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
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:
Staff
Ma 2 ab
Differential Equations, Probability and Statistics
9 units (4-0-5)
|
first, second terms
Prerequisites: 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
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:
Wilson, Balachandran, Epstein
Ma 8
Problem Solving in Calculus
3 units (3-0-0)
|
first term
Prerequisites: 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
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
Prerequisites: 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
Prerequisites: 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
Prerequisites: 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
Prerequisites: 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
Prerequisites: 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
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, Lp-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 Hp-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
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. 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
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.
Instructor:
Lorden
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.
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
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.
Instructor:
Kechris
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 2008-09.
Ma 120 abc
Abstract Algebra
9 units (3-0-6)
|
first, second, third terms
Prerequisites: 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
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:
Balachandran, van de Bult
Ma 122 ab
Topics in Group Theory
9 units (3-0-6)
|
second, third terms
Prerequisites: 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
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 2008-09.
EE/Ma 126 ab
Information Theory
9 units (3-0-6)
|
first, second terms
Prerequisites: Ma 2.
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, first- order Markov, ergodic, and Gaussian. Calculation of capacity-cost and rate-distortion functions. Kolmogorov complexity and 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 127 ab, EE 161, and/or EE 167 should prepare the student for research in information theory, coding theory, wireless communications, and/or data compression.
Instructors:
Effros, staff
EE/Ma 127 ab
Error-Correcting Codes
9 units (3-0-6)
|
second, third terms
Prerequisites: Ma 2.
This course, which is a sequel to EE/Ma 126 a, but which may be taken independently, will develop from first principles the theory and practical implementation of the most important techniques for combatting errors in digital transmission or storage systems. Topics include algebraic block codes, e.g., Hamming, Golay, Fire, BCH, Reed-Solomon (including a self-contained introduction to the theory of finite fields); convolutional codes; and concatenated coding systems. Emphasis will be placed on the associated encoding and decoding algorithms, and students will be asked to demonstrate their understanding of these algorithms with software projects. In the third term, the modern theory of "turbo" and related codes (e.g., regular and irregular LDPC codes), with suboptimal iterative decoding based on belief propagation, will be presented. Not offered 2008-09.
CS/EE/Ma 129 abc
Information and Complexity
9 units (3-0-6), first and second terms
|
(1-4-4) third term
Prerequisites: basic knowledge of probability and discrete mathematics.
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. Parts b, c not offered 2008-09.
Instructor:
Abu-Mostafa
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. Part c not offered 2008-09.
Instructor:
Morin
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
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. Not offered 2008-09.
Ma 140 abc
Functional Analysis
9 units (3-0-6)
|
first, second, third terms
Prerequisites: 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
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. Part c not offered 2008-09.
Instructors:
Zinchenko, Kang
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. Only part c offered 2008-09 in winter.
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. Part c not offered 2008-09.
Instructor:
Makarov
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
Prerequisites: 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
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. Not offered 2008-09.
Instructor:
Gholampour
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:
Balasubramanyam, Mantovan
Ma 162 abc
Topics in Number Theory
9 units (3-0-6)
|
first, second, third 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 2008-09.
Ma 191 ab
Topics in Orthogonal Polynomials
9 units (3-0-6)
|
first, second terms
Prerequisites: 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
Prerequisites: 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
Prerequisites: 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
Prerequisites: 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
Prerequisites: 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
Prerequisites: 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)
|
second term
A course on fundamentals of the mathematical modeling of stock prices and interest rates, the theory of option pricing, risk management, and optimal portfolio selection. Students will be introduced to the stochastic calculus of various continuous-time models, including diffusion models and models with jumps. Not offered 2008-09.
Ma 290
Reading
Hours and units by arrangement
Occasionally, advanced work is given through a reading course under the direction of an instructor.
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
Published Date:
July 28, 2022