# Mathematics (Ma) Graduate Courses (2021-22)

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: Dunn, Wu, Karpukhin.

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: Marcolli, Karpukhin, Smillie.

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: Karpukhin, Rains, Radziwill.

Ma 111 abc.
Topics in Analysis.
9 units (3-0-6):
third term.
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. Part a (first term) and part b (second term) will not be offered 2021-2022.
Instructor: Katz.

Ma 112 ab.
Statistics.
9 units (3-0-6):
second term.
Prerequisites: 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 2021-22.

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.
Instructor: Vidnyanszky.

Ma/CS 117 abc.
Computability Theory.
9 units (3-0-6):
first, second, third terms.
Prerequisites: 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. Not offered 2021-22.

Ma 118.
Topics in Mathematical Logic: Geometrical Paradoxes.
9 units (3-0-6):
second term.
Prerequisites: 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 2021-22.

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: Szumowicz, Zhu, Flach.

Ma 121 ab.
Combinatorial Analysis.
9 units (3-0-6):
first, third terms.
Prerequisites: 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: Schülke, Conlon.

Ma 123.
Classification of Simple Lie Algebras.
9 units (3-0-6):
third term.
Prerequisites: 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 2021-22.

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 2021-22.

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 2021-22.

EE/Ma/CS 126 ab.
Information Theory.
9 units (3-0-6):
first, second terms.
Prerequisites: Ma 3.
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/IDS 127, EE/CS 161, and EE/CS/IDS 167, should prepare the student for research in information theory, coding theory, wireless communications, and/or data compression. EE/Ma/CS 126a offered 2021-2022; EE/Ma/CS 126b not offered 2021-2022.
Instructor: Effros.

EE/Ma/CS/IDS 127.
Error-Correcting Codes.
9 units (3-0-6):
third term.
Prerequisites: EE 55 or Ma 3.
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 highly symmetric linear codes, such as Hamming, Reed-Muller, and Polar codes; algebraic block codes, e.g., BCH, Reed-Solomon (including a self-contained introduction to the theory of finite fields); and sparse graph codes with iterative decoding, i.e., LDPC code and turbo codes. Students will become acquainted with encoding and decoding algorithms, design principles and performance evaluation of codes.
Instructor: Kostina.

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. Not offered 2021-22.

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

Ma 132 abc.
Topics in Algebraic Geometry.
9 units (3-0-6):
.
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. Previous topics have included geometric invariant theory, moduli of curves, logarithmic geometry, Hodge theory, and toric varieties. This course can be repeated for credit. Not offered 2021-22.

Ma 135 ab.
Arithmetic Geometry.
9 units (3-0-6):
first term.
Prerequisites: 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.
Instructor: Flach.

EE/Ma/CS/IDS 136.
Topics in Information Theory.
9 units (3-0-6):
third term.
Prerequisites: Ma 3 or ACM/EE/IDS 116 or CMS 117 or Ma/ACM/IDS 140a.
This class introduces information measures such as entropy, information divergence, mutual information, information density from a probabilistic point of view, and discusses the relations of those quantities to problems in data compression and transmission, statistical inference, language modeling, game theory and control. Topics include information projection, data processing inequalities, sufficient statistics, hypothesis testing, single-shot approach in information theory, large deviations. Not offered 2021-2022.
Instructor: Kostina.

Ma/ACM/IDS 140 ab.
Probability.
9 units (3-0-6):
second, third terms.
Prerequisites: For 140 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.
Instructors: Vigneaux, Tamuz.

Ma/ACM 142 ab.
Ordinary and Partial Differential Equations.
9 units (3-0-6):
second, third terms.
Prerequisites: 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: Angelopoulos, Isett.

Ma 145 abc.
Topics in Representation Theory.
9 units (3-0-6):
second term.
Prerequisites: Ma 5.
This course will discuss the study of representations of a group (or related algebra) by linear transformations of a vector space. Topics will vary from year to year, and may include modular representation theory (representations of finite groups in finite characteristic), complex representations of specific families of groups (esp. the symmetric group) and unitary representations (and structure theory) of compact groups. Not offered in 2021-22.

Ma 147 abc.
Dynamical Systems.
9 units (3-0-6):
first term.
Prerequisites: Ma 108, Ma 109, or equivalent.
First term: real dynamics and ergodic theory. Second term: Hamiltonian dynamics. Third term: complex dynamics. Part b and c not offered in 2021-2022.
Instructor: Wu.

Ma 148 ab.
Topics in Mathematical Physics.
9 units (3-0-6):
first 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. Part b not offered in 2021-22.
Instructor: Marcolli.

Ma 151 abc.
Algebraic and Differential Topology.
9 units (3-0-6):
first, second, third terms.
Prerequisites: Ma 109 abc or equivalent.
Part a: Homology Theory. Homology and calculation of homology groups, exact sequences, cohomology rings, Poincaré duality. Part b: Homotopy Theory and K-theory. Fibrations, higher homotopy groups, and exact sequences of fibrations. Fiber bundles, Eilenberg-MacLane spaces, classifying spaces. K-theory, generalized cohomology theory, Bott periodicity. Part c: Characteristic classes. Stiefel-Whitney classes, Chern classes, Pontryagin classes, cobordism theory, Chern-Weil theory.
Instructors: Mazel-Gee, Ni.

Ma 157 abc.
Riemannian Geometry.
9 units (3-0-6):
first, second terms.
Prerequisites: 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 2021-22.
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: Dunn, Burungale Radziwill.

Ma 162 ab.
Topics in Number Theory.
9 units (3-0-6):
first, second terms.
Prerequisites: 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 2021-22.

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: Burungale, Katz, Kechris, Zhu, Szumowicz, Dunn, Hutchcroft, Karpukhin, Schülke.

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:
.

### Please Note

The online version of the Caltech Catalog is provided as a convenience; however, the printed version is the only authoritative source of information about course offerings, option requirements, graduation requirements, and other important topics.