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: Breuer, Graber, Aschbacher, Wilson, Flach, Rains.

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: Balasubramanyam.

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: Ryckman.

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: Aschbacher, Morin.

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, Balachandran, Epstein.

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: Lyons.

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 may also be elementary lectures from members of the mathematics faculty on topics of their own research interest. Instructor: Rains.

Ma 11. Mathematical Writing. 3 units (0-0-3); third term. Freshmen must have instructor’s permission to enroll. 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 in a classroom setting. Some help with typesetting in TeX may be available. Students are encouraged to take advantage of the Hixon Writing Center’s facilities. The mentor and the topic are to be 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 (mathematics students at Caltech). Fulfills the Institute scientific writing requirement. Not offered on a pass/fail basis. Instructor: Wilson.

Ma 12. Chance. 9 units (4-0-5); second term. Prerequisite: Ma 2 b (probability and statistics). This course will explore examples of the use and misuse of notions of probability and statistics in popular culture and in scientific research. Basic ideas about random fluctuations will be introduced, along with simple techniques like nonparametric statistics and the bootstrap. Not offered 2008–09.

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: Nelson.

Ma 91 a. Homological Algebra. 9 units (3-0-6); first term. Prerequisite: Ma 5 or instructor’s permission. This course will be a first introduction to homological algebra, covering generalities on additive and abelian categories; the category of complexes, and the long exact sequence of cohomology; cones and homotopies; the homotopic category of complexes; projective and injective resolutions, and the derived category; derived functors; double complexes; spectral sequences; and further topics as time permits. Instructor: Aluffi.

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. Instructor: Rains.

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, Ryckman, van de Bult.

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: Wang, Gholampour.

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 real analysis: topological spaces, Hilbert space basics, Fejer’s theorem, measure theory, measures as functionals, product measures, L^p-spaces, Baire category, Hahn- Banach theorem, Alaoglu’s theorem, Krein-Millman theorem, countably normed spaces, tempered distributions and the Fourier transform. Second term: basic complex analysis: analytic functions, conformal maps and fractional linear transformations, idea of Riemann surfaces, elementary and some special functions, infinite sums and products, entire and meromorphic functions, elliptic functions. Third term: harmonic analysis; operator theory. Harmonic analysis: maximal functions and the Hardy-Littlewood maximal theorem, the maximal and Birkoff ergodic theorems, harmonic and subharmonic functions, theory of H^p-spaces and boundary values of analytic functions. Operator theory: compact operators, trace and determinant on a Hilbert space, orthogonal polynomials, the spectral theorem for bounded operators. If time allows, the theory of commutative Banach algebras. Instructors: Duits, Simon, Breuer.

Ma 111 ab. Analysis, II. 9 units (3-0-6); first, third terms. Prerequisite: Ma 110 or instructor’s permission. This course will discuss advanced topics in analysis, which vary from year to year. Topics from previous years include potential theory, bounded analytic functions in the unit disk, probabilistic and combinatorial methods in analysis, operator theory, C*-algebras. First term: classical special functions. Third term: Riemann surface theory. Instructors: van de Bult, Zinchenko.

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 2008–09.

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. Instructor: Kechris.

Ma 118. Topics in Mathematical Logic: Geometrical Paradoxes. 9 units (3-0-6); second term. Prerequisite: Ma 5 or equivalent, or instructor’s permission. This course will provide an introduction to the striking paradoxes that challenge our geometrical intuition. Topics to be discussed include geometrical transformations, especially rigid motions; free groups; amenable groups; group actions; equidecomposability and invariant measures; Tarski’s theorem; the role of the axiom of choice; old and new paradoxes, including the Banach-Tarski paradox, the Laczkovich paradox (solving the Tarski circle-squaring problem), and the Dougherty-Foreman paradox (the solution of the Marczewski problem). Not offered 2008–09.

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: Graber, Flach, Mantovan.

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: Balachandran, van de Bult.

Ma 122 ab. Topics in Group Theory. 9 units (3-0-6); second, third 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 2008–09.

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. Not offered 2008–09.

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. Instructor: Morin. Part c not offered 2008–09.

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 2008–09.

Ma 135 ab. Arithmetic Geometry. 9 units (3-0-6); first, second 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 2008–09.

Ma 140 abc. Functional Analysis. 9 units (3-0-6); first, second, third terms. Prerequisite: Ma 110. First term: introduction to C*-algebras and von Neumann algebras. Second term: von Neumann algebras arising from countable groups and from measure-preserving actions of countable groups. Third term: introduction to spectral theory with applications to Schrödinger operators. Not offered 2008–09.

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. Instructors: Zinchenko, Kang. Part c not offered 2008–09.

Ma/ACM 144 ab. Probability. 9 units (3-0-6); second, third 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 2008–09.

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. Instructor: Borodin. Parts a, b not offered 2008–09.

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. Instructor: Makarov. Part c not offered 2008–09.

Ma 148. Topics in Mathematical Physics: Hamiltonian Dynamics. 9 units (3-0-6); first term. This course will study the Hamiltonian formalism of classical mechanics. Topics will include symplectic structures on finite-dimensional vector spaces, Hamiltonian vector fields, Poisson brackets, symplectic manifolds, Darboux’s theorem, canonical transformations, normal forms near a critical point of the Hamiltonian, completely integrable systems, and some elements of KAM theory. Instructor: Ryckman.

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: Calegari, Gholampour, Staff.

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. Instructor: Gholampour. Part b not offered 2008–09.

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: Balasubramanyam, 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. Not offered 2008–09.

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 ab. Topics in Orthogonal Polynomials. 9 units (3-0-6); first, second terms. Prerequisite: Ma 110. A comprehensive study of the spectral theory and general asymptotics of orthogonal polynomials, especially on the real line and unit circle. Topics may include general techniques: recursion relations and transfer matrices, Jacobi and CMV matrices, Green’s function and connection to spectral measures, co-efficient stripping and continued fractions, CD kernel and formulae. Examples: classical OPs, ergodic families, potential theory and regular OPs, clock behavior of zeros. Ratio asymptotics: Nevai class, Weyl and Denisov–Rahkmanov theorems, Remling theory. Sum rules: Szego’s theorem, Killip–Simon theorem, matrix OPs. Periodic Jacobi matrices: isospectral tori, finite gap analysis. Instructor: Simon.

Ma 191 c. Topics in Schramm-Loewner Evolutions. 9 units (3-0-6); third term. Prerequisite: Ma 110 or instructor’s permission. The course will discuss the basic notions and results in Schramm-Loewner evolutions (SLEs). Topics will cover the proof of Mandelbrot conjecture on the planar Brownian frontier, the scaling limits of several lattice models from statistical physics, and certain path properties of SLEs including the regularity, reversibility, and duality. Applications in conformal field theory will be presented. Instructor: Kang.

Ma 192 a. Geometry and Arithmetic of Quantum Fields. 9 units (3-0-6); first term. The course will focus on mathematical structures of renormalization in perturbative quantum field theory and of the standard model of elementary particle physics. The main themes will be the mysterious relation between renormalization in quantum field theory and the theory of motives in arithmetic geometry, as well as the models of particle physics obtained using noncommutative geometry. Instructor: Marcolli.

Ma 192 b. Topics in Low-Dimensional Topology: Stable Commutator Length. 9 units (3-0-6); second term. Prerequisite: Ma 151 or instructor’s permission. In this course we present and discuss some elements of the geometric theory of two-dimensional (bounded) homology from several points of view, making contact with low-dimensional geometry and topology, geometric group theory, and group dynamics. Instructor: Calegari.

Ma 192 c. Riemann-Hilbert Problems and Orthogonal Polynomials. 9 units. (3-0-6); third term. Prerequisite: Ma 110. This course will discuss the Riemann-Hilbert approach to obtain asymptotics for orthogonal polynomials. The course will mainly focus on orthogonal polynomials on the real line with respect to a varying exponential weight, but other types of orthogonality will be discussed as well. The methods that will be used involve elements of complex analysis, special functions, and operator theory. The course also includes applications to random matrix theory. Instructor: Duits.

Ma 193 a. Random Matrix Theory. 9 units (3-0-6); first term. Prerequisite: Ma 108. Wigner matrices, Gaussian and circular ensem-bles of random matrices. Dyson’s threefold way: orthogonal, unitary, and symplectic ensembles. Correlation functions; determinantal and Pfaffian random point processes. Scaling limits. Fredholm determinant approach to gap probabilities. Instructor: Borodin.

Ma 193 b. Dimer Models. 9 units (3-0-6); second term. Prerequisite: Ma 108. Height function for dimers on bipartite graphs. Kasteleyn theory. Gibbs measures. Honeycomb and square lattices: inverse Kasteleyn matrix, decay of correlations, height fluctuations. Domino tilings of the Aztec diamond, lozenge tilings of the hexagon. Correlations via classical orthogonal polynomials. Gaussian free field fluctuations. Instructor: Borodin.

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. Instructor: Aschbacher.

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. Instructor: Calegari.

Ma 352 abc. Seminar in Algebraic Geometry. Instructor: Graber.

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