Tufts University  |  School of Arts and Sciences  |  School of Engineering  |  Find People  | 
   

Courses

Regularly Offered Courses

Frequencies reflect recent history and are not a guarantee about future offerings. For planning purposes, please consult an advisor in the mathematics department.

UNDERGRADUATE COURSES

Math 4 - Fundamentals of Mathematics
Every Fall semester.
A review of basic algebra: fractions, exponents, polynomials. Equations: linear, quadratic, simultaneous equations, word problems. Functions and their graphs; logarithms. Trigonometry.

Math 10 - Introductory Special Topics
Irregular.
(Recently: Statistics, Financial Math.)

Math 14 - Introduction to Finite Mathematics (formerly Math 6)
Every Spring semester.
Topics selected from financial mathematics, matrix algebra, linear inequalities and linear programming, counting arguments, and statistics and probability.

Math 15 - Mathematics in Antiquity (formerly Math 7)
Irregular.
History of mathematics in four ancient civilizations: Babylonian, Egyptian, Greek, and Chinese. Number systems and computational techniques; achievements in elementary algebra, geometry, and number theory; famous results, proofs and constructions. Emphasis on solving problems in the style and spirit of each culture.

Math 16 - Symmetry (formerly Math 8)
Every Spring semester.
A mathematical treatment of the symmetries of wallpaper patterns. The main goal is to prove that the symmetries of these patterns fall into seventeen distinct types. In addition, students will learn to identify the symmetries of given patterns (with special emphasis on the periodic drawings of M.C. Escher) and to draw such patterns.

Math 19 - The Mathematics of Social Choice (formerly Math 9)
Every semester.
Introduction to mathematical methods for dealing with questions arising from social decision making. Topics vary but usually include ranking, determining the strength of, and choosing participants in multicandidate and two-candidate elections, and apportioning votes and rewards to candidates.

Math 30 - Introduction to Calculus (formerly Math 5)
Every semester.
Functions and their graphs, limits, derivatives, techniques of differentiation. Applications of derivatives, curve sketching, extremal problems. Integration: indefinite and definite integrals, some techniques of integration, Fundamental Theorem of Calculus. Logarithmic and exponential functions with applications.

Math 32 - Calculus I (formerly Math 11)
Every semester.
Differential and integral calculus: limits and continuity, the derivative and techniques of differentiation, extremal problems, related rates, the definite integral, Fundamental Theorem of Calculus, derivatives and integrals of trigonometric functions, logarithmic and exponential functions.

Math 34 - Calculus II (formerly Math 12)
Every semester.
Applications of the integral, techniques of integration, separable differential equations, improper integrals. Sequences, series, convergence tests, Taylor series. Polar coordinates, complex numbers.

Math 36 - Applied Calculus II
Every Fall semester.
Symbolic anti-differentiation (substitution, integration by parts), introduction to numerical integration, use of Riemann sums to understand how integrals arise from problems in the natural sciences, probability, and geometry, infinite sums, Taylor expansions and applications, polar coordinates, complex numbers.

Math 39 - Honors Calculus I-II (formerly Math 17)
Every Fall semester.
The first course of the two-semester sequence of honors calculus. Intended for students who have had at least the AB syllabus of advanced placement mathematics in secondary school. Stresses the theoretical aspects of the subject, including proofs of basic results. Topics include: convergence of sequences and series; continuous functions, Intermediate Value and Extreme Value Theorems; definition of the derivative, formal differentiation, finding extrema, curve-sketching, Mean Value Theorems; basic theory of the Riemann integral, Fundamental Theorem of Calculus and formal integration, improper integrals; Taylor series, power series and analytic functions.

Math 42 - Calculus III (formerly Math 13)
Every semester.
Vectors in two and three dimensions, applications of the derivative of vector-valued functions of a single variable. Functions of several variables, continuity, partial derivatives, the gradient, directional derivatives. Multiple integrals and their applications. Line integrals, Green's theorem and related results.

Math 44 - Honors Calculus III (formerly Math 18)
Every Spring semester.
Analysis of real- and vector-valued functions of one or more variables using tools from linear and multilinear algebra; stress is on theoretical aspects of the subject, including proofs of basic results. Topics include: geometry and algebra of vectors in 3-space, parametrized curves and arc length, linear transformations and matrices; Jacobian and gradient of a real-valued function, Implicit Function Theorem, extrema, Taylor's Theorem and Lagrange multipliers; multiple integrals, differential forms and vector fields, line integrals, parametrized surfaces and surface integrals, exact and closed forms, vector calculus.

Math 50 - Special Topics
Every Fall semester.
(Recently: Applied calculus, Problem-solving seminar, Modeling and computing, Nonlinear dynamics and chaos, Transition to advanced mathematics, Financial mathematics.)

Math 51 - Differential Equations (formerly Math 38)
Every semester.
An introduction to linear differential equations with constant coefficients, linear algebra, and Laplace transforms.

Math 61 - Discrete Mathematics (formerly Math 22; cross-listed as Comp Sci 22)
Every semester.
Sets, relations and functions, logic and methods of proof, combinatorics, graphs and digraphs.

Math 63 - Number Theory (formerly Math 41)
Irregular.
An introduction to number theory, including the Euclidean algorithm, congruences, primitive roots, and the law of quadratic reciprocity.

Math 70 - Linear Algebra (formerly Math 46)
Every semester.
An introduction to the theory of vector spaces and linear transformations over the real or complex numbers, including linear independence, dimension, matrix multiplication, similarity and change of basis, and some applications. Topics such as eigenvalues and eigenvectors, the Cayley-Hamilton theorem, and inner product spaces may be included.

Math 72 - Abstract Linear Algebra (formerly Math 54)
Once per year.
An introduction to the theory of linear algebra starting with vector spaces. Subspaces, bases, dimension. Linear transformations, invariant subspaces, eigenvalues, eigenspaces, and diagonalizing linear operators. Inner product spaces and orthogonal projections. The emphasis will, in general, be on proofs rather than computation. Additional topics may include normal forms, minimal polynomials, determinant of an operator, normal operators over complex vector spaces as well as applications of linear algebra.

Math 87 - Mathematical Modeling and Computation
Every Fall semester.
A survey of major techniques in the use of mathematics to model physical, biological, economic, and other systems: topics may include derivative-based optimization and sensitivity analysis, linear programming, graph algorithms, probabilistic modeling, Monte-Carlo methods, difference equations, and statistical data fitting. This course includes an introduction to computing using a high-level programming language, and studies the transformation of mathematical objects into computational algorithms.

Math 112 - Topics in the History of Mathematics
Every Spring semester.
The evolution of mathematical concepts and techniques from antiquity to modern times.

Math 126 - Numerical Analysis
Every other year.
Analysis of algorithms involving computation with real numbers. Interpolation, methods for solving linear and nonlinear systems of equations, numerical integration, numerical methods for solving ordinary differential equations.

Math 128 - Numerical Linear Algebra
Every other year.
The two basic computational problems of linear algebra: solution of linear systems and computation of eigenvalues and eigenvectors.

Math 135 - Real Analysis I
Every Fall semester.
An introduction to analysis. Metric spaces (with Euclidean spaces as the primary example), compactness, connectedness, continuity and uniform continuity, uniform convergence, the space of continuous functions on a compact set, contraction mapping lemma with applications.

Math 136 - Real Analysis II
Every Spring semester.
Applications of ideas from MATH 135 to further, in-depth study of functions on Euclidean spaces. Derivatives as linear maps, differentiable mappings, inverse and implicit function theorems. Further topics such as theory of the Riemann and Lebesgue integral, Hilbert spaces, and Fourier series.

Math 145 - Abstract Algebra I
Every Fall semester.
An introduction to the basic concepts of abstract algebra, including groups and rings.

Math 146 - Abstract Algebra II
Every Spring semester.
Further topics in groups and rings. Field extensions and Galois theory.

Math 150 - Advanced Special Topics
Every Fall semester.
(Recently: Mathematical neuroscience, Affine and projective geometry, Nonlinear dynamics and chaos, Knot theory.)

Math 151 - Applications of Advanced Calculus (cross-listed as Mech Eng 150)
Every Fall semester.
The solutions of certain boundary-value problems in mathematical physics, including the following topics: inner-product spaces, function spaces, orthogonalization, Fourier series, orthogonal families of polynomials, Sturm-Liouville problems, separation of variables in partial differential equations, applications.

Math 152 - Nonlinear Partial Differential Equations
Alternate years in Spring semester.
Introduction to the theory of nonlinear partial differential equations, including the method of characteristics, weak solutions, shocks and jump conditions, hodograph transformations, nonlinear wave equations and solutions, nonlinear diffusion and reaction-diffusion equations, fluid dynamics, combustion and detonation.

Math 158 - Complex Variables
Every Spring semester.
Introduction to the theory of analytic functions of a single complex variable, analytic functions, Cauchy's integral theorem and formula, residues, series expansions of analytic functions, conformal representation, entire and meromorphic functions, multivalued functions.

Math 161 - Probability
Every Fall semester.
Probability, conditional probability, random variables and distributions, expectation, special distributions, joint distributions, laws of large numbers, and the central limit theorem.

Math 162 - Statistics
Every Spring semester.
Statistical estimation, sampling distributions of estimators, hypothesis testing, regression, analysis of variance, and nonparametric methods.

Math 163 - Computational Geometry (cross-listed as Comp Sci 163)
Irregular.
Design and analysis of algorithms for geometric problems. Topics include proof of lower bounds, convex hulls, searching and point location, plane sweep and arrangements of lines, Voronoi diagrams, intersection problems, decomposition and partitioning, farthest-pairs and closest-pairs, rectilinear computational geometry.

Math 167 - Differential Geometry
Irregular.
Study of basic notations of differential geometry in the context of curves and surfaces. Curvature and torsion, implicit function theorem, coordinate systems, first and second fundamental forms, geodesics, Gauss-Bonnet theorem.

Math 168 - Algebraic Topology
Irregular.
Applications of algebra to the study of topological objects, with emphasis on surfaces. Surfaces as manifolds, homotopy of curves, fundamental group, simple connectedness, covering spaces, genus, Euler Characteristic, orientability, and the classification of compact surfaces.

Math 191/192 - Seminar in Mathematics
Ad hoc.

Math 195 - Senior Honors Thesis
A few per year.

GRADUATE COURSES

Math 211 - Analysis
Every Fall semester.
An introduction to modern analysis in abstract spaces, including point-set topology and measure and integration. Topological spaces, compactness, completeness, Baire category, function spaces, measures, integration, convergence theorems, Fubini theorem, Radon-Nikodym theorem, Riesz representation theorem, Banach and Hilbert spaces.

Math 212 - Topics in Analysis
Most Spring semesters.
(Recently: Functional and harmonic analysis, Measure theoretic probability, Ergodicity and hyperbolicity, Functional analysis.)

Math 213 - Complex Analysis
Irregular.
Analytic functions, power series. Integration in the complex plane, Cauchy's integral theorem and formulas. Entire functions. Singularities. Conformal mapping, Riemann mapping theorem.

Math 215 - Algebra
Every Fall semester.
General properties of groups, rings, especially polynomial rings, modules over a principal ideal ring, field extensions and Galois theory.

Math 216 - Topics in Algebra
Every Spring semester.
(Recently: Fields and buildings, Homology and algebra, Commutative algebra, Fields and algebraic curves, The theory of fields.)

Math 217 - Geometry and Topology
Every Fall semester.
An introduction to the underlying geometric concepts of contemporary mathematics. Manifolds, vector fields, integral curves, differential forms, wedge product, exterior derivative, integration of differential forms, some classical Lie groups and their Lie algebras, and possibly fundamental groups and covering spaces or de Rham cohomology.

Math 218 - Topics in Geometry and Topology
Every Fall semester.
(Recently: Algebraic topology, Differential geometry, Differential forms in algebraic topology, Algebraic topology and surface theory, Lie groups.)

Math 250 - Graduate Special Topics
Every Fall semester.
(Recently: Numerical analysis, Algebraic geometry, Geometric group theory, Nonlinear dynamics, New hyperbolic dynamics, Nonlinear PDEs, Linear PDEs, Stochastic processes, Algebraic topology, Numerical methods for PDEs, Dynamical systems, Dynamics of surface homeomorphisms, Advanced PDEs.)

Math 263 - Advanced Computational Geometry (cross-listed as Comp Sci 263)
Irregular.
Design and analysis of sequential, parallel, probabilistic, and approximation algorithms for geometry problems. Geometric data structures, complexity, searching, computation, and applications. Selected advanced topics.

Math 291-293 - Graduate Seminar
Ad hoc.
©2013 Tufts University. All rights reserved.  Site designed & maintained by Tufts Technology Services (TTS).  Major artwork contributions by Lun-Yi Tsai.