Mathematics

Professor Christoph Börgers, Interim Chair; Mathematical biology
Professor Bruce Boghosian, Fluid dynamics
Professor Mauricio Gutierrez, Group Theory
Professor Marjorie Hahn, Probability
Professor Boris Hasselblatt, Dynamical systems
Professor Misha Kilmer, Numerical linear algebra
Emeritus Professor George Leger, Algebra
Professor Zbigniew H. Nitecki, Dynamical systems
Professor Eric Todd Quinto, Robinson Professorship in Mathematics; Functional analysis
Professor Montserrat Teixidor i Bigas, Algebraic geometry
Professor Richard Weiss, William Walker Professor of Mathematics; Group theory
Associate Professor Fulton Gonzalez, Harmonic analysis
Associate Professor David F. Isles, Logic
Associate Professor Loring Tu, Algebraic geometry
Associate Professor/Coordinator Lenore Feigenbaum, History of mathematics
Assistant Professor Christian Benes, Stochastic analysis
Assistant Professor Tawanda Gwena, Algebraic geometry
Assistant Professor George McNinch, Algebraic groups
Assistant Professor Lisa Perrone, Numerical analysis
Assistant Professor Adam Piggott, Geometric group theory
Assistant Professor Kim Ruane, Geometric group theory
Lecturer Mary Glaser, Combinatorics
Adjunct Professor Lenore J. Cowen, Computational biology, theory of computation, algorithm design and analysis
Adjunct Professor Diane Souvaine, Design and analysis of algorithms, computational geometry
Visiting Research Professor Mary Beth Ruskai, Mathematical physics, quantum computing
 

Our experience up to date justifies us in feeling sure that in Nature is actualized the ideal of mathematical simplicity.
Albert Einstein
Spencer Lecture, Oxford, 1933

Undergraduate Concentration Requirements
To major in mathematics, a student must take ten courses in the department. Up to two of these courses may be replaced by certain courses in related fields approved by the department. The courses in mathematics may not include Mathematics 4 through 12, or 17, and must include Mathematics 13 or 18, 46, 135, and 145. For depth, students must complete at least one of the four major year-long sequences (Mathematics135/136, 145/146, 151/152, and 161/162), and for breadth, students must either complete a second one of these sequences or else take an additional course chosen from among Mathematics 126, 128, 151, 158, 161, 167, 168, 161. Majors are advised to complete Mathematics 13 or 18 and Mathematics 46 by the end of their sophomore year. Majors must demonstrate knowledge of a computer language by including an approved course in computer science in their program.

Students in the School of Engineering may, with certain exceptions, choose mathematics as a second area of concentration. To do so a student must notify the dean of engineering through the Department of Mathematics at least one semester before graduation. The student must complete a program that simultaneously satisfies the conditions for a degree from the School of Engineering and the concentration recommendations of the Department of Mathematics.

Undergraduate Minor Program
To minor in mathematics, a student must take six courses in the department beyond the level of Mathematics 12 (or 17). These must include Mathematics 13 (or 18) and 46, as well as Mathematics 135 or 145 (or both).

Graduate Program
The Department of Mathematics offers programs leading to the degrees of master of science or arts, and doctor of philosophy. Applicants for the master's degree are expected to have a preparation equivalent to the usual major in mathematics, including courses in analysis and modern algebra. Applicants for admission directly into the doctoral program will be considered only if they have shown exceptional ability. Often doctoral candidates are drawn from the Tufts master's program.

Master's Degree
To qualify for the master's degree a student completes an approved program of at least nine courses numbered above 100. At least seven of these courses must be numbered 136, 146, 158, or above 164; this may include 295 and 296. A student must either write a master's thesis, which counts as two of the nine required courses, or pass oral examinations on analysis, algebra, and geometry as in the Ph.D. program.

For breadth, the nine courses taken to fulfill the master's degree course requirement must include at least one course from each of the following four categories:

Real analysis: 136, 211, 212
Complex analysis: 158, 213
Algebra: 146, 215, 216
Geometry or topology: 167, 168, 217, 218

No more than two of these four courses can be at the 100 level. Any part of this requirement may be fulfilled by equivalent courses with prior approval by the mathematics department graduate committee.

Doctor of Philosophy
A student who has been admitted to the doctoral program must first pass oral examinations on analysis, algebra, and geometry, and then pass an oral qualifying examination on material studied in preparation for work on a dissertation.

The major task of a doctoral student is to write a dissertation under the direction of a department member. This must be a substantial original contribution to the field of the student's specialty and must meet standards of quality as exemplified by current mathematical research journals.

In addition, a student must demonstrate proficiency in reading mathematics written in French, German, or Russian. A student who is not a native speaker of English must also demonstrate proficiency in oral and written mathematical communication in English. A student is also required to have at least one semester's teaching experience at Tufts.

Undergraduate Courses
Please note: Students who wish to begin calculus should register for either Mathematics 5, 11, or 17. Those interested in computer science, engineering, mathematics, and the physical sciences should begin the three-course sequence Mathematics 11, 12, 13 or the two-course accelerated honors sequence Mathematics 17, 18. Mathematics 5 and 11 will not both count toward a degree in any college at Tufts. Other combinations of these courses may not be granted full credit. Mathematics 4, 5, 6, 8, 9, and 10 do not count toward a degree in the School of Engineering.

4 Fundamental Mathematics. A review of basic algebra: fractions, exponents, polynomials. Equations: linear, quadratic, simultaneous equations, word problems. Functions and their graphs; logarithms. Trigonometry. Prerequisite: consent. Normally a student who has passed Mathematics 5 or 11 will not be allowed to take Mathematics 4 for credit. Mathematics 4 counts toward the mathematical sciences distribution requirement only for students with a mathematics SAT score below 560; see Quantitative Reasoning Requirement. Fall. Members of the department

5 Introduction to Calculus. 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. Logarithmic and exponential functions with applications. Prerequisites: high school geometry and algebra. Mathematics 5 is a one-term calculus course and is not adequate preparation for Mathematics 12. Engineering students are not permitted to take Mathematics 5 for credit; in no case will both Mathematics 5 and Mathematics 11 be counted for credit. Members of the department

6 Introduction to Finite Mathematics. Topics selected from financial mathematics, matrix algebra, linear inequalities and linear programming, counting arguments, and statistics and probability. Prerequisites: high school geometry and algebra. (Mathematics 5 is not a prerequisite.) Engineering students are not permitted to take Mathematics 6 for credit. Spring. Members of the department

8 Symmetry. 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. Three lectures, one section. Prerequisite: high school geometry. Engineering students are not permitted to take Mathematics 8 for credit. Members of the department

9 The Mathematics of Social Choice. 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. Prerequisite: high school algebra. Engineering students are not permitted to take Mathematics 9 for credit. Members of the department

10 Introductory Special Topics. Content and prerequisites vary. Topics covered in recent years have included chance and mathematics in antiquity. Engineering students are not permitted to take Mathematics 10 for credit. Members of the department

11 Calculus I. Differential and integral calculus: limits and continuity, the derivative and techniques of differentiation, extremal problems, related rates, the definite integral, the fundamental theorem of calculus, the derivatives and integrals of trigonometric functions, logarithmic and exponential functions. Prerequisites: high school geometry, algebra, and trigonometry. In no case will both Mathematics 5 and 11 be counted for credit. Members of the department

12 Calculus II. Applications of the integral, techniques of integration, separable differential equations, improper integrals. Sequences, series, convergence tests, Taylor series. Polar coordinates, complex numbers. Prerequisite: Mathematics 11. Members of the department

13 Calculus III. 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. Prerequisite: Mathematics 12 or 17. Members of the department

17 Honors Calculus I-II. 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. Beginning with a rapid review of differential and integral calculus, this course continues with a review of transcendental functions and the techniques of integration. The second half of the semester is devoted to the study of sequences, series, and complex numbers. Students who have not received acceleration credit by receiving a 4 or better on an advanced placement exam will nevertheless receive one acceleration credit (without grade) and one course credit (with grade) after successfully completing this course. Prerequisite: AB syllabus of advanced placement mathematics. Students who receive credit for Mathematics 17 cannot receive credit for Mathematics 11 or 12. Fall. Members of the department

18 Honors Calculus III. The derivatives of vector-valued functions of several variables and the matrices associated with these derivatives. Applications to finding affine approximations and the tangent space of curves or surfaces. Multiple integrals, line integrals, Green's theorem, the divergence theorem, and Stoke's theorem. Prerequisite: Mathematics 17. Students who receive credit for Mathematics 18 cannot receive credit for Mathematics 13. Spring. Members of the department

22 Discrete Mathematics. Sets, relations and functions, logic and methods of proof, combinatorics, graphs and digraphs. Prerequisite: Mathematics 11 or Computer Science 11 or consent. Members of the department

38 Differential Equations. An introduction to linear differential equations with constant coefficients, linear algebra, and Laplace transforms. Prerequisite: Mathematics 13 or 18. Members of the department

41 Number Theory. An introduction to number theory, including the Euclidean algorithm, congruences, primitive roots, and the law of quadratic reciprocity. Prerequisite: Mathematics 11 or consent. Members of the department

46 Linear Algebra. 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. Prerequisite: Mathematics 12 or 17 or consent. Members of the department

50 Elementary Special Topics. Content and prerequisites vary from semester to semester. Topics covered in recent years have included combinatorics, set theory, and chaos. Members of the department

Courses for Undergraduate and Graduate Students

112 Topics in the History of Mathematics. The evolution of mathematical concepts and techniques from antiquity to modern times. Prerequisite: Mathematics 12 or consent. Members of the department

126 Numerical Analysis. (Cross-listed as Computer Science 126.) Analysis of algorithms involving computation with real numbers. Interpolation, methods for solving linear and nonlinear systems of equations, numerical integration, methods for ordinary differential equations. Prerequisites: Mathematics 38 and programming ability in a language such as C, C++, Fortran, Pascal, or Matlab. Members of the department

128 Numerical Linear Algebra. (Cross-listed as Computer Science 128.) The two basic computational problems of linear algebra: solution of linear systems and computation of eigenvalues and eigenvectors. Prerequisites: Mathematics 46 and Computer Science 11. Members of the department

135 Real Analysis I. 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. Prerequisites: Mathematics 13 or 18, and 46, or consent. Fall. Members of the department

136 Real Analysis II. Applications of ideas from Mathematics 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. Prerequisite: Mathematics 135 or consent. Spring. Members of the department

145 Abstract Algebra I. An introduction to the basic concepts of abstract algebra, including groups and rings. Prerequisite: Mathematics 46. Fall. Members of the department

146 Abstract Algebra II. Further topics in groups and rings. Field extensions and Galois theory. Prerequisite: Mathematics 145 or 215, or consent. Spring. Members of the department

150 Special Topics. Content and prerequisites vary from semester to semester. Topics covered in recent years have included fractals, logical systems, advanced linear algebra, and nonlinear dynamics and chaos. Members of the department

151 Applications of Advanced Calculus. (Cross-listed as Mechanical Engineering 150.) 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. Prerequisite: Mathematics 38. Members of the department

152 Nonlinear Partial Differential Equations. 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. Prerequisite:  Mathematics 151 or consent.

158 Complex Variables. 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. Prerequisite: Mathematics 13 or 18, or consent. Members of the department

161 Probability. Probability, conditional probability, random variables and distributions, expectation, special distributions, and joint distributions. Prerequisite: Mathematics 13 or 18, or consent. Fall. Members of the department

162 Statistics. Statistical estimation, sampling distributions of estimators, hypothesis testing, regression, analysis of variance, and nonparametric methods. Prerequisite: Mathematics 161 or consent. Spring. Members of the department

163 Computational Geometry. (Cross-listed as Computer Science 163.) 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. Prerequisite: Computer Science 160 or consent. Souvaine

167 Differential Geometry. 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. Prerequisites: Mathematics 46 and 135 or consent. Members of the department

168 Algebraic Topology. 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. Prerequisites: Mathematics 135 and 145. Members of the department

171 Computational Mathematical Logic. Topics selected from Gentzen and natural deduction systems; proof theory: normal forms, normalization, Herbrand's theorem, ordinal rotations; Gödel incompleteness and related results; lambda-calculi: properties and relations to computation and logical systems. Prerequisite: consent. Members of the department

191, 192 Seminar in Mathematics. Student lectures, discussion, and criticism. Prerequisite: consent. Credit as arranged. Members of the department

193, 194 Special Topics. Guided study of an approved topic. Prerequisite: consent. Credit as arranged. Members of the department

195, 196 Senior Honors Thesis. Thesis course for thesis honors candidates; see Thesis Honors Program for details. Open to seniors. Normally two credits. Prerequisite: consent. Members of the department

211, 212 Analysis. 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. Other topics in analysis selected by the instructor. Two courses: 211 in Fall; 212 in Spring. Prerequisites: Mathematics 135 and 136 or equivalent, or consent. Members of the department

213 Complex Analysis. Analytic functions, power series. Integration in the complex plane, Cauchy's integral theorem and formulas. Entire functions. Singularities. Conformal mapping, Riemann mapping theorem. Prerequisite: Mathematics 135 or consent. Members of the department

215, 216 Algebra. General properties of groups, rings, and modules. Divisibility theory in integral domains. Field extensions. Other topics in abstract algebra selected by the instructor. Two courses: 215 in Fall; 216 in Spring. Prerequisite: Mathematics 145 or consent.  Members of the department

217, 218 Geometry and Topology. An introduction to the underlying geometric concepts of contemporary mathematics. Multivariate calculus on manifolds, fundamental groups, elementary homology and covering spaces, Riemannian metrics, curvature, geodesics, some classical Lie groups and their Lie algebras. Other topics in geometry and topology selected by the instructor. Two courses: 217 in Fall; 218 in Spring. Prerequisites: Mathematics 135 and 145, or consent. Members of the department

250 Advanced Special Topics. Content and prerequisites vary with semester. Topics covered in recent years have included stochastic optimization, differential topology, algebraic geometry, dynamical systems, and numerical methods. Members of the department

263 Advanced Computational Geometry. (Cross-listed as Computer Science 263.) Design and analysis of sequential, parallel, probabilistic, and approximation algorithms for geometry problems. Geometric data structures, complexity, searching, computation, and applications. Selected advanced topics. Prerequisites: Computer Science 163 or consent. Souvaine

291, 292 Graduate Seminar. Presentation of individual reports on basic topics to a seminar group for discussion and criticism. Credit as arranged. Members of the department

293, 294 Special Topics. Guided individual study of an approved topic. Credit as arranged. Members of the department

295, 296 Thesis. Guided research on a topic that has been approved as suitable for a master's thesis. Members of the department

297, 298 Graduate Research. Guided research on a topic suitable for doctoral dissertation. Credit as arranged. Members of the department

401PT Master's Continuation, Part-time.

402FT Master's Continuation, Full-time.

501PT Doctoral Continuation, Part-time.

502FT Doctoral Continuation, Full-time.