Mathematics

Ma 1 abc. Calculus of One and Several Variables and Linear Algebra. 9 units (4-0-5); first, second, third terms. Prerequisites: high-school algebra, trigonometry, and calculus. Special section of Ma 1 a, 12 units (5-0-7). Review of calculus. Complex numbers, Taylor polynomials, infinite series. Comprehensive presentation of linear algebra. Derivatives of vector functions, multiple integrals, line and path integrals, theorems of Green and Stokes. Ma 1 b, c is divided into two tracks: analytic and practical. Students will be given information helping them to choose a track at the end of the fall term. There will be a special section or sections of Ma 1 a for those students who, because of their background, require more calculus than is provided in the regular Ma 1 a sequence. These students will not learn series in Ma 1 a and will be required to take Ma 1 d. Instructors: Groves, Borodin, Aschbacher, Wales, Ramakrishnan, Dunfield.

Ma 1 d. Series. 5 units (2-0-3); second term only. Prerequisite: special section of Ma 1 a. This is a course intended for those students in the special calculus-intensive sections of Ma 1 a who did not have complex numbers, Taylor polynomials, and infinite series during Ma 1 a. It may not be taken by students who have passed the regular Ma 1 a. Instructor: Staff.

Ma 2 ab. Differential Equations, Probability and Statistics. 9 units (4-0-5); first, second terms. Prerequisite: Ma 1 abc. Ordinary differential equations, probability, statistics. Instructors: Makarov, Calegari, Lorden.

Ma 3. Number Theory for Beginners. 9 units (3-0-6); third term. Some of the fundamental ideas, techniques, and open problems of basic number theory will be introduced. Examples will be stressed. Topics include Euclidean algorithm, primes, Diophantine equations, including aⁿ + bⁿ = cⁿ and a² - db² = ^±1, constructible numbers, composition of binary quadratic forms, and congruences. Instructor: Keevash.

Ma 4. Introduction to Mathematical Chaos. 9 units (3-0-6); third term. An introduction to the mathematics of “chaos.” Period doubling universality, and related topics; interval maps, symbolic itineraries, stable/unstable manifold theorem, strange attractors, iteration of complex analytic maps, applications to multidimensional dynamics systems and real-world problems. Possibly some additional topics, such as Sarkovski’s theorem, absolutely continuous invariant measures, sensitivity to initial conditions, and the horseshoe map. Instructor: Gorodetski.

Ma 5 abc. Introduction to Abstract Algebra. 9 units (3-0-6); first, second, third terms. Freshmen must have instructor’s permission to register. Introduction to groups, rings, fields, and modules. The first term is devoted to groups and includes treatments of semidirect products and Sylow’s theorem. The second term discusses rings and modules and includes a proof that principal ideal domains have unique factorization and the classification of finitely generated modules over principal ideal domains. The third term covers field theory and Galois theory, plus some special topics if time permits. Instructors: Wambach, Wales, Flach.

Ma/CS 6 abc. Introduction to Discrete Mathematics. 9 units (3-0-6); first, second, third terms. Prerequisite: for Ma/CS 6 c, Ma/CS 6 a or Ma 5 a or instructor’s permission. First term: a survey emphasizing graph theory, algorithms, and applications of algebraic structures. Graphs: paths, trees, circuits, breadth-first and depth-first searches, colorings, matchings. Enumeration techniques; formal power series; combinatorial interpretations. Topics from coding and cryptography, including Hamming codes and RSA. Second term: directed graphs; networks; combinatorial optimization; linear programming. Permutation groups; counting nonisomorphic structures. Topics from extremal graph and set theory, and partially ordered sets. Third term: elements of computability theory and computational complexity. Discussion of the P=NP problem, syntax and semantics of propositional and first-order logic. Introduction to the Gödel completeness and incompleteness theorems. Instructors: Wilson, Ku, Kechris.

Ma 8. Problem Solving in Calculus. 3 units (3-0-0); first term. Prerequisite: simultaneous registration in Ma 1 a. A three-hour per week hands-on class for those students in Ma 1 needing extra practice in problem solving in calculus. Instructor: Vuletic.

Ma 10. Oral Presentation. 3 units (2-0-1); first term. Open for credit to anyone. Freshmen must have instructor’s permission to enroll. In this course, students will receive training and practice in presenting mathematical material before an audience. In particular, students will present material of their own choosing to other members of the class. There will also be elementary lectures from members of the mathematics faculty on topics of their own research interest. Instructor: Borodin.

Ma 11. Mathematical Writing. 3 units (0-0-3); third term. Students will work with the instructor and a mentor to write and revise a self-contained paper dealing with a topic in mathematics. In the first week, an introduction to some matters of style and format will be given. Some help with typesetting in TeX may be available. Students are encouraged to take advantage of the Hixon Writing Center. The mentor and the topic are selected in consultation with the instructor. It is expected that in most cases the paper will be in the style of a textbook or journal article, at the level of the student’s peers. Fulfills the Institute scientific writing requirement. Graded pass/fail. Instructor: Wilson.

Ma 12. Chance. 9 units (4-0-5); second term. This course will explore the use and misuse of notions of probability and statistics in popular culture and in science. The course will be structured around case studies chosen from mass media and from the scientific literature. Not offered 2006–07.

Ma 17. How to Solve It. 4 units (2-0-2); first term. There are many problems in elementary mathematics that require ingenuity for their solution. This is a seminar-type course on problem solving in areas of mathematics where little theoretical knowledge is required. Students will work on problems taken from diverse areas of mathematics; there is no prerequisite and the course is open to freshmen. May be repeated for credit. Graded pass/fail. Instructor: Tsankov.

Ma 92 abc. Senior Thesis. 9 units (0-0-9); first, second, third terms. Prerequisite: To register, the student must obtain permission of the mathematics undergraduate representative, Richard Wilson. Open only to senior mathematics majors who are qualified to pursue independent reading and research. This research must be supervised by a faculty member. The research must begin in the first term of the senior year and will normally follow up on an earlier SURF or independent reading project. Two short presentations to a thesis committee are required: the first at the end of the first term and the second at the midterm week of the third term. A draft of the written thesis must be completed and distributed to the committee one week before the second presentation. Graded pass/fail in the first and second terms; a letter grade will be given in the third term.

Ma 98. Independent Reading. 3–6 units by arrangement. Occasionally a reading course will be offered after student consultation with a potential supervisor. Topics, hours, and units by arrangement. Graded pass/fail.

Ma 105. Elliptic Curves. 9 units (3-0-6); first term. Prerequisite: Ma 5, Ma 3, or equivalents. The ubiquitous elliptic curves will be analyzed from elementary, geometric, and arithmetic points of view. Possible topics are the group structure via the chord-and-tangent method, the Nagel-Lutz procedure for finding division points, Mordell’s theorem on the finite generation of rational points, points over finite fields through a special case treated by Gauss, Lenstra’s factoring algorithm, integral points. Other topics may include diophantine approximation and complex multiplication. Not offered 2006–07.

Ma 108 abc. Classical Analysis. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 2 or equivalent, or instructor’s permission. May be taken concurrently with Ma 109. First term: structure of the real numbers, topology of metric spaces, a rigorous approach to differentiation in Rⁿ. Second term: brief introduction to ordinary differential equations; Lebesgue integration and an introduction to Fourier analysis. Third term: the theory of functions of one complex variable. Instructors: Zinchenko, Christiansen.

Ma 109 abc. Introduction to Geometry and Topology. 9 units (3-0-6); first, second, third terms. Prerequisites: Ma 2 or equivalent, and Ma 108 must be taken previously or concurrently. First term: aspects of point set topology, and an introduction to geometric and algebraic methods in topology. Second term: the differential geometry of curves and surfaces in two- and three-dimensional Euclidean space. Third term: an introduction to differentiable manifolds. Transversality, differential forms, and further related topics. Instructors: Gorodnik, McReynolds, Graber.

Ma 110 abc. Analysis, I. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 108 or previous exposure to metric space topology, Lebesgue measure. First term: integration theory and basic functional analysis: measure spaces, integral, L^p-spaces, Fejer’s theorem, measures as functionals, product measures, Baire category, Hahn-Banach theorem, Alaoglu’s theorem, differential theory including maximal functions. Second term: basic complex analysis: analytic functions, conformal maps, idea of Riemann surfaces, elementary and some special functions, residue calculus, infinite sums and products, entire and meromorphic functions, asymptotic analysis. Third term: real and harmonic analysis. Real analysis: convexity, Hausdorff measures, arc length and surface measures, integral and convolution operators, L^p-estimates, interpolation theorems, singular integrals, generalized functions, Sobolev spaces. Harmonic analysis: Fourier analysis of periodic functions, harmonic functions in the unit disk and Fourier series, harmonic analysis in R^d, Fourier integral, boundary behavior of harmonic functions, convergence of Fourier series and integrals, applications to analysis of convolution operators. Instructor: Simon.

Ma 111 ab. Analysis, II. 9 units (3-0-6); second, third terms. Prerequisite: Ma 110 or instructor’s permission. This course will discuss advanced topics in analysis, including zeros of analytic functions and functions on the unit disk, Riemann surfaces, probabilistic methods in analysis, combinatorial methods in analysis, operator theory, C*-algebras. First term: advanced topics in the theory of a complex variable, including elliptic functions, Picard’s theorems, zeros, properties of bounded analytic functions on the disk. Second term: potential theory. Part a not offered 2006–07. Instructor: Makarov.

Ma 112 ab. Statistics. 9 units (3-0-6); first, second terms. Prerequisite: Ma 2 a probability and statistics or equivalent. The first term covers general methods of testing hypotheses and constructing confidence sets, including regression analysis, analysis of variance, and nonparametric methods. The second term covers permutation methods and the bootstrap, point estimation, Bayes methods, and multistage sampling. Instructor: Lorden.

Ma 116 abc. Mathematical Logic and Axiomatic Set Theory. 9 units(3-0-6); first, second, third terms. Prerequisite: Ma 5 or equivalent, or instructor’s permission. Propositional logic, predicate logic, formal proofs, Gödel completeness theorem, the method of resolution, elements of model theory. Computability, undecidability, Gödel incompleteness theorems. Axiomatic set theory, ordinals, transfinite induction and recursion, iterations and fixed points, cardinals, axiom of choice. Not offered 2006–07.

Ma/CS 117 abc. Computability Theory. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 5 or equivalent, or instructor’s permission. Various approaches to computability theory, e.g., Turing machines, recursive functions, Markov algorithms; proof of their equivalence. Church’s thesis. Theory of computable functions and effectively enumerable sets. Decision problems. Undecidable problems: word problems for groups, solvability of Diophantine equations (Hilbert’s 10th problem). Relations with mathematical logic and the Gödel incompleteness theorems. Decidable problems, from number theory, algebra, combinatorics, and logic. Complexity of decision procedures. Inherently complex problems of exponential and superexponential difficulty. Feasible (polynomial time) computations. Polynomial deterministic vs. nondeterministic algorithms, NP-complete problems and the P = NP question. Instructors: Kechris, Caicedo.

Ma 118. Topics in Mathematical Logic. 9 units (3-0-6); first term. Prerequisite: Ma 116 or Ma 117 or equivalent. Consistency and independence in set theory: Goedel’s constructible universe L, and consistency of the axiom of choice and the continuum hypothesis CH with the other axioms of set theory. Cohen’s method of forcing and independence of CH and the axiom of choice. Solovay’s theorem on the consistency of “all sets of reals are Lebesgue measurable.” Additional topics in set theory depending on the audience. Contents vary from year to year so that students may take the course in successive years. Instructor: Caicedo.

Ma 120 abc. Abstract Algebra. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 5 or equivalent. Undergraduates who have not taken Ma 5 must have instructor’s permission. Basic theory of groups, rings, modules, and fields, including free groups; Sylow’s theorem; solvable and nilpotent groups; factorization in commutative rings; integral extensions; Wedderburn theorems; Jacobson radical; semisimple, projective, and injective modules; tensor products; chain conditions; Galois theory; cyclotomic extensions; separability; transcendental extensions. Instructors: Aschbacher, Dimitrov.

Ma 121 abc. Combinatorial Analysis. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 5. A survey of modern combinatorial mathematics, starting with an introduction to graph theory and extremal problems. Flows in networks with combinatorial applications. Counting, recursion, and generating functions. Theory of partitions. (0, 1)-matrices. Partially ordered sets. Latin squares, finite geometries, combinatorial designs, and codes. Algebraic graph theory, graph embedding, and coloring. Instructors: Keevash, Wilson, Ku.

Ma 122 ab. Topics in Group Theory. 9 units (3-0-6); first, second terms. Prerequisite: Ma 5 abc or instructor’s permission. Groups of Lie type: classical groups, Coxeter groups, root systems, Chevalley groups, weight theory, linear algebraic groups, buildings. Not offered 2006–07.

Ma 123. Classification of Simple Lie Algebras. 9 units (3-0-6); third term. Prerequisite: Ma 5 or equivalent. This course is an introduction to Lie algebras and the classification of the simple Lie algebras over the complex numbers. This will include Lie’s theorem, Engel’s theorem, the solvable radical, and the Cartan Killing trace form. The classification of simple Lie algebras proceeds in terms of the associated reflection groups and a classification of them in terms of their Dynkin diagrams. Instructor: Wales.

EE/Ma 126 ab. Information Theory. 9 units (3-0-6). For course description, see Electrical Engineering.

EE/Ma 127 ab. Error-Correcting Codes. 9 units (3-0-6). For course description, see Electrical Engineering.

CS/EE/Ma 129 abc. Information and Complexity. 9 units (3-0-6) first, second terms; (1-4-4) third term. For course description, see Computer Science.

Ma 130 abc. Algebraic Geometry. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 120 (or Ma 5 plus additional reading). Plane curves, rational functions, affine and projective varieties, products, local properties, birational maps, divisors, differentials, intersection numbers, schemes, sheaves, general varieties, vector bundles, coherent sheaves, curves and surfaces. Instructors: Arinkin, Graber.

Ma 131. Algebraic Geometry of Curves. 9 units (3-0-6); second term. Prerequisites: Ma 5, Ma 108, and Ma 109, or equivalent. The theory of algebraic curves is a central branch of mathematics, having relations to fields as diverse as complex analysis, number theory, combinatorics, codes, topology, representation theory, and physics. The aim of the course is to give a substantial introduction to this subject. The topics will include affine and projective plane curves, mappings, differentials, divisors and line bundles, Jacobians, sheaves, cohomology, and moduli. Important results such as Riemann-Roch theorem, Hurwitz’s theorem, and Abel’s theorem will be discussed. Not offered 2006–07.

Ma 135 ab. Arithmetic Geometry. 9 units (3-0-6); first, third terms. Prerequisite: Ma 130. The course deals with aspects of algebraic geometry that have been found useful for number theoretic applications. Topics will be chosen from the following: general cohomology theories (étale cohomology, flat cohomology, motivic cohomology, or p-adic Hodge theory), curves and Abelian varieties over arithmetic schemes, moduli spaces, Diophantine geometry, algebraic cycles. Not offered 2006–07.

Ma 140 abc. Functional Analysis. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 110. First term: theory of operators. Second term: symmetric and self-adjoint operators, spectral analysis of differential operators. Third term: theory of trace ideals on a Hilbert space, moment problems, and orthogonal polynomials. Not offered 2006–07.

Ma/ACM 142 abc. Ordinary and Partial Differential Equations. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 108. Ma 109 is desirable. The mathematical theory of ordinary and partial differential equations, including a discussion of elliptic regularity, maximal principles, solubility of equations. The method of characteristics. Part c not offered 2006–07. Instructor: Christiansen.

Ma/ACM 144 ab. Probability. 9 units (3-0-6); first, second terms. Overview of measure theory. Random walks and the Strong law of large numbers via the theory of martingales and Markov chains. Characteristic functions and the central limit theorem. Poisson process and Brownian motion. Not offered 2006–07.

Ma 145 abc. Introduction to Unitary Group Representations. 9 units (3-0-6); first, second, third terms. The study of representations of a group by unitary operators on a Hilbert space, including finite and compact groups, and, to the extent that time allows, other groups. First term: general representation theory of finite groups. Frobenius’s theory of representations of semidirect products. The Young tableaux and the representations of symmetric groups. Second term: the Peter-Weyl theorem. The classical compact groups and their representation theory. Weyl character formula. Third term: introduction to the representation theory of big groups (injective limits of finite and compact groups). Classification of irreducible characters for the infinite symmetric group and the infinite-dimensional unitary group. Generalized regular representations and their decomposition on irreducible components. Parts a, b not offered 2006–07. Instructor: Borodin.

Ma 147 abc. Dynamical Systems. 9 units (3-0-6); first, second, third terms. Prerequisites: Ma 108, Ma 109, or equivalent. First term: real dynamics and ergodic theory. Second term: Hamiltonian dynamics. Third term: complex dynamics. Instructors: Gorodetski, Makarov.

Ma 148. Topics in Mathematical Physics. 9 units (3-0-6); third term. The course will discuss the moment problem, inverse spectral theory for one-dimensional Schrödinger operators, and the connections between them. May be taken for credit in multiple years. Not offered 2006–07.

Ma 151 abc. Algebraic and Differential Topology. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 108 ab or equivalent. A basic graduate core course. Fundamental groups and covering spaces, homology and calculation of homology groups, exact sequences. Fibrations, higher homotopy groups, and exact sequences of fibrations. Bundles, Eilenberg-Maclane spaces, classifying spaces. Structure of differentiable manifolds, transversality, degree theory, De Rham cohomology, spectral sequences. Instructors: Dunfield, Calegari, Groves.

Ma 157 ab. Riemannian Geometry. 9 units (3-0-6); second, third terms. Prerequisite: Ma 151 or equivalent, or instructor’s permission. Part a: basic Riemannian geometry: geometry of Riemannian manifolds, connections, curvature, Bianchi identities, completeness, geodesics, exponential map, Gauss’s lemma, Jacobi fields, Lie groups, principal bundles, and characteristic classes. Part b: basic topics may vary from year to year and may include elements of Morse theory and the calculus of variations, locally symmetric spaces, special geometry, comparison theorems, relation between curvature and topology, metric functionals and flows, geometry in low dimensions. Instructors: Dunfield, Gorodnik.

Ma 160 abc. Number Theory. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 5. In this course, the basic structures and results of algebraic number theory will be systematically introduced. Topics covered will include the theory of ideals/divisors in Dedekind domains, Dirichlet unit theorem and the class group, p-adic fields, ramification, Abelian extensions of local and global fields. Instructors: Dimitrov, Mantovan.

Ma 162 abc. Topics in Number Theory. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 160. The course will discuss in detail some advanced topics in number theory, selected from the following: Galois representations, elliptic curves, modular forms, L-functions, special values, automorphic representations, p-adic theories, theta functions, regulators. Instructors: Wambach, Ramakrishnan.

Note: The courses labeled Ma 191, Ma 192, etc., are topics courses. Different courses are offered each year, reflecting the interests of faculty, visiting faculty, and students. Those offered in the fall term have an “a” designation, and “b” and “c” denote winter and spring. None of these courses is a prerequisite for any other.

Ma 191 a. Combinatorics of Finite Sets. 9 units (3-0-6); first term. Properties of subsets of finite sets. Topics include LYM inequality, AZ identity, shadows, shifting operation, the Erdos-Ko-Rado theorem, Katona’s intersection theorem, the complete intersection theorem of Ahlswede and Khachatrian, Hilton-Milner theorem, Turan’s problems for cancellative hypergraphs, Turan’s density for triple systems. Instructor: Ku.

Ma 191 b. Topics in Logic, Ergodic Theory, and Topological Dynamics. 9 units (3-0-6); second term. Some possible topics include the descriptive set theory of Borel equivalence relations and group actions, a theory of complexity of classification problems in mathematics, countable equivalence relations and ergodic theory, topological dynamics of the infinite symmetric group and automorphism groups of countable structures with applications to model theory and combinatorics, the Urysohn space and its group of isometries. Instructor: Kechris.

Ma 191 c. Topics in Spectral and Inverse Spectral Theory. 9 units (3-0-6); third term. Study of spectral and inverse spectral problems that arise in mathematical physics. Continuous (Schrödinger operators) and discrete models (Jacobi operators) will be discussed. Possible topics: spectral theorem, Weyl-Titchmarsh theory, eigenfunction expansion, Borg’s inverse theorems, Gel’fand-Levitan and Marchenko methods in inverse problems. Instructor: Zinchenko.

Ma 192 a. Riemann-Hilbert Problems, Their Asymptotics and Applications. 9 units (3-0-6); first term. Prerequisites: basic probability theory, complex variables, functional analysis. Basic theory of Riemann-Hilbert problems. Applications to problems in PDEs, orthogonal polynomials, random matrix theory, and combinatorics. Steepest descent method for Riemann-Hilbert problems. Solution of various asymptotic problems such as the longtime behavior of integrable systems, the asymptotics of orthogonal polynomials, and universality for random matrix ensembles, and the solution of Ulam’s problem in combinatorics. Instructor: Deift.

Ma 193 a. Discrete Groups and Geometry. 9 units (3-0-6); first term. This course will focus on groups that arise in geometry. Topics will include discrete subgroups of Lie groups, arithmetic groups, automorphism groups of geometric objects like manifolds and varieties, and properties of these groups. The production of interesting examples will pull ideas from many areas of mathematics. However, the primary focus will be on the interaction of groups with geometry, and in particular, on building geometric objects with interesting properties. Instructor: McReynolds.

SS/Ma 214. Mathematical Finance. 9 units (3-0-6). For course description, see Social Science.

Ma 290. Reading. Hours and units by arrangement. Occasionally, advanced work is given through a reading course under the direction of an instructor.

Note: The following research courses and seminars, intended for advanced graduate students, are offered according to demand. They cover selected topics of current interest. The courses offered, and the topics covered, will be announced at the beginning of each term.

Ma 316 abc. Seminar in Mathematical Logic. Instructor: Kechris.

Ma 324 abc. Seminar in Combinatorics. Instructor: Wilson.

Ma 325 abc. Seminar in Algebra. Instructors: Aschbacher, Wales.

Ma 345 abc. Seminar in Analysis and Dynamics. Instructors: Borodin, Makarov.

Ma 348 abc. Seminar in Mathematical Physics. Instructor: Simon.

Ma 351 abc. Seminar in Geometry and Topology. Instructors: Calegari, Dunfield.

Ma 360 abc. Seminar in Number Theory. Instructors: Flach, Graber, Mantovan, Ramakrishnan.

Ma 390. Research. Units by arrangement.

Ma 392. Research Conference. Three terms.

See also the list of courses in Applied and Computational Mathematics.