MATH 19:
Calculus
Introduction to differential calculus of functions of one variable. Topics: review of elementary functions including exponentials and logarithms, limits, rates of change, the derivative, and applications. Math 19, 20, and 21 cover the same material as Math 41 and 42, but in three quarters rather than two. Prerequisites: precalculus, including trigonometry, advanced algebra, and analysis of elementary functions.
Terms: Aut, Win, Sum
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 20:
Calculus
Continuation of 19. Applications of differential calculus; introduction to integral calculus of functions of one variable, including: the definite integral, methods of symbolic and numerical integration, applications of the definite integral. Prerequisites: 19 or equivalent.
Terms: Win, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 21:
Calculus
Continuation of 20. Applications of integral calculus, introduction to differential equations, infinite series. Prerequisite: 20 or equivalent.
Terms: Spr
| Units: 4
| UG Reqs: GER:DB-Math, WAY-FR
MATH 41:
Calculus (accelerated)
Introduction to differential and integral calculus of functions of one variable. Topics: limits, rates of change, the derivative and applications, introduction to the definite integral and integration. Math 41 and 42 cover the same material as Math 19-20-21, but in two quarters rather than three. Prerequisites: trigonometry, advanced algebra, and analysis of elementary functions, including exponentials and logarithms.
Terms: Aut
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 41A:
Calculus ACE
Students attend MATH 41 lectures with different recitation sessions, four hours instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html.
Terms: Aut
| Units: 6
| UG Reqs: GER:DB-Math, WAY-FR
MATH 42:
Calculus (Accelerated)
Continuation of 41. Methods of symbolic and numerical integration, applications of the definite integral, introduction to differential equations, infinite series. Prerequisite: 41 or equivalent.
Terms: Aut, Win
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 42A:
Calculus ACE
Students attend MATH 41 lectures with different recitation sessions, four hours instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html.
Terms: Aut, Win
| Units: 6
| UG Reqs: GER:DB-Math, WAY-FR
MATH 50V:
Multivariable Differential Calculus
Differential calculus for functions of two or more variables. Topics: vectors and vector-valued functions in 2-space and 3-space, tangent and normal vectors, curvature, functions of two or more variables, partial derivatives and differentiability, directional derivatives and gradients, maxima and minima, and optimization using Lagrange multipliers. Prerequisites: two quarters of single variable calculus, or consent of instructor.
Terms: Sum
| Units: 4
| UG Reqs: GER:DB-Math
MATH 51:
Linear Algebra and Differential Calculus of Several Variables
Geometry and algebra of vectors, systems of linear equations, matrices and linear transformations, diagonalization and eigenvectors, vector valued functions and functions of several variables, parametric curves, partial derivatives and gradients, the derivative as a matrix, chain rule in several variables, constrained and unconstrained optimization. Prerequisite: 21, or 42, or a score of 4 on the BC Advanced Placement exam or 5 on the AB Advanced Placement exam, or consent of instructor.
Terms: Aut, Win, Spr, Sum
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
Instructors: ;
Church, T. (PI);
Ding, J. (PI);
Lucianovic, M. (PI);
Luu, M. (PI);
Maydanskiy, M. (PI);
Mazzeo, R. (PI);
McNamara, P. (PI);
Mukamel, R. (PI);
Rubinstein, Y. (PI);
Van Horn-Morris, J. (PI);
Wang, Y. (PI);
White, B. (PI);
Lucianovic, M. (SI)
MATH 51A:
Linear Algebra and Differential Calculus of Several Variables, ACE
Students attend MATH 51 lectures with different recitation sessions: four hours per week instead of two, emphasizing engineering applications. Prerequisite: application; see http://soe.stanford.edu/edp/programs/ace.html.
Terms: Aut, Win, Spr
| Units: 6
| UG Reqs: GER:DB-Math, WAY-FR
MATH 51H:
Honors Multivariable Mathematics
For prospective Mathematics majors in the honors program and students from other areas of science or engineering who have a strong mathematics background. Three quarter sequence covers the material of 51, 52, 53, and additional advanced calculus and ordinary and partial differential equations. Unified treatment of multivariable calculus, linear algebra, and differential equations with a different order of topics and emphasis from standard courses. Students should know one-variable calculus and have an interest in a theoretical approach to the subject. Prerequisite: score of 5 on BC Advanced Placement exam, or consent of instructor.
Terms: Aut
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 51M:
Introduction to MATLAB for Multivariable Mathematics
Corequisite: MATH 51.
Terms: Aut
| Units: 1
MATH 52:
Integral Calculus of Several Variables
Iterated integrals, line and surface integrals, vector analysis with applications to vector potentials and conservative vector fields, physical interpretations. Divergence theorem and the theorems of Green, Gauss, and Stokes. Prerequisite: 51 and 42 or equivalents.
Terms: Aut, Win, Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 52H:
Honors Multivariable Mathematics
Continuation of 51H. Prerequisite: 51H.
Terms: Win
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 52V:
Multivariable Integral Calculus
Integral calculus for functions of two or more variables. Topics: double and triple integrals, change of variables and the Jacobian, vector fields, line integrals, independence of path and the fundamental theorem of line integrals, Green's theorem, divergence theorem, Stokes's theorem. Prerequisites: one quarter of multivariable differential calculus (comparable to 50V or 51), or consent of instructor.
Terms: Sum
| Units: 5
| UG Reqs: GER:DB-Math
MATH 53:
Ordinary Differential Equations with Linear Algebra
Ordinary differential equations and initial value problems, systems of linear differential equations with constant coefficients, applications of second-order equations to oscillations, matrix exponentials, Laplace transforms, stability of non-linear systems and phase plane analysis, numerical methods. Prerequisite: 51 and 42 or equivalents.
Terms: Aut, Win, Spr, Sum
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 53H:
Honors Multivariable Mathematics
Continuation of 52H. Prerequisite: 52H.
Terms: Spr
| Units: 5
| UG Reqs: GER:DB-Math, WAY-FR
MATH 70SI:
The Game of Go: Strategy, Theory, and History
Strategy and mathematical theories of the game of Go, with guest appearance by a professional Go player.
Terms: Win
| Units: 1
MATH 80Q:
Capillary Surfaces: Explored and Unexplored Territory
Preference to sophomores. Capillary surfaces: the interfaces between fluids that are adjacent to each other and do not mix. Recently discovered phenomena, predicted mathematically and subsequently confirmed by experiments, some done in space shuttles. Interested students may participate in ongoing investigations with affinity between mathematics and physics.
Last offered: Winter 2011
| Units: 3
| UG Reqs: WAY-FR, WAY-SMA
MATH 87Q:
Mathematics of Knots, Braids, Links, and Tangles
Preference to sophomores. Types of knots and how knots can be distinguished from one another by means of numerical or polynomial invariants. The geometry and algebra of braids, including their relationships to knots. Topology of surfaces. Brief summary of applications to biology, chemistry, and physics.
Terms: Win
| Units: 3
| UG Reqs: WAY-FR
MATH 102:
Mathematics and magic tricks
Performance magic tricks based on combinatorics, number theory, and topology; open mathematical problems related to magic tricks; history of magic and methods of inventing new tricks (and new mathematics.) Students are expected to learn performable skills.
Terms: Spr
| Units: 3
MATH 104:
Applied Matrix Theory
Linear algebra for applications in science and engineering: orthogonality, projections, the four fundamental subspaces of a matrix, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares, the condition number of a matrix, algorithms for solving linear systems. Prerequisites: MATH 51 and MATH 52 or 53.
Terms: Aut, Win, Sum
| Units: 3
| UG Reqs: GER:DB-Math
MATH 106:
Functions of a Complex Variable
Complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings. Prerequisite: 52.
Terms: Win, Sum
| Units: 3
| UG Reqs: GER:DB-Math
MATH 108:
Introduction to Combinatorics and Its Applications
Topics: graphs, trees (Cayley's Theorem, application to phylogony), eigenvalues, basic enumeration (permutations, Stirling and Bell numbers), recurrences, generating functions, basic asymptotics. Prerequisites: 51 or 103 or equivalent.
Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math
MATH 109:
Applied Group Theory
Applications of the theory of groups. Topics: elements of group theory, groups of symmetries, matrix groups, group actions, and applications to combinatorics and computing. Applications: rotational symmetry groups, the study of the Platonic solids, crystallographic groups and their applications in chemistry and physics. WIM.
Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 110:
Applied Number Theory and Field Theory
Number theory and its applications to modern cryptography. Topics: congruences, finite fields, primality testing and factorization, public key cryptography, error correcting codes, and elliptic curves, emphasizing algorithms. WIM.
Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 111:
Computational Commutative Algebra
Introduction to the theory of commutative rings, ideals, and modules. Systems of polynomial equations in several variables from the algorithmic viewpoint. Groebner bases, Buchberger's algorithm, elimination theory. Applications to algebraic geometry and to geometric problems.
| Units: 3
| UG Reqs: GER:DB-Math
MATH 113:
Linear Algebra and Matrix Theory
Algebraic properties of matrices and their interpretation in geometric terms. The relationship between the algebraic and geometric points of view and matters fundamental to the study and solution of linear equations. Topics: linear equations, vector spaces, linear dependence, bases and coordinate systems; linear transformations and matrices; similarity; eigenvectors and eigenvalues; diagonalization.
Terms: Win, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 113V:
Linear Algebra
Topics: matrices, linear equations, vector spaces, linear dependence, bases and coordinates, linear transformations, similarity, eigenvectors and eigenvalues, and diagonalization. Prerequisites: two quarters of single variable calculus, or consent of instructor.
Terms: Sum
| Units: 3
| UG Reqs: GER:DB-Math
MATH 115:
Functions of a Real Variable
The development of real analysis in Euclidean space: sequences and series, limits, continuous functions, derivatives, integrals. Basic point set topology. Honors math majors and students who intend to do graduate work in mathematics should take 171. Prerequisite: 51.
Terms: Aut, Spr, Sum
| Units: 3
| UG Reqs: GER:DB-Math
MATH 116:
Complex Analysis
Analytic functions, Cauchy integral formula, power series and Laurent series, calculus of residues and applications, conformal mapping, analytic continuation, introduction to Riemann surfaces, Fourier series and integrals. Prerequisites: 52, and 115 or 171.
Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math
MATH 120:
Modern Algebra
Groups acting on sets, examples of ¿nite groups, Sylow theorems, solvable and simple groups. Fields, rings, and ideals; polynomial rings over a field; PID and non-PID. Unique factorization domains. WIM.
Terms: Aut, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 121:
Modern Algebra II
Continuation of 120. Field of fractions, ¿eld theory and Galois theory. Modules over a PID, quotient modules, non-free modules. Canonical forms, generalized eigenspaces, quotients and duality for vector spaces. Prerequisite: Math 120.
Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math
MATH 122:
GROUP REPRESENTATIONS
Group representations and group rings, tensor algebra, character theory. Bilinear and quadratic forms, semisimplicity. Induced representations. Classi¿cation of representations, applications. Prerequisite: Math 120, and either Math 121 or permission of instructor.
Terms: Spr
| Units: 3
MATH 131P:
Partial Differential Equations I
An introduction to PDE; particularly suitable for non-Math majors. Topics include physical examples of PDE's, method of characteristics, D'Alembert's formula, maximum principles, heat kernel, Duhamel's principle, separation of variables, Fourier series, Harmonic functions, Bessel functions, spherical harmonics. Students who have taken MATH 171 should consider taking MATH 173 rather than 131p. Prerequisite: 53.
Terms: Aut, Win
| Units: 3
| UG Reqs: GER:DB-Math
MATH 132:
Partial Differential Equations II
Laplace's equation and properties of harmonic functions. Green's functions. Distributions and Fourier transforms. Eigenvalue problems and generalized Fourier series. Numerical solutions. Prerequisite: 131P.
Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math
MATH 136:
Stochastic Processes (STATS 219)
Introduction to measure theory, Lp spaces and Hilbert spaces. Random variables, expectation, conditional expectation, conditional distribution. Uniform integrability, almost sure and Lp convergence. Stochastic processes: definition, stationarity, sample path continuity. Examples: random walk, Markov chains, Gaussian processes, Poisson processes, Martingales. Construction and basic properties of Brownian motion. Prerequisite: STATS 116 or MATH 151 or equivalent. Recommended: MATH 115 or equivalent.
Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math
MATH 143:
Differential Geometry
Geometry of curves and surfaces in three-space and higher dimensional manifolds. Parallel transport, curvature, and geodesics. Surfaces with constant curvature. Minimal surfaces.
Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math
MATH 145:
Algebraic Geometry
Hilbert's nullstellensatz, complex affine and projective curves, Bezout's theorem, the degree/genus formula, blow-up, Riemann-Roch theorem. Prerequisites: 120, and 121 or knowledge of fraction fields. Recommended: familiarity with surfaces equivalent to 143, 146, 147, or 148.
Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math
MATH 146:
Analysis on Manifolds
Differentiable manifolds, tangent space, submanifolds, implicit function theorem, differential forms, vector and tensor fields. Frobenius' theorem, DeRham theory. Prerequisite: 52 or 52H.
Last offered: Autumn 2010
| Units: 3
| UG Reqs: GER:DB-Math
MATH 147:
Differential Topology
Smooth manifolds, transversality, Sards' theorem, embeddings, degree of a map, Borsuk-Ulam theorem, Hopf degree theorem, Jordan curve theorem. Prerequisite: 115 or 171.
Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math
MATH 148:
Algebraic Topology
Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.
Last offered: Spring 2011
| Units: 3
| UG Reqs: GER:DB-Math
MATH 149:
APPLIED ALGEBRAIC TOPOLOGY
Introduction to algebraic topology and its applications, in particular persistent homologynnas a tool for shape and pattern recognition from high dimensional data sets, with examplesnnanalyzed using state-of-the-art software. Prerequisite: linear algebra.
Terms: Spr
| Units: 3
MATH 151:
Introduction to Probability Theory
Counting; axioms of probability; conditioning and independence; expectation and variance; discrete and continuous random variables and distributions; joint distributions and dependence; central limit theorem and laws of large numbers. Prerequisite: 52 or consent of instructor.
Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math
MATH 152:
Elementary Theory of Numbers
Euclid's algorithm, fundamental theorems on divisibility; prime numbers; congruence of numbers; theorems of Fermat, Euler, Wilson; congruences of first and higher degrees; quadratic residues; introduction to the theory of binary quadratic forms; quadratic reciprocity; partitions.
Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math
MATH 154:
Algebraic Number Theory
Properties of number fields and Dedekind domains, quadratic and cyclotomic fields, applications to some classical Diophantine equations; introduction to elliptic curves. Prerequisites: 120 and 121, especially modules over principal ideal domains and Galois theory of finite fields.
Last offered: Spring 2011
| Units: 3
| UG Reqs: GER:DB-Math
MATH 155:
Analytic Number Theory
Topics in analytic number theory such as the distribution of prime numbers, the prime number theorem, twin primes and Goldbach's conjecture, the theory of quadratic forms, Dirichlet's class number formula, Dirichlet's theorem on primes in arithmetic progressions, and the fifteen theorem. Prerequisite: 152, or familiarity with the Euclidean algorithm, congruences, residue classes and reduced residue classes, primitive roots, and quadratic reciprocity.
Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math
MATH 159:
Discrete Probabilistic Methods
Modern discrete probabilistic methods suitable for analyzing discrete structures of the type arising in number theory, graph theory, combinatorics, computer science, information theory and molecular sequence analysis. Prerequisite: STATS 116/MATH 151 or equivalent.
Last offered: Winter 2011
| Units: 3
MATH 161:
Set Theory
Informal and axiomatic set theory: sets, relations, functions, and set-theoretical operations. The Zermelo-Fraenkel axiom system and the special role of the axiom of choice and its various equivalents. Well-orderings and ordinal numbers; transfinite induction and transfinite recursion. Equinumerosity and cardinal numbers; Cantor's Alephs and cardinal arithmetic. Open problems in set theory. Prerequisite: students should be comfortable doing proofs.
Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math
MATH 162:
Philosophy of Mathematics (PHIL 162, PHIL 262)
(Graduate students register for PHIL 262.) 20th-century approaches to the foundations and philosophy of mathematics. The background in mathematics, set theory, and logic. Schools and programs of logicism, predicativism, platonism, formalism, and constructivism. Readings from leading thinkers. Prerequisite: PHIL151 or consent of instructor.
Terms: Spr
| Units: 4
| UG Reqs: GER:DB-Math
MATH 171:
Fundamental Concepts of Analysis
Recommended for Mathematics majors and required of honors Mathematics majors. Similar to 115 but altered content and more theoretical orientation. Properties of Riemann integrals, continuous functions and convergence in metric spaces; compact metric spaces, basic point set topology. Prerequisites: 51 and 52, or 51H and 52H. WIM
Terms: Aut, Spr
| Units: 3
| UG Reqs: GER:DB-Math, WAY-FR
MATH 172:
Lebesgue Integration and Fourier Analysis
Similar to 205A, but for undergraduate Math majors and graduate students in other disciplines. Topics include Lebesgue measure on Euclidean space, Lebesgue integration, L^p spaces, the Fourier transform, the Hardy-Littlewood maximal function and Lebesgue differentiation. Prerequisite: 171 or consent of instructor.
Terms: Win
| Units: 3
| UG Reqs: GER:DB-Math
MATH 173:
Theory of Partial Differential Equations
A rigorous introduction to PDE accessible to advanced undergraduates. Elliptic, parabolic, and hyperbolic equations in many space dimensions including basic properties of solutions such as maximum principles, causality, and conservation laws. Methods include the Fourier transform as well as more classical methods. The Lebesgue integral will be used throughout, but a summary of its properties will be provided to make the course accessible to students who have not had 172 or 205A. Prerequisite: 171 or equivalent.
Terms: Win
| Units: 3
MATH 174A:
Topics in Analysis and Differential Equations with Applications
For students planning graduate work in mathematics or physics, and for honors math majors and other students at ease with rigorous proofs and qualitative discussion. Topics may include: geometric theory of ODE¿s with applications to dynamics; mathematical foundations of classical mechanics including variational principles, Lagrangian and Hamiltonian formalisms, theory of integrable systems; theorems of existence and uniqueness; Sturm-Liouville theory. Prerequisite: 53H or 171, or consent of instructor.
Last offered: Winter 2008
| Units: 3
| UG Reqs: GER:DB-Math
MATH 175:
Elementary Functional Analysis
Linear operators on Hilbert space. Spectral theory of compact operators; applications to integral equations. Elements of Banach space theory. Prerequisite: 115 or 171.
Terms: Spr
| Units: 3
| UG Reqs: GER:DB-Math
MATH 180:
Introduction to Financial Mathematics
Financial derivatives: contracts and options. Hedging and risk management. Arbitrage, interest rate, and discounted value. Geometric random walk and Brownian motion as models of risky assets. Initial boundary value problems for the heat and related partial differential equations. Self-financing replicating portfolio. Black-Scholes pricing of European options. Dividends. Implied volatility. Optimal stopping and American options. Prerequisite: 53. Corequisites: 131, 151 or STATS 116.
Terms: Aut
| Units: 3
| UG Reqs: GER:DB-Math
MATH 193:
Polya Problem Solving Seminar
Topics in mathematics and problem solving strategies with an eye towards the Putnam Competition. Topics may include parity, the pigeonhole principle, number theory, recurrence, generating functions, and probability. Students present solutions to the class. Open to anyone with an interest in mathematics.
Terms: Aut
| Units: 1
| Repeatable
5 times
(up to 5 units total)
MATH 196:
Undergraduate Colloquium
Weekly lectures by different experts on topics in pure and applied mathematics that go beyond the standard curriculum. May be repeated for credit for up to 3 units. Does not count toward the math major or minor.
Terms: Aut, Win, Spr
| Units: 1
| Repeatable
3 times
(up to 3 units total)
MATH 197:
Senior Honors Thesis
(Staff) May be repeated 3 times for a max of 9 units.
Terms: Aut, Win, Spr
| Units: 1-6
| Repeatable
3 times
(up to 9 units total)
Instructors: ;
Blumberg, A. (PI);
Brendle, S. (PI);
Brubaker, B. (PI);
Brumfiel, G. (PI);
Bump, D. (PI);
Candes, E. (PI);
Cantarero Lopez, J. (PI);
Carlsson, G. (PI);
Clingher, A. (PI);
Coffey, J. (PI);
Cohen, P. (PI);
Cohen, R. (PI);
Conrad, B. (PI);
Demanet, L. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Durrleman, V. (PI);
Elgart, A. (PI);
Eliashberg, Y. (PI);
Feferman, S. (PI);
Fetecau, R. (PI);
Finn, R. (PI);
Galatius, S. (PI);
Gromoll, H. (PI);
Guth, L. (PI);
Ionel, E. (PI);
Iyer, G. (PI);
Kahle, M. (PI);
Karlin, S. (PI);
Katznelson, Y. (PI);
Keller, J. (PI);
Kerckhoff, S. (PI);
Khuri, M. (PI);
Kiem, Y. (PI);
Levy, D. (PI);
Li, J. (PI);
Li, T. (PI);
Liu, T. (PI);
Lucianovic, M. (PI);
Mazzeo, R. (PI);
Meckes, M. (PI);
Milanov, T. (PI);
Milgram, R. (PI);
Mints, G. (PI);
Munson, B. (PI);
Nedelec, L. (PI);
Ng, L. (PI);
Oprea, D. (PI);
Ornstein, D. (PI);
Osserman, R. (PI);
Papanicolaou, G. (PI);
Papikian, M. (PI);
Pettet, A. (PI);
Ryzhik, L. (PI);
Schoen, R. (PI);
Shirokova, N. (PI);
Siefring, R. (PI);
Simon, L. (PI);
Soundararajan, K. (PI);
Storm, P. (PI);
Thiem, F. (PI);
Vakil, R. (PI);
Vasy, A. (PI);
Venkatesh, A. (PI);
White, B. (PI);
Wieczorek, W. (PI);
Young, M. (PI)
MATH 199:
Independent Work
Undergraduates pursue a reading program; topics limited to those not in regular department course offerings. Credit can fulfill the elective requirement for math majors. Approval of Undergraduate Affairs Committee is required to use credit for honors majors area requirement.
Terms: Aut, Win, Spr, Sum
| Units: 1-3
| Repeatable
3 times
(up to 9 units total)
Instructors: ;
Blumberg, A. (PI);
Brendle, S. (PI);
Brubaker, B. (PI);
Brumfiel, G. (PI);
Bump, D. (PI);
Camilier, I. (PI);
Carlsson, G. (PI);
Clingher, A. (PI);
Coffey, J. (PI);
Cohen, P. (PI);
Cohen, R. (PI);
Conrad, B. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Durrleman, V. (PI);
Elgart, A. (PI);
Eliashberg, Y. (PI);
Feferman, S. (PI);
Fetecau, R. (PI);
Finn, R. (PI);
Galatius, S. (PI);
Gromoll, H. (PI);
Guth, L. (PI);
Ionel, E. (PI);
Iyer, G. (PI);
Kahle, M. (PI);
Karlin, S. (PI);
Katznelson, Y. (PI);
Keller, J. (PI);
Kerckhoff, S. (PI);
Khuri, M. (PI);
Kiem, Y. (PI);
Levy, D. (PI);
Li, J. (PI);
Li, T. (PI);
Liu, T. (PI);
Lucianovic, M. (PI);
Mazzeo, R. (PI);
Meckes, M. (PI);
Milanov, T. (PI);
Milgram, R. (PI);
Munson, B. (PI);
Nedelec, L. (PI);
Ng, L. (PI);
Oprea, D. (PI);
Ornstein, D. (PI);
Osserman, R. (PI);
Papanicolaou, G. (PI);
Papikian, M. (PI);
Pettet, A. (PI);
Ryzhik, L. (PI);
Schoen, R. (PI);
Shirokova, N. (PI);
Siefring, R. (PI);
Simon, L. (PI);
Soundararajan, K. (PI);
Storm, P. (PI);
Thiem, F. (PI);
Toussaint, A. (PI);
Vakil, R. (PI);
Vasy, A. (PI);
Venkatesh, A. (PI);
White, B. (PI);
Wieczorek, W. (PI);
Young, M. (PI)
MATH 205A:
Real Analysis
Basic measure theory and the theory of Lebesgue integration. Prerequisite: 171 or equivalent.
Terms: Aut
| Units: 3
MATH 205B:
Real Analysis
Point set topology, basic functional analysis, Fourier series, and Fourier transform. Prerequisites: 171 and 205A or equivalent.
Terms: Win
| Units: 3
MATH 210A:
Modern Algebra
Basic commutative ring and module theory, tensor algebra, homological constructions, linear and multilinear algebra, introduction to representation theory. Prerequisite: 121 or equivalent.
Terms: Aut
| Units: 3
MATH 210B:
Modern Algebra
Continuation of 210A. Topics in group theory, Galois theory, commutative algebra, introductions to algebraic number theory and algebraic geometry.
Terms: Win
| Units: 3
MATH 210C:
Modern Algebra
Continuation of 210B. Semisimple rings and modules, representation theory, Lie algebras, Lie groups and their representations.
Terms: Spr
| Units: 3
MATH 215A:
Complex Analysis, Geometry, and Topology
Analytic functions, complex integration, Cauchy's theorem, residue theorem, argument principle, conformal mappings, Riemann mapping theorem, Picard's theorem, elliptic functions, analytic continuation and Riemann surfaces.
Terms: Aut
| Units: 3
MATH 215B:
Complex Analysis, Geometry, and Topology
Topics: fundamental group and covering spaces, homology, cohomology, products, basic homotopy theory, and applications. Prerequisites: 113, 120, and 171, or equivalent; 215A is not a prerequisite for 215B.
Terms: Win
| Units: 3
MATH 215C:
Complex Analysis, Geometry, and Topology
Differentiable manifolds, transversality, degree of a mapping, vector fields, intersection theory, and Poincare duality. Differential forms and the DeRham theorem. Prerequisite: 215B or equivalent.
Terms: Spr
| Units: 3
MATH 216A:
Introduction to Algebraic Geometry
Algebraic curves, algebraic varieties, sheaves, cohomology, Riemann-Roch theorem. Classification of algebraic surfaces, moduli spaces, deformation theory and obstruction theory, the notion of schemes. May be repeated for credit. Prerequisites: 210ABC or equivalent.
Terms: Aut
| Units: 3
| Repeatable
for credit
MATH 216B:
Introduction to Algebraic Geometry
Continuation of 216A. May be repeated for credit.
Terms: Win
| Units: 3
| Repeatable
for credit
MATH 216C:
Introduction to Algebraic Geometry
Continuation of 216B. May be repeated for credit.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 217A:
Differential Geometry
Smooth manifolds and submanifolds, tensors and forms, Lie and exterior derivative, DeRham cohomology, distributions and the Frobenius theorem, vector bundles, connection theory, parallel transport and curvature, affine connections, geodesics and the exponential map, connections on the principal frame bundle. Prerequisite: 215C or equivalent.
Terms: Spr
| Units: 3
MATH 220:
Partial Differential Equations of Applied Mathematics (CME 303)
First-order partial differential equations; method of characteristics; weak solutions; elliptic, parabolic, and hyperbolic equations; Fourier transform; Fourier series; and eigenvalue problems. Prerequisite: foundation in multivariable calculus and ordinary differential equations.
Terms: Aut
| Units: 3
MATH 227:
Partial Differential Equations and Diffusion Processes
Parabolic and elliptic partial differential equations and their relation to diffusion processes. First order equations and optimal control. Emphasis is on applications to mathematical finance. Prerequisites: MATH 131 and MATH 136/STATS 219, or equivalents.
Terms: Win
| Units: 3
MATH 228:
Stochastic Methods in Engineering (CME 308)
Review of basic probability; Monte Carlo simulation; state space models and time series; parameter estimation, prediction, and filtering; Markov chains and processes; stochastic control; and stochastic differential equations. Examples from various engineering disciplines. Prerequisites: exposure to probability; background in real variables and analysis.
Terms: Spr
| Units: 3
MATH 230A:
Theory of Probability (STATS 310A)
Mathematical tools: sigma algebras, measure theory, connections between coin tossing and Lebesgue measure, basic convergence theorems. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Large deviations. Weak convergence; central limit theorems; Poisson convergence; Stein's method. Prerequisites: 116, MATH 171.
Terms: Aut
| Units: 2-4
MATH 230B:
Theory of Probability (STATS 310B)
Conditional expectations, discrete time martingales, stopping times, uniform integrability, applications to 0-1 laws, Radon-Nikodym Theorem, ruin problems, etc. Other topics as time allows selected from (i) local limit theorems, (ii) renewal theory, (iii) discrete time Markov chains, (iv) random walk theory,nn(v) ergodic theory. Prerequisite: 310A or MATH 230A.
Terms: Win
| Units: 2-3
MATH 230C:
Theory of Probability (STATS 310C)
Continuous time stochastic processes: martingales, Brownian motion, stationary independent increments, Markov jump processes and Gaussian processes. Invariance principle, random walks, LIL and functional CLT. Markov and strong Markov property. Infinitely divisible laws. Some ergodic theory. Prerequisite: 310B or MATH 230B.
Terms: Spr
| Units: 2-4
MATH 231B:
The Spectrum of Large Random Matrices
Asymptotics of eigenvalues of large random matrices, focusing on Wigner matrices and the Gaussian unitary ensemble: the combinatorics of non-crossing partitions and word graphs, concentration inequalities, Cauchy-Stieltjes transform, Hermite polynomials, Fredholm determinants, Laplace asymptotic method, special functions (Airy, Painleve), and stochastic calculus. Prerequisities: STATS 310A or MATH 205A.
Terms: Spr
| Units: 3
| Repeatable
1 times
(up to 3 units total)
MATH 232:
Topics in Probability: Malliavin Calculus, Fractional Brownian Motion and Applications
Malliavin calculus: derivative and divergence operators, Skorohod integral. Fractional Brownian motion: relavance for financial mathematics, Ito and Tanaka formula, driving force for the heat equation. Ito formula for irregular Gaussian processes and other applications of Malliavin calculus. May be repeated for credit. Prerequisites: MATH 236, STATS 310C or equivalent.
Last offered: Winter 2009
| Units: 3
| Repeatable
for credit
MATH 236:
Introduction to Stochastic Differential Equations
Brownian motion, stochastic integrals, and diffusions as solutions of stochastic differential equations. Functionals of diffusions and their connection with partial differential equations. Random walk approximation of diffusions. Prerequisite: 136 or equivalent and differential equations.
Terms: Win
| Units: 3
MATH 237:
Default and Systemic Risk
Introduction to mathematical models of complex static and dynamic stochastic systems that undergo sudden regime change in response to small changes in parameters. Examples from materials science (phase transitions), power grid models, financial and banking systems. Special emphasis on mean field models and their large deviations, including computational issues of uncertainty quantification.
Terms: Spr
| Units: 3
MATH 238:
Mathematical Finance (STATS 250)
Stochastic models of financial markets. Forward and futures contracts. European options and equivalent martingale measures. Hedging strategies and management of risk. Term structure models and interest rate derivatives. Optimal stopping and American options. Corequisites: MATH 236 and 227 or equivalent.
Terms: Win
| Units: 3
MATH 239:
Computation and Simulation in Finance
Monte Carlo, finite difference, tree, and transform methods for the numerical solution of partial differential equations in finance. Emphasis is on derivative security pricing. Prerequisite: 238 or equivalent.
Terms: Spr
| Units: 3
MATH 240:
Topics in Financial Mathematics: Fixed Income Models
Introduction to continuous time models for arbitrage-free pricing of interest rate derivatives. Bonds, yields, and the construction of yield curves. Caps, floors, swaps, swaptions, and bond options. Short rate models. Yield curve models. Forward measures. Forward and futures. LIBOR and swap market models. Prerequisite: MATH 238.
Terms: Spr
| Units: 3
MATH 248:
Ergodic Theory and Szemeredi's Theorem
An introduction to ergodic theory leading to (and proving) Szemeredi's theorem and its multidimensional extension. Prerequisite: 205a and some knowledge of Hilbert spaces.
Terms: Aut, Win
| Units: 3
| Repeatable
for credit
MATH 249A:
Topics in number theory
Terms: Aut
| Units: 3
| Repeatable
3 times
(up to 9 units total)
MATH 249B:
Topics in Number Theory
Terms: Win
| Units: 3
| Repeatable
3 times
(up to 9 units total)
MATH 249C:
Topics in Number Theory
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 256A:
Partial Differential Equations
The theory of linear and nonlinear partial differential equations, beginning with linear theory involving use of Fourier transform and Sobolev spaces. Topics: Schauder and L2 estimates for elliptic and parabolic equations; De Giorgi-Nash-Moser theory for elliptic equations; nonlinear equations such as the minimal surface equation, geometric flow problems, and nonlinear hyperbolic equations.
Terms: Spr
| Units: 3
MATH 256B:
Partial Differential Equations
Continuation of 256A.
Terms: Win
| Units: 3
| Repeatable
2 times
MATH 258:
Topics in Geometric Analysis
May be repeated for credit.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 259:
mirror symmetry
Terms: Aut
| Units: 3
| Repeatable
3 times
(up to 9 units total)
MATH 263A:
Lie Groups and Lie Algebras
Definitions, examples, properties. Semi-simple Lie algebras, their structure and classification. Cartan decomposition: real Lie algebras. Representation theory: Cartan-Stiefel diagram, weights. Weyl character formula. Orthogonal and symplectic representations. May be repeated for credit. Prerequisite: 210 or equivalent.
Last offered: Winter 2011
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 263B:
Lie Groups and Lie Algebras
Continuation of 263A. May be repeated for credit.
Terms: Win
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 271:
The H-Principle
The language of jets. Thom transversality theorem. Holonomic approximation theorem. Applications: immersion theory and its generaliazations. Differential relations and Gromov's h-principle for open manifolds. Applications to symplectic geometry. Microflexibility. Mappings with simple singularities and their applications. Method of convex integration. Nash-Kuiper C^1-isometric embedding theorem.
Terms: Win
| Units: 3
MATH 280:
Evolution Equations in Differential Geometry
Terms: Aut
| Units: 3
| Repeatable
for credit
MATH 282A:
Low Dimensional Topology
The theory of surfaces and 3-manifolds. Curves on surfaces, the classification of diffeomorphisms of surfaces, and Teichmuller space. The mapping class group and the braid group. Knot theory, including knot invariants. Decomposition of 3-manifolds: triangulations, Heegaard splittings, Dehn surgery. Loop theorem, sphere theorem, incompressible surfaces. Geometric structures, particularly hyperbolic structures on surfaces and 3-manifolds.
Terms: Aut
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 282B:
Homotopy Theory
Homotopy groups, fibrations, spectral sequences, simplicial methods, Dold-Thom theorem, models for loop spaces, homotopy limits and colimits, stable homotopy theory.
Terms: Win
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 282C:
Fiber Bundles and Cobordism
Possible topics: principal bundles, vector bundles, classifying spaces. Connections on bundles, curvature. Topology of gauge groups and gauge equivalence classes of connections. Characteristic classes and K-theory, including Bott periodicity, algebraic K-theory, and indices of elliptic operators. Spectral sequences of Atiyah-Hirzebruch, Serre, and Adams. Cobordism theory, Pontryagin-Thom theorem, calculation of unoriented and complex cobordism. May be repeated for credit.
Terms: Spr
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 283:
Topics in Algebraic and Geometric Topology
May be repeated for credit.
Terms: Aut, Win, Spr
| Units: 3
| Repeatable
for credit
MATH 286:
Topics in Differential Geometry
May be repeated for credit.
Terms: Win
| Units: 3
| Repeatable
for credit
MATH 293A:
Proof Theory (PHIL 353A)
Gentzen's natural deduction and sequential calculi for first-order propositional and predicate logics. Normalization and cut-elimination procedures. Relationships with computational lambda calculi and automated deduction. Prerequisites: 151, 152, and 161, or equivalents.
Terms: Aut
| Units: 3
MATH 355:
Graduate Teaching Seminar
Required of and limited to first-year Mathematics graduate students.
Terms: Spr
| Units: 1
MATH 360:
Advanced Reading and Research
(Staff)
Terms: Aut, Win, Spr, Sum
| Units: 1-9
| Repeatable
for credit
Instructors: ;
Brendle, S. (PI);
Brumfiel, G. (PI);
Bump, D. (PI);
Candes, E. (PI);
Carlsson, G. (PI);
Cohen, R. (PI);
Conrad, B. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Durrleman, V. (PI);
Eliashberg, Y. (PI);
Feferman, S. (PI);
Fetecau, R. (PI);
Finn, R. (PI);
Galatius, S. (PI);
Ionel, E. (PI);
Iyer, G. (PI);
Kahle, M. (PI);
Katznelson, Y. (PI);
Kerckhoff, S. (PI);
Levy, D. (PI);
Li, J. (PI);
Li, T. (PI);
Liu, T. (PI);
Mazzeo, R. (PI);
Milgram, R. (PI);
Mints, G. (PI);
Mirzakhani, M. (PI);
Munson, B. (PI);
Ornstein, D. (PI);
Papanicolaou, G. (PI);
Papikian, M. (PI);
Rubin, K. (PI);
Rubinstein, Y. (PI);
Ryzhik, L. (PI);
Schoen, R. (PI);
Simon, L. (PI);
Soundararajan, K. (PI);
Storm, P. (PI);
Thiem, F. (PI);
Vakil, R. (PI);
Vasy, A. (PI);
Venkatesh, A. (PI);
White, B. (PI)
MATH 361:
Research Seminar Participation
Participation in a faculty-led seminar which has no specific course number.nn (Staff)
Terms: Aut, Win, Spr, Sum
| Units: 1-3
| Repeatable
for credit
Instructors: ;
Brendle, S. (PI);
Brumfiel, G. (PI);
Bump, D. (PI);
Candes, E. (PI);
Carlsson, G. (PI);
Cohen, R. (PI);
Conrad, B. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Durrleman, V. (PI);
Eliashberg, Y. (PI);
Galatius, S. (PI);
Ionel, E. (PI);
Kahle, M. (PI);
Kerckhoff, S. (PI);
Levy, D. (PI);
Li, J. (PI);
Liu, T. (PI);
Mazzeo, R. (PI);
Mirzakhani, M. (PI);
Papanicolaou, G. (PI);
Rubinstein, Y. (PI);
Ryzhik, L. (PI);
Schoen, R. (PI);
Simon, L. (PI);
Soundararajan, K. (PI);
Vakil, R. (PI);
Vasy, A. (PI);
Venkatesh, A. (PI);
White, B. (PI)
MATH 391:
Research Seminar in Logic and the Foundations of Mathematics (PHIL 391)
Contemporary work. May be repeated a total of three times for credit.
Terms: Aut, Win, Spr
| Units: 1-3
| Repeatable
3 times
(up to 9 units total)
MATH 394:
classics in analysis
Original papers in analysis.
Terms: Win
| Units: 3
| Repeatable
3 times
(up to 9 units total)
MATH 395:
Classics in Geometry and Topology
Original papers in geometry and in algebraic and geometric topology. May be repeated for credit.
Terms: Spr
| Units: 3
| Repeatable
for credit
MATH 78SI:
Speedcubing: HIstory, Theory, and Practice
History of the Rubik's cube; the current cubing community; basic mathematical theory; concepts to improve speed solving skill. Prior ability to solve cube not required.
| Units: 1
MATH 802:
TGR Dissertation
Terms: Aut, Win, Spr, Sum
| Units: 0
| Repeatable
for credit
Instructors: ;
Brendle, S. (PI);
Brumfiel, G. (PI);
Bump, D. (PI);
Candes, E. (PI);
Carlsson, G. (PI);
Cohen, R. (PI);
Conrad, B. (PI);
Dembo, A. (PI);
Diaconis, P. (PI);
Eliashberg, Y. (PI);
Galatius, S. (PI);
Ionel, E. (PI);
Kerckhoff, S. (PI);
Levy, D. (PI);
Li, J. (PI);
Liu, T. (PI);
Mazzeo, R. (PI);
Mirzakhani, M. (PI);
Papanicolaou, G. (PI);
Ryzhik, L. (PI);
Schoen, R. (PI);
Simon, L. (PI);
Soundararajan, K. (PI);
Vakil, R. (PI);
Vasy, A. (PI);
Venkatesh, A. (PI);
White, B. (PI)
MATH 88Q:
The Mathematics of the Rubik's Cube
Preference to sophomores. Group theory through topics that can be illustrated with the Rubik's cube: subgroups, homomorphisms and quotient groups, the symmetric and alternating groups, conjugation, commutators, and Sylow subgroups.
| Units: 3
MATH 100:
Mathematics for Elementary School Teachers
Mathematics and pedagogical strategies. Core mathematical content in grades K-6, classroom presentation, how to handle student errors, and mathematical issues that come up during instruction.
| Units: 4
MATH 118:
Mathematics of Computation
Notions of analysis and algorithms central to modern scientific computing: continuous and discrete Fourier expansions, the fast Fourier transform, orthogonal polynomials, interpolation, quadrature, numerical differentiation, analysis and discretization of initial-value and boundary-value ODE, finite and spectral elements. Prerequisites: MATH 51 and 53.
| Units: 3
| UG Reqs: GER:DB-Math
MATH 137:
Mathematical Methods of Classical Mechanics
Newtonian mechanics. Lagrangian formalism. E. Noether's theorem. Oscillations. Rigid bodies. Introduction to symplectic geometry. Hamiltonian formalism. Legendre transform. Variational principles. Geometric optics. Introduction to the theory of integrable systems. Prerequisites: 51, 52, 53, or 51H, 52H, 53H.
| Units: 3
| UG Reqs: GER:DB-Math
MATH 138:
Celestial Mechanics
Mathematically rigorous introduction to the classical N-body problem: the motion of N particles evolving according to Newton's law. Topics include: the Kepler problem and its symmetries; other central force problems; conservation theorems; variational methods; Hamilton-Jacobi theory; the role of equilibrium points and stability; and symplectic methods. Prerequisites: 53, and 115 or 171.
| Units: 3
| UG Reqs: GER:DB-Math
MATH 174:
Calculus of Variations
An introductory course emphasizing the historical development of the theory, its connections to physics and mechanics, its independent mathematical interest, and its contacts with daily life experience. Applications to minimal surfaces and to capillary surface interfaces. Prerequisites: Math 171 or equivalent.
| Units: 3
MATH 174B:
Honors Analysis
Continuation of 174A. Topics may include: introduction to PDEs including transport equations, Laplace, wave, and heat equations; techniques of solution including separation of variables and Green¿s functions; Fourier series and integrals; introduction to the theory of distributions; mathematical foundations of quantum mechanics. Prerequisite: 174A.
| Units: 3
| UG Reqs: GER:DB-Math
MATH 198:
Category Theory and Functional Programming
An introduction to category theory and its applications in computer science: in particular to Haskell, a programming language with many language elements inspired by category theory. Students should have some familiarity with linear algebra, discrete mathematics and functional programming. Recommended corequisite: CS 242.
| Units: 1
MATH 221:
Mathematical Methods of Imaging
Mathematical methods of imaging: array imaging using Kirchhoff migration and beamforming, resolution theory for broad and narrow band array imaging in homogeneous media, topics in high-frequency, variable background imaging with velocity estimation, interferometric imaging methods, the role of noise and inhomogeneities, and variational problems that arise in optimizing the performance of imaging algorithms and the deblurring of images. Prerequisite: 220.
| Units: 3
MATH 222:
Computational Methods for Fronts, Interfaces, and Waves
High-order methods for multidimensional systems of conservation laws and Hamilton-Jacobi equations (central schemes, discontinuous Galerkin methods, relaxation methods). Level set methods and fast marching methods. Computation of multi-valued solutions. Multi-scale analysis, including wavelet-based methods. Boundary schemes (perfectly matched layers). Examples from (but not limited to) geometrical optics, transport equations, reaction-diffusion equations, imaging, and signal processing.
| Units: 3
MATH 224:
Topics in Mathematical Biology
Mathematical models for biological processes based on ordinary and partial differential equations. Topics: population and infectious diseases dynamics, biological oscillators, reaction diffusion models, biological waves, and pattern formation. Prerequisites: 53 and 131, or equivalents.
| Units: 3
MATH 231A:
An Introduction to Random Matrix Theory (STATS 351A)
Patterns in the eigenvalue distribution of typical large matrices, which also show up in physics (energy distribution in scattering experiments), combinatorics (length of longest increasing subsequence), first passage percolation and number theory (zeros of the zeta function). Classical compact ensembles (random orthogonal matrices). The tools of determinental point processes.
| Units: 3
MATH 231C:
Free Probability
Background from operator theory, addition and multiplication theorems for operators, spectral properties of infinite-dimensional operators, the free additive and multiplicative convolutions of probability measures and their classical counterparts, asymptotic freeness of large random matrices, and free entropy and free dimension. Prerequisite: STATS 310B or equivalent.
| Units: 3
MATH 233:
Probabilistic Methods in Analysis
Proofs and constructions in analysis obtained from basic results in Probability Theory and a 'probabilistic way of thinking.' Topics: Rademacher functions, Gaussian processes, entropy.
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 234:
Large Deviations
Combinatorial estimates and the method of types. Large deviation probabilities for partial sums and for empirical distributions, Cramer's and Sanov's theorems and their Markov extensions. Applications in statistics, information theory, and statistical mechanics. Prerequisite: MATH 230A or STATS 310.
| Units: 3
MATH 243:
Functions of Several Complex Variables
| Units: 3
| Repeatable
for credit
MATH 244:
Riemann Surfaces
Compact Riemann surfaces and algebraic curves; cohomology of sheaves; Serre duality; Riemann-Roch theorem and application; Jacobians; Abel's theorem. May be repeated for credit.
| Units: 3
| Repeatable
for credit
MATH 245A:
Topics in Algebraic Geometry: Moduli Theory
Topics in the study of moduli spaces: Grothendieck's Quot scheme, Hilbert scheme, boundedness, semistable reduction, fine and coarse moduli space, geometric invariant theory, deformation theory, and universal families. Examples of moduli spaces including moduli of curves, moduli of vector bundles and moduli of maps. May be repeated for credit.
| Units: 3
| Repeatable
3 times
(up to 9 units total)
MATH 245B:
Topics in Algebraic Geometry: Intersection Theory
Topics such as intersection theory on surfaces, toric varieties, and homogeneous spaces; numerical criteria for positivity; Chow groups and rings. May be repeated for credit.
| Units: 3
| Repeatable
3 times
(up to 9 units total)
MATH 245C:
Topics in Algebraic Geometry: Alterations
| Units: 3
| Repeatable
for credit
MATH 247:
Topics in Group Theory
Topics include the Burnside basis theorem, classification of p-groups, regular and powerful groups, Sylow theorems, the Frattini argument, nilpotent groups, solvable groups, theorems of P. Hall, group cohomology, and the Schur-Zassenhaus theorem. The classical groups and introduction to the classification of finite simple groups and its applications. May be repeated for credit.
| Units: 3
| Repeatable
for credit
MATH 248A:
Algebraic Number Theory
Structure theory and Galois theory of local and global fields, finiteness theorems for class numbers and units, adelic techniques. Prerequisites: MATH 210A,B.
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 250:
Elliptic Curves in Cryptography (CS 259C)
Discusses the mathematics of elliptic curves and their applications in cryptography. Studies crypto-systems based on elliptic curves and discuss their security. Studies algorithmic aspects of elliptic curves related to crypto-system construction and security. Topics include: elliptic curves over finite fields, attacks on elliptic curve crypto-systems, point counting, pairing-based cryptography. Suggested background: Math 120 or 152, or CS255.
| Units: 3
MATH 252:
Algebraic Groups
Smooth affine groups over general fields, quotients, tori, solvable groups, reductive groups, root systems, Existence and Isomorphism theorem, structure theory. If time permits, classification theory over interesting fields. Prerequisites: 210A, 210B, and familiarity with algebraic varieties over general fields.
| Units: 3
| Repeatable
for credit
MATH 254:
Geometric Methods in the Theory of Ordinary Differential Equations
Topics may include: structural stability and perturbation theory of dynamical systems; hyperbolic theory; first order PDE; normal forms, bifurcation theory; Hamiltonian systems, their geometry and applications. May be repeated for credit.
| Units: 3
| Repeatable
for credit
MATH 257A:
Symplectic Geometry and Topology
Linear symplectic geometry and linear Hamiltonian systems. Symplectic manifolds and their Lagrangian submanifolds, local properties. Symplectic geometry and mechanics. Contact geometry and contact manifolds. Relations between symplectic and contact manifolds. Hamiltonian systems with symmetries. Momentum map and its properties. May be repeated for credit.
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 257B:
Symplectic Geometry and Topology
Continuation of 257A. May be repeated for credit.
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 261A:
Functional Analysis
Geometry of linear topological spaces. Linear operators and functionals. Spectral theory. Calculus for vector-valued functions. Operational calculus. Banach algebras. Special topics in functional analysis. May be repeated for credit.
| Units: 3
| Repeatable
2 times
(up to 6 units total)
MATH 264:
Infinite Dimensional Lie Algebra
| Units: 3
| Repeatable
for credit
MATH 266:
Computational Signal Processing and Wavelets
Theoretical and computational aspects of signal processing. Topics: time-frequency transforms; wavelet bases and wavelet packets; linear and nonlinear multiresolution approximations; estimation and restoration of signals; signal compression. May be repeated for credit.
| Units: 3
MATH 269:
Topics in symplectic geometry
May be repeated for credit.
| Units: 3
| Repeatable
for credit
MATH 270:
Geometry and Topology of Complex Manifolds
Complex manifolds, Kahler manifolds, curvature, Hodge theory, Lefschetz theorem, Kahler-Einstein equation, Hermitian-Einstein equations, deformation of complex structures. May be repeated for credit.
| Units: 3
| Repeatable
for credit
MATH 272:
Topics in Partial Differential Equations
| Units: 3
| Repeatable
for credit
MATH 283A:
Topics in Topology
| Units: 3
MATH 284:
Topics in Geometric Topology: Heegaard splittings and three-manifolds
Incompressible surfaces, irreducible manifolds, prime decomposition, Morse theory, Heegaard diagrams, Heegaard splittings, the Thurston norm, sutured manifold theory, Heegaard Floer homology, sutured Floer homology.
| Units: 3
| Repeatable
for credit
MATH 284A:
Geometry and Topology in Dimension 3
The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.
| Units: 3
| Repeatable
for credit
MATH 284B:
Geometry and Topology in Dimension 3
The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.
| Units: 3
| Repeatable
for credit
MATH 287:
Introduction to optimal transportation
This will be an introductory course on Optimal Transportation theory. We will study Monge's problem, Kantorovich's problem, c-concave functions (also in the Riemannian setting), Wasserstein distance and geodesics (including a PDE formulation), applications to inequalities in convex analysis, as well as other topics, time permitting.
| Units: 3
MATH 290B:
Model Theory B (PHIL 350B)
Decidable theories. Model-theoretic background. Arithmetic of addition, real closed and algebraically closed fields, weak second order arithmetic, theories of terms, theories of arrays, temporal logic. Combining decision procedures. May be repeated for credit. Prerequisite: 151,152 or equivalents.
| Units: 1-3
| Repeatable
for credit
MATH 292A:
Set Theory (PHIL 352A)
The basics of axiomatic set theory; the systems of Zermelo-Fraenkel and Bernays-Gödel. Topics: cardinal and ordinal numbers, the cumulative hierarchy and the role of the axiom of choice. Models of set theory, including the constructible sets and models constructed by the method of forcing. Consistency and independence results for the axiom of choice, the continuum hypothesis, and other unsettled mathematical and set-theoretical problems. Prerequisites: PHIL151 and MATH 161, or equivalents.
| Units: 3
MATH 295:
Computation and Algorithms in Mathematics
Use of computer and algorithmic techniques in various areas of mathematics. Computational experiments. Topics may include polynomial manipulation, Groebner bases, computational geometry, and randomness. May be repeated for credit.
| Units: 3
| Repeatable
for credit
MATH 301:
Advanced Topics in Convex Optimization
Modern developments in convex optimization: semidefinite programming; novel and efficient first-order algorithms for smooth and nonsmooth convex optimization. Emphasis on numerical methods suitable for large scale problems arising in science and engineering. Prerequisites: convex optimization (EE 364), linear algebra (Math 104), numerical linear algebra (CME 302); background in probability, statistics, real analysis and numerical optimization.
| Units: 3
| Repeatable
for credit
MATH 381:
Seminar in Analysis
| Units: 1-3
| Repeatable
3 times
MATH 384:
Seminar in Geometry
| Units: 1
| Repeatable
3 times
(up to 9 units total)
MATH 385:
Seminar in Topology
| Units: 1-3
| Repeatable
3 times
(up to 9 units total)
MATH 388:
Seminar in Probability and Stochastic Processes
| Units: 1-3
| Repeatable
3 times
(up to 9 units total)
MATH 389:
Seminar in Mathematical Biology
| Units: 1-3
| Repeatable
3 times
(up to 9 units total)
MATH 396:
Graduate Progress
Results and current research of graduate and postdoctoral students. May be repeated for credit.
| Units: 1
| Repeatable
for credit
