitative sciences. Mathematical theory of sampling; normal populations and distributions; chi-square, t, and F distributions; hypothesis testing; estimation; confidence intervals; sequential analysis; correlation, regression; analysis of variance. Applications to the sciences.
V63.0250 Mathematics of Finance
Introduction to the mathematics of finance. Topics include: Linear programming with application pricing and quadratic. Interest rates and present value. Basic probability: random walks, central limit theorem, Brownian motion, lognormal model of stock prices. Black-Scholes theory of options. Dynamic programming with application to portfolio optimization.
V63.0252 Numerical Analysis
In numerical analysis one explores how mathematical problems can be analyzed and solved with a computer. As such, numerical analysis has very broad applications in mathematics, physics, engineering, finance, and the life sciences. This course gives an introduction to this subject for mathematics majors. Theory and practical examples using Matlab will be combined to study a range of topics ranging from simple root-finding procedures to differential equations and the finite element method.
V63.0262 Ordinary Differential Equations
First and second order equations. Series solutions. Laplace transforms. Introduction to partial differential equations and Fourier series.
V63.0263 Partial Differential Equations
Many laws of physics are formulated as partial differential equations. This course discusses the simplest examples, such as waves, diffusion, gravity, and static electricity. Non-linear conservation laws and the theory of shock waves are discussed. Further applications to physics, chemistry, biology, and population dynamics.
V63.0282 Functions of a Complex Variable
Complex numbers and complex functions. Differentiation and the Cauchy-Riemann equations. Cauchy's theorem and the Cauchy integral formula. Singularities, residues, and Laurent series. Fractional Linear transformations and conformal mapping. Analytic continuation. Applications to fluid flow etc.
V63.0325 Analysis I
The real number system. Convergence of sequences and series. Rigorous study of functions of one real variable: continuity, connectedness, compactness, metric spaces, power series, uniform convergence and continuity.
V63.0326 Analysis II
Functions of several variables. Limits and continuity. Partial derivatives. The implicit function theorem. Transformation of multiple integrals. The Riemann integral and its extensions.
G63.1410.001, 1420.001 INTRODUCTION TO MATHEMATICAL ANALYSIS I, II Fall term
Functions of one variable: rigorous treatment of limits and continuity. Derivatives. Riemann integral. Taylor series. Convergence of infinite series and integrals. Absolute and uniform convergence. Infinite series of functions. Fourier series. Spring term
Functions of several variables and their derivatives. Topology of Euclidean spaces. The implicit function theorem, optimization and Lagrange multipliers. Line integrals, multiple integrals, theorems of Gauss, Stokes, and Green.
G63.2450.001, 2460.001 COMPLEX VARIABLES I, II Fall Term
Complex numbers; analytic functions, Cauchy-Riemann equations; linear fractional transformations; construction and geometry of the elementary functions; Green's theorem, Cauchy's theorem; Jordan curve theorem, Cauchy's formula; Taylor's theorem, Laurent expansion; analytic continuation; isolated singularities, Liouville's theorem; Abel's convergence theorem and the Poisson integral formula.
Text: Introduction to Complex Variables and Applications, Brown & Churchill Spring Term
The fundamental theorem of algebra, the argument principle; calculus of residues, Fourier transform; the Gamma and Zeta functions, product expansions; Schwarz principle of reflection and Schwarz-Christoffel transformation; elliptic functions, Riemann surfaces; conformal mapping and univalent functions; maximum principle and Schwarz's lemma; the Riemann mapping theorem.}
Text: Complex Analysis, Alfors
G63.2470.001 ORDINARY DIFFERENTIAL EQUATIONS
Existence theorem: finite differences; power series. Uniqueness. Linear systems: stability, resonance. Linearized systems: behavior in the neighborhood of fixed points. Linear systems with periodic coefficients. Linear analytic equations in the complex domain: Bessel and hypergeometric equations.
Recommended text: Ordinary Differential Equations, Coddington & Levinson
G63.2490.001 PARTIAL DIFFERENTIAL EQUATIONS (one-term format)
Basic constant-coefficient linear examples: Laplace's equation, the heat equation, and the wave equation, analyzed from many viewpoints including solution formulas, maximum principles, and energy inequalities. Key nonlinear examples such as scalar conservation laws, Hamilton-Jacobi equations, and semilinear elliptic equations, analyzed using appropriate tools including the method of characteristics, variational principles, and viscosity solutions. Simple numerical schemes: finite differences and finite elements. Important PDE from mathematical physics, including the Euler and Navier-Stokes equations for incompressible flow.
Suggested texts: Partial Differential Equations, Paul R. Garabedian, AMS; Partial Differential Equations, L. C. Evans, AMS; Partial Differential Equations, Fritz John, Springer
G63.2550.001 FUNCTIONAL ANALYSIS
The course will concentrate on concrete aspects of the subject and on the spaces most commonly used in practice and their duals. Working knowledge of Lebesgue measure and integral is expected. Special attention to Hilbert space (L2, Hardy spaces, Sobolev spaces, etc.), to the general spectral theorem there, and to its application to ordinary and partial differential equations. Fourier series and integrals in that setting. Compact operators and Fredholm determinants with an application or two. Introduction to measure/volume in infinite-dimensional spaces (Brownian motion). Some indications about non-linear analysis in an infinite-dimensional setting. General theme: How does ordinary linear algebra and calculus extend to infinite dimensions?
Mandatory text: Functional Analysis, P. Lax, (Pure & Applied Mathematics, New York), Wiley-Interscience, John Wiley & Sons, 2002
Rec. text: Methods of Modern Mathematical physics Vol. I: Functional Analysis, M. Reed & B. Simon, Academic Press, New York-London, 1972
G63.2012.002 ADVANCED TOPICS IN NUMERICAL ANALYSIS (Numerical Methods with Probability)
A continuation of Numerical Methods I, introducing statistical and scientific applications of numerical linear algebra (including randomized algorithms), digital signal processing (including stochastic processes), spectral and adaptive schemes for numerical integration, Monte-Carlo techniques (including Metropolis and Hastings), the enhancement of accuracy via postprocessing, and other fundamentals. The focus is on basic methods for solving problems encountered frequently in modern science and technology.
Cross-listed as G22.2945.002
G63.2902.001 STOCHASTIC CALCULUS
Review of basic probability and useful tools. Bernoulli trials and random walk. Law of large numbers and central limit theorem. Conditional expectation and martingales. Brownian motion and its simplest properties. Diffusion in general: forward and backward Kolmogorov equations, stochastic differential equations and the Ito calculus. Feynman-Kac and Cameron-Martin Formulas. Applications as time permits.
Text: Stochastic Calculus, A Practical Introduction, Richard Durrett, CRC Press, Probability & Stochastics Series
G63.2911.001, 2912.001 PROBABILITY: LIMIT THEOREMS I, II
Newman, (fall); H. McKean (spring). Fall term
Probability, independence, laws of large numbers, limit theorems including the central limit theorem. Markov chains (discrete time). Martingales, Doob inequality, and martingale convergence theorems. Ergodic theorem. Spring term
Independent increment processes, including Poisson processes and Brownian motion. Markov chains (continuous time). Stochastic differential equations and diffusions, Markov processes, semigroups,
generators and connection with partial differential equations.
Spring text: Stochastic Processes, S. R. S. Varadhan, CIMS - AMS, 2007
G63.2931.001 ADVANCED TOPICS IN PROBABILITY (Markov Processes and Diffusions).
In the first part of the course, we will give an introduction to the general theory of Markov processes for both discrete and continuous time. Our main focus will be the study of their long-time behavior (transience, recurrence, ergodicity, mixing) in the classical context of Harris chains, but also for a larger class of processes that doesn't fit into this context. The second part of the course will be aimed at applying the abstract results from the first part to the more concrete framework of elliptic diffusion processes. Lyapunov function techniques will play a prominent role in this part of the course. The final part of the course will be an introduction to the theory of hypoelliptic diffusion processes. We will give a short introduction to Mallivain calculus and use it to give a probabilistic proof of Hormander's famous "sums of squares" theorem.
Recommended texts: Markov Chains and Stochastic Stability, Meyn and Tweedie ; Introduction to the Theory of Diffusion Processes, Krylov; The Malliavin Calculus and Related Topics, Nualart
G63.2932.001 ADVANCED TOPICS IN PROBABILITY (Large Deviations and Applications)
Prerequisites: Probability: Limit Theorems I and II; familiarity with some Markov Processes, Brownian motion, SDE, diffusions.
Standard Cramer Theory for sums of iid random variables, Ventcel Freidlin theory for ordinary differential equations with small noise and the exit problem. Donsker-Varadhan theory of large time behavior of Markov Processes. Applications to interacting particle systems.
Recommended Texts on Large Deviations: Dembo & Zeitouni, Deuschel & Stroock, Weiss & Schwartz
G63.2044 Monte Carlo Methods and Simulation of Physical Systems Principles of Monte Carlo: sampling methods and statistics, importance sampling and variance reduction, Markov chains and the Metropolis algorithm. Advanced topics such as acceleration strategies, data analysis, and quantum Monte Carlo and the fermion problem.
G63.2830, 2840 Advanced Topics in Applied Mathematics Recent topics: mathematical models of crystal growth; math adventures in data mining; ice dynamics; vortex dynamics; applied stochastic analysis; developments in statistical learning; fluctuation dissipation theorems and climate change; theory and modeling of rare events.