Rigorous and intensive treatment of standard topics from algebra and trigonometry as a preparation for further courses in calculus or statistics. Fulfills the QR requirement. Fall Only.

# All courses in Fall 2012

Second semester of the 3-semester calculus sequence 103/104/201. Main topics are Integration and Series. Offered Fall and Spring.

Second semester of the standard 3-semester calculus sequence 103/104/201 for science, engineering and finance. Topics include techniques and applications of integration, convergence of infinite series and improper integrals, Taylor's theorem, introduction to differential equations and complex numbers. Emphasizes concrete computations over more theoretical considerations. Offered both Fall and Spring. Prerequisite: MAT103 or equivalent.

Taken concurrently with EGR/MAT/PHY 192, this course offers an integrated presentation of the material from PHY 103 (General Physics: Mechanics and Thermodynamics) and MAT 201 (Multivariable Calculus) with an emphasis on applications to engineering. Physics topics include: mechanics with applications to fluid mechanics; wave phenomena; and thermodynamics. Open only to BSE freshmen and adminstered by the Keller Center.

Taken concurrently with EGR/MAT/PHY 191, this course offers an integrated presentation of the material from PHY 103 (General Physics: Mechanics and Thermodynamics) and MAT 201 (Multivariable Calculus) with an emphasis on applications to engineering.
Open only to BSE freshmen and administered by the Keller Center.

A continuation of MAT103/104, the third semester in the calculus sequence gives a thorough introduction to multivariable calculus. Topics include limits, continuity and differentiability in several variables, extrema, Lagrange multipliers, Taylor's theorem, multiple integrals, integration on curves and surfaces, Green's theorem, Stokes' theorem, divergence theorem. Emphasizes concrete computations over more theoretical considerations. Offered both Fall and Spring. Prerequisite: MAT104 or equivalent.

Linear Algebra, mostly in real n-space. Companion course to 201. Main topics are matrices, linear transformations, linear independence and dimension, bases and coordinates, determinants, orthogonal projection, least squares, eigenvalues, eigenvectors and their applications to quadratic forms and dynamical systems. Offered Fall and Spring.

Advanced multivariable calculus. More theoretical treatment of limits, continuity, differentiation and integration for functions of several variables than that found in MAT201, but less theoretical than MAT218. A course for those with a strong mathematical background and interest. Recommended for physics majors and students interested in applied math. Offered Fall only.

Rigorous, proof-based introduction to classical number theory, to prepare for higher-level courses in the department. Topics include Pythagorean triples and sums of squares, unique factorization, Chinese remainder theorem, arithmetic of Gaussian integers, finite fields and cryptography, arithmetic functions and quadratic reciprocity. There will be a topic, chosen by the instructor, from more advanced or more applied number theory: possibilities include p-adic numbers, cryptography, and Fermat's Last Theorem. Suitable both for students preparing to enter the Mathematics Department and for non-majors interested in exposure to higher mathematics. Fall Only.

Rigorous course in multivariable analysis. Continuation of MAT215/217 (or MAT203/204, with instructor's permission). Topics include metric spaces, completeness, compactness, total derivatives, partial derivatives, inverse function theorem, implicit function theorem, Riemann integrals in several variables, Fubini's theorem, change of variables theorem, and the theorems of Green, Gauss, and Stokes.

Introduction to real analysis, including the theory of Lebesgue measure and integration on the line and n-dimensional space, and the theory of Fourier series. Prerequisite: MAT201 and MAT202 or equivalent.
(Replaces MAT314 beginning AY2012-13)

Introduction to numerical methods with emphasis on algorithms, applications and computer implementation issues. Topics covered include solution of nonlinear equations; numerical differentiation, integration and interpolation; direct and iterative methods for solving linear systems; numerical solutions of differential equations; two-point boundary value problems; and approximation theory. Lectures are supplemented with numerical examples using MATLAB. Prerequisites: MAT201 and MAT202; or MAT203 and MAT204; or equivalent.

Draws problems from the sciences & engineering for which mathematical models have been developed and analyzed to describe, understand and predict natural and man-made phenomena. Emphasizes model building strategies, analytical and computational methods, and how scientific problems motivate new mathematics.
This interdisciplinary course in collaboration with Molecular Biology, Psychology and the Program in Neuroscience is directed toward upperclass undergraduate students and first-year graduate students with knowledge of linear algebra and differential equations.

Calculus of functions of one complex variable, power series expansions, residues, and conformal mapping. (Replaces MAT317 beginning AY 2012-13)

Study of functions of a complex variable, with emphasis on interrelations with other parts of mathematics. Cauchy's theorems, singularities, contour integration, power series, infinite products. The gamma and zeta functions and the prime number theorem. Elliptic functions, theta functions, Jacobi's triple product and combinatorics. An overall view of Special Functions via the hypergeometric series. The second course in a four-semester sequence, but may be taken independently.
(Replaces MAT331 beginning AY 2012-13)

Covers the basics of symmetry and group theory, with applications. Topics include the fundamental theorem of finitely generated abelian groups, Sylow theorems, group actions and the representation theory of finite groups. Prerequisites: MAT202 or MAT204 or MAT217.
(Replaces MAT322 beginning AY 2012-13)

Riemannian geometry of surfaces. Surfaces in Euclidan space, second fundamental form, minimal surfaces, geodesics, Gauss curvature, Gauss-Bonnet Theorem, uniformization of surfaces. Prerequisites: MAT218 or equivalent.
(Replaces MAT327 beginning AY 2012-13)

An introduction to point set topology, the fundamental group, covering spaces, methods of calculation and applications. Prerequisites: MAT202 or MAT204 or MAT218 or equivalent.
(Replaces MAT325 beginning AY 2012-13)

This course introduces students to Combinatorics, a fundamental mathematical discipline as well as an essential component of many mathematical areas. (Replaces MAT307 beginning AY 2012-13)

Sequences of independent trials, applications to number theory and analysis, Monte Carlo method. Markov chains, ergodic theorem for Markov chains, Entropy and McMillan theorem. Random walks, recurrence and non-recurrence; connection with linear difference equations. Strong laws of large numbers, random series and products. Weak convergence of probability measures, weak Helly theorems, Fourier transforms of distributions. Limit theorems of probability theory. Prerequisites: MAT203 or MAT218 or equivalent.
(Replaces MAT390 beginning AY 2012-13)

An introduction to classical results in analytic number theory, presenting fundamental theorems with detailed proofs and highlighting the tight connections between them. Topics covered might include: the prime number theorem, Dirichlet L-functions, zero-free regions, sieve methods, representation by quadratic forms, and Gauss sums. Prerequisites: MAT335 (Complex Analysis) and MAT345 (Algebra I).

The course concerns explicit solution of simple differential equations. Methods of proving that one has found all the solutions are discussed. For this purpose, a brief review of foundational concepts in real analysis is provided. The second part concerns explicit solutions of simultaneous linear differential equations with constant coefficients, a topic closely connected with linear algebra (assumed prerequisite knowledge). The third part concerns the proof of the basic existence and uniqueness theorem for ordinary differential equations. Students will do simple proofs.
(Replaces MAT303 beginning AY 2012-13)

This course will cover the standard material in a first course on commutative algebra. Topics include: ideals in and modules over commutative rings, localization, primary decomposition, integral dependence, Noetherian rings and chain conditions, discrete valuation rings and Dedekind domains, completion; dimension theory. Prerequisites: Algebra I & II, MAT345-6.

Introductory course in modern Analysis with applications to Partial Differential Equations,Distribution Theory, Maximal Functions, Littlewood/Paley decompositions and applications, Strichartz inequalities, Bilinear Estimates, Concentrated compactness and applications. (Replaces MAT433 beginning AY 2012-13)

The course is intended as a basic introductory course to the modern methods of Analysis. Specific applications of these methods to problems in other fields, such as Partial Differential Equations, Probability, and Number Theory, will also be presented. Topics will include Lp spaces, tempered distributions and the Fourier transform, the Riesz interpolation theorem, the Hardyâ€“Littlewood maximal function, Calderonâ€“Zygmund theory, the spaces H1 and BMO, oscillatory integrals, almost orthogonality, restriction theorems and applications to dispersive equations, the law of large numbers and ergodic theory. We will also discuss applications of Fourier methods to discrete counting problems, using the Poisson summation formula.

Introduction to the techniques necessary for the formulation and solution of problems involving partial differential equations in the natural sciences and engineering, with detailed study of the heat and wave equations. Topics include method of eigenfunction expansions, Fourier series, the Fourier transform, inhomogeneous problems, the method of variation of parameters. Prerequisite MAT202 or MAT204 or MAT218.

Random Walks - Junior Seminar with Michael Damron

Geometry - Junior Seminary with Sophie Chen

Lie-groups, Lie-algebras and their representations - Junior Seminar with Tasho Kaletha