# 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: Amir Khosravi, Ramakrishnan, Kechris, Rains, Flach, Katz.

**Ma 1 d. Series. ***4 units (2-0-2); second term only. Prerequisites: 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: Mantovan.

**Ma 2/102. Differential Equations.** 9 units (4-0-5); first term. Prerequisites: Ma 1 abc.* *The course is aimed at providing an introduction to the theory of ordinary differential equations, with a particular emphasis on equations with well known applications ranging from physics to population dynamics. The material covered includes some existence and uniqueness results, first order linear equations and systems, exact equations, linear equations with constant coefficients, series solutions, regular singular equations, Laplace transform, and methods for the study of nonlinear equations (equilibria, stability, predator-prey equations, periodic solutions and limiting cycles). Instructor: Isett.

**Ma 3/103. Introduction to Probability and Statistics. ***9 units (4-0-5); second term. Prerequisites: Ma 1 abc.*** **Randomness is not anarchy-it follows mathematical laws that we can understand and use to clarify our knowledge of the universe. This course is an introduction to the main ideas of probability and statistics. The first half is devoted to the fundamental concepts of probability theory, including distributions and random variables, independence and conditional probability, expectation, the Law of Averages (Laws of Large Numbers), and “the bell curve” (Central Limit Theorem). The second half is devoted to statistical reasoning: given our observations of the world, what can we infer about the stochastic mechanisms generating our data? Major themes include estimation of parameters (e.g. maximum likelihood), hypothesis testing, confidence intervals, and regression analysis (least squares). Students will be expected to be able to carry out computer-based analyses. Instructor: Border.

**Ma 4/104. 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: Parikh.

**Ma 5/105 abc. Introduction to Abstract Algebra. ***9 units (3-0-6); first, second, third terms.*** **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: Graber, Mantovan, Xu.

**Ma/CS 6/106 abc. Introduction to Discrete Mathematics. ***9 units (3-0-6); first, second, third terms. Prerequisites: 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: Katz, Durcik, Kechris.

**Ma 7/107. 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 = Â±1, constructible numbers, composition of binary quadratic forms, and congruences. Instructor: Burungale.

**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: Mantovan.

**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: Graber.

**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: Ni.

**FS/Ma 12. The Mathematics of Enzyme Kinetics. ***6 units (2-0-4); third term. Prerequisites: Ma 1a, b. *For course description, see Freshman Seminars.

**Ma 13. Problem Solving in Vector Calculus.** *2 units (2-0-0); second term. Prerequisites: Concurrent registration in Ph 1b. *A two-hour per week, hands-on class for those students enrolled in Ph 1b needing extra practice with problem solving in vector calculus. Instructor: Mantovan.

**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: Katz.

**Ma 20. Frontiers in Mathematics.*** 1 unit (1-0-0); first term. Prerequisites: Open for credit to freshman and sophomores.* Weekly seminar by a member of the math department or a visitor, to discuss his or her research at an introductory level. The course aims to introduce students to research areas in mathematics and help them gain an understanding of the scope of the field. Graded pass/fail. Instructor: Graber.

**Ma 92 abc. Senior Thesis. ***9 units (0-0-9); first, second, third terms. Prerequisites: To register, the student must obtain permission of the mathematics undergraduate representative.* 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 97. Research in Mathematics. ***Units to be arranged in accordance with work accomplished. *This course is designed to allow students to continue or expand summer research projects and to work on new projects. Students registering for more than 6 units of Ma 97 must submit a brief (no more than 3 pages) written report outlining the work completed to the undergraduate option rep at the end of the term. Approval from the research supervisor and student’s advisor must be granted prior to registration. Graded pass/fail.

**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 108 abc. Classical Analysis. ***9 units (3-0-6); first, second, third terms. Prerequisites: Ma 1 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 R^n. 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: Lazebnik, Durcik, Ivrii.

**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: Markovic, Chen.

**Ma 110 abc. Analysis.** *9 units (3-0-6); first, second, third terms. Prerequisites: 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: Makarov, Durcik.

**Ma 111 ab. Topics in Analysis. ***9 units (3-0-6); second, third terms. Prerequisites: 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, functional analysis. The third term will cover special functions: gamma functions, hypergeometric functions, beta/Selberg integrals and $q$-analogues. Time permitting: orthogonal polynomials, Painleve transcendents and/or elliptic analogues Instructors: Ivrii, Lazebnik.

**Ma 112 ab. Statistics. ***9 units (3-0-6); second term. 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. Not offered 2018–19.

**Ma 116 abc. Mathematical Logic and Axiomatic Set Theory. ***9 units (3-0-6); first, second, third terms. Prerequisites: Ma 5 or equivalent, or instructor’s permission.* First term: Introduction to first-order logic and model theory. The Godel Completeness Theorem and the Completeness Theorem. Definability, elementary equivalence, complete theories, categoricity. The Skolem-Lowenheim Theorems. The back and forth method and Ehrenfeucht-Fraisse games. Farisse theory. Elimination of quantifiers, applications to algebra and further related topics if time permits. Second and third terms: Axiomatic set theory, ordinals and cardinals, the Axiom of Choice and the Continuum Hypothesis. Models of set theory, independence and consistency results. Topics in descriptive set theory, combinatorial set theory and large cardinals. Not offered 2018–19.

**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, Panagiotopolous.

**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 2018–19.

**Ma 120 abc. Abstract Algebra.*** 9 units (3-0-6); first, second, third terms. Prerequisites: Ma 5 or equivalent or instructor’s permission.* This course will discuss advanced topics in algebra. Among them: an introduction to commutative algebra and homological algebra, infinite Galois theory, Kummer theory, Brauer groups, semisimiple algebras, Weddburn theorems, Jacobson radicals, representation theory of finite groups. Instructors: Ramakrishnan, Zhu, Yom Din.

**Ma 121 ab. Combinatorial Analysis. ***9 units (3-0-6); first, second 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: Krause, Katz.

**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 2018–2019.

**Ma 124. Elliptic Curves. ***9 units (3-0-6); second term. Prerequisites: Ma 5 or equivalent. *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 2018–19.

**Ma 125. Algebraic Curves. 9*** units (3-0-6); third term. Prerequisites: Ma 5. *An elementary introduction to the theory of algebraic curves. Topics to be covered will include affine and projective curves, smoothness and singularities, function fields, linear series, and the Riemann-Roch theorem. Possible additional topics would include Riemann surfaces, branched coverings and monodromy, arithmetic questions, introduction to moduli of curves. Not offered 2018–19.

**EE/Ma/CS 126 ab. Information Theory. ***9 units (3-0-6); first, second terms. Prerequisites: Ma 3 *For course description, see Electrical Engineering.

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

**Ma 128. Homological Algebra. ***9 units (3-0-6); second term. Prerequisites: Math 120 abc or instructor’s permission. *This course introduces standard concepts and techniques in homological algebra. Topics will include Abelian and additive categories; Chain complexes, homotopies and the homotopy category; Derived functors; Yoneda extension and its ring structure; Homological dimension and Koszul complexe; Spectral sequences; Triangulated categories, and the derived category. Instructor: Qi.

**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: Campbell, Xu.

**Ma 132 ab. Topics in Algebraic Geometry. ***9 units (3-0-6); second, third terms. Prerequisites: Ma 130 or instructor’s permission. *This course will cover advanced topics in algebraic geometry that will vary from year to year. Topics will be listed on the math option website prior to the start of classes. Instructors: Graber, Zhu.

**Ma 135 ab. Arithmetic Geometry.*** 9 units (3-0-6); first term. 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 2018-2019.

**EE/Ma/CS/IDS 136. Topics in Information Theory. ***9 units (3-0-6). *For course description, see Electrical Engineering.

**Ma/ACM 142. Ordinary and Partial Differential Equations.** *9 units (3-0-6); second term. 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. Instructor: Isett.

**Ma/ACM/IDS 144 ab. Probability. ***9 units (3-0-6); first term. Prerequisites: For 144 a, Ma 108 b is strongly recommended.* 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. Topics in statistics. Part b not offered in 2018–19. Instructor: Tamuz.

**Ma 145ab. Representation Theory.** *9 units (3-0-6); second term. Prerequisites: Ma 5.*** **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. Part a not offered in 2018–19. Instructor: Rains.

**Ma 147 abc. Dynamical Systems. ***9 units (3-0-6); first, second terms. Prerequisites: Ma 108, Ma 109, or equivalent. *First term: real dynamics and ergodic theory. Second term: Hamiltonian dynamics. Third term: complex dynamics. Part c not offered in 2018–19. Instructors: Makarov, Lazebnik.

**Ma 148. Topics in Mathematical Physics. ***9 units (3-0-6); second term.* This course covers a range of topics in mathematical physics. The content will vary from year to year. Topics covered will include some of the following: Lagrangian and Hamiltonian formalism of classical mechanics; mathematical aspects of quantum mechanics: Schroedinger equation, spectral theory of unbounded operators, representation theoretic aspects; partial differential equations of mathematical physics (wave, heat, Maxwell, etc.); rigorous results in classical and/or quantum statistical mechanics; mathematical aspects of quantum field theory; general relativity for mathematicians. Geometric theory of quantum information and quantum entanglement based on information geometry and entropy. Instructors: Parikh.

**Ma 151 abc. Algebraic and Differential Topology. ***9 units (3-0-6); first, second, third terms. Prerequisite: Ma 109 abc 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: Markovic, Qi, Ni.

**Ma 157 abc. 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. Part c not offered in 2018-2019. Instructor: Smillie.

**Ma 160 abc. Number Theory. ***9 units (3-0-6); first, second, third terms. Prerequisites: 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: Burungale, Campbell, Flach.

**Ma 162 ab. Topics in Number Theory.*** 9 units (3-0-6); first term. 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. Part b not offered in 2018–19. Instructor: Radziwill.

**Ma 191 abc. Selected Topics in Mathematics.** *9 units (3-0-6); first, second, third terms.* Each term we expect to give between 0 and 6 (most often 2-3) topics courses in advanced mathematics covering an area of current research interest. These courses will be given as sections of 191. Students may register for this course multiple times even for multiple sections in a single term. The topics and instructors for each term and course descriptions will be listed on the math option website each term prior to the start of registration for that term. Instructors: Panagiotopolous, Ni, Yom Din, Flach, Radziwill, Zhu, Parikh, Ramakrishnan, Amir Khosravi, Rains, Krause, Campbell.

**SS/Ma 214. Mathematical Finance. ***9 units (3-0-6); second term. *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.

**Ma 390. Research.*** Units by arrangement.*

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