# Mathematics (Ma) Graduate Courses (2017-18)

Ma 106.
Elliptic Curves.
9 units (3-0-6):
second term.
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.
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.
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: Marcolli, Li, Vafaee.

Ma 110 abc.
Analysis.
9 units (3-0-6):
first, second, third terms.
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, Rains, Katz.

Ma 111 ab.
Topics in Analysis.
9 units (3-0-6):
second, third terms.
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: Makarov, Lazebnik.

Ma 112 ab.
Statistics.
9 units (3-0-6):
second term.
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 2017-18.

Ma 116 abc.
Mathematical Logic and Axiomatic Set Theory.
9 units (3-0-6):
first, second, third terms.
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.
Instructor: Panagiotopoulos.

Ma/CS 117 abc.
Computability Theory.
9 units (3-0-6):
first, second, third terms.
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. Not offered 2017-18.

Ma 118.
Topics in Mathematical Logic: Geometrical Paradoxes.
9 units (3-0-6):
second term.
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 2017-18.

Ma 120 abc.
Abstract Algebra.
9 units (3-0-6):
first, second, third terms.
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.
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: Rains, Conlon.

Ma 123.
Classification of Simple Lie Algebras.
9 units (3-0-6):
third term.
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 2017-18.

Ma 125.
Algebraic Curves.
9 units (3-0-6):
third term.
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.
Instructor: Zhu.

EE/Ma/CS 126 ab.
Information Theory.
9 units (3-0-6):
first, second terms.
Shannon's mathematical theory of communication, 1948-present. Entropy, relative entropy, and mutual information for discrete and continuous random variables. Shannon's source and channel coding theorems. Mathematical models for information sources and communication channels, including memoryless, Markov, ergodic, and Gaussian. Calculation of capacity and rate-distortion functions. Universal source codes. Side information in source coding and communications. Network information theory, including multiuser data compression, multiple access channels, broadcast channels, and multiterminal networks. Discussion of philosophical and practical implications of the theory. This course, when combined with EE 112, EE/Ma/CS 127, EE/CS 161, EE 167, and/or EE 226 should prepare the student for research in information theory, coding theory, wireless communications, and/or data compression.
Instructor: Effros.

EE/Ma/CS 127.
Error-Correcting Codes.
9 units (3-0-6):
first term.
This course develops from first principles the theory and practical implementation of the most important techniques for combating errors in digital transmission or storage systems. Topics include algebraic block codes, e.g., Hamming, BCH, Reed-Solomon (including a self-contained introduction to the theory of finite fields); and the modern theory of sparse graph codes with iterative decoding, e.g. LDPC codes, turbo codes. The students will become acquainted with encoding and decoding algorithms, design principles and performance evaluation of codes.
Instructor: Kostina.

CS/EE/Ma 129 abc.
Information and Complexity.
9 units (3-0-6), first and second terms:
(1-4-4) third term.
A basic course in information theory and computational complexity with emphasis on fundamental concepts and tools that equip the student for research and provide a foundation for pattern recognition and learning theory. First term: what information is and what computation is; entropy, source coding, Turing machines, uncomputability. Second term: topics in information and complexity; Kolmogorov complexity, channel coding, circuit complexity, NP-completeness. Third term: theoretical and experimental projects on current research topics. Not offered 2017-18.

Ma 130 abc.
Algebraic Geometry.
9 units (3-0-6):
first, second, third terms.
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: Graber, Xu.

Ma 132 c.
Topics in Algebraic Geometry.
9 units (3-0-6):
third term.
This course will cover advanced topics in algebraic geometry that will vary from year to year. This year, the topic will be deformation theory. Not offered 2017-18.

Ma 135 ab.
Arithmetic Geometry.
9 units (3-0-6):
first term.
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. Part b not offered in 2017-18
Instructor: Flach.

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

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. Topics in statistics.
Instructors: Tamuz, Ivrii.

Ma 145 ab.
Representation Theory.
9 units (3-0-6):
first term.
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.
Instructor: Yom Din.

Ma 147 abc.
Dynamical Systems.
9 units (3-0-6):
first, second, third terms.
First term: real dynamics and ergodic theory. Second term: Hamiltonian dynamics. Third term: complex dynamics. Not offered 2017-18.

Ma 148 abc.
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. Parts b & c not offered in 2017-18.
Instructor: Makarov.

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

Ma 157 abc.
Riemannian Geometry.
9 units (3-0-6):
second, third terms.
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 2017-018.
Instructor: Li.

Ma 160 ab.
Number Theory.
9 units (3-0-6):
first, second terms.
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. Not offered 2017-18.

Ma 162 ab.
Topics in Number Theory.
9 units (3-0-6):
second, third terms.
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 2017-18.

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: Qi, Ni, Kechris, Xu, Lazebnik, Amir Khosravi, Yom Din, Zhu, Zhang, Ramakrishnan, Durcik, Ivrii, Lupini.

SS/Ma 214.
Mathematical Finance.
9 units (3-0-6):
second term.
A course on pricing financial derivatives, risk management, and optimal portfolio selection using mathematical models. Students will be introduced to methods of Stochastic, Ito Calculus for models driven by Brownian motion. Models with jumps will also be discussed.
Instructor: Cvitanic.

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