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MATH 19: Calculus

Introduction to differential calculus of functions of one variable. Review of elementary functions (including exponentials and logarithms), limits, rates of change, the derivative and its properties, applications of the derivative. Prerequisites: periodic trigonometric functions, advanced algebra, and analysis of elementary functions (including exponentials and logarithms). You must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR

MATH 19ACE: Calculus, ACE

Additional problem solving session for Math 19 guided by a course assistant. Concurrent enrollment in Math 19 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Terms: Aut, Win | Units: 1

MATH 20: Calculus

The definite integral, Riemann sums, antiderivatives, the Fundamental Theorem of Calculus. Integration by substitution and by parts. Area between curves, and volume by slices, washers, and shells. Initial-value problems, exponential and logistic models, direction fields, and parametric curves. Prerequisite: Math 19 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr | Units: 3 | UG Reqs: GER:DB-Math, WAY-FR

MATH 20ACE: Calculus, ACE

Additional problem solving session for Math 20 guided by a course assistant. Concurrent enrollment in Math 20 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Terms: Aut, Win, Spr | Units: 1

MATH 21: Calculus

This course addresses a variety of topics centered around the theme of "calculus with infinite processes", largely the content of BC-level AP Calculus that isn't in the AB-level syllabus. It is needed throughout probability and statistics at all levels, as well as to understand approximation procedures that arise in all quantitative fields (including economics and computer graphics). After an initial review of limit rules, the course goes on to discuss sequences of numbers and of functions, as well as limits "at infinity" for each (needed for any sensible discussion of long-term behavior of a numerical process, such as: iterative procedures and complexity in computer science, dynamic models throughout economics, and repeated trials with data in any field). Integration is discussed for rational functions (a loose end from Math 20) and especially (improper) integrals for unbounded functions and "to infinity": this shows up in contexts as diverse as escape velocity for a rocket, the present value of a perpetual yield asset, and important calculations in probability (including the famous "bell curve" and to understand why many statistical tests work as they do). The course then turns to infinite series (how to "sum" an infinite collection of numbers), some useful convergence and divergence rests for these, and the associated killer app: power series and their properties, as well as Taylor approximations, all of which provide the framework that underlies virtually all mathematical models used in any quantitative field. Prerequisite: Math 20 or equivalent. If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr, Sum | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 21ACE: Calculus, ACE

Additional problem solving session for Math 21 guided by a course assistant. Concurrent enrollment in Math 21 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Terms: Aut, Win, Spr | Units: 1

MATH 51: Linear Algebra, Multivariable Calculus, and Modern Applications

This course provides unified coverage of linear algebra and multivariable differential calculus, and the free course e-text connects the material to many fields. Linear algebra in large dimensions underlies the scientific, data-driven, and computational tasks of the 21st century. The linear algebra portion includes orthogonality, linear independence, matrix algebra, and eigenvalues with applications such as least squares, linear regression, and Markov chains (relevant to population dynamics, molecular chemistry, and PageRank); the singular value decomposition (essential in image compression, topic modeling, and data-intensive work in many fields) is introduced in the final chapter of the e-text. The multivariable calculus portion includes unconstrained optimization via gradients and Hessians (used for energy minimization), constrained optimization (via Lagrange multipliers, crucial in economics), gradient descent and the multivariable Chain Rule (which underlie many machine learning algorithms, such as backpropagation), and Newton's method (an ingredient in GPS and robotics). The course emphasizes computations alongside an intuitive understanding of key ideas. The widespread use of computers makes it important for users of math to understand concepts: novel users of quantitative tools in the future will be those who understand ideas and how they fit with examples and applications. This is the only course at Stanford whose syllabus includes nearly all the math background for CS 229, which is why CS 229 and CS 230 specifically recommend it (or other courses resting on it). For frequently asked questions about the differences between Math 51 and CME 100, see the FAQ on the placement page on the Math Department website. Prerequisite: Math 21 or equivalent (e.g. 5 on the AP Calculus BC test or suitable score on certain international exams: https://studentservices.stanford.edu/my-academics/earn-my-degree/undergraduate-degree-progress/test-transfer-credit/external-test-2). If you have not previously taken a calculus course at Stanford then you must have taken the math placement diagnostic (offered through the Math Department website: https://mathematics.stanford.edu/academics/math-placement) in order to register for this course.
Terms: Aut, Win, Spr, Sum | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 51ACE: Linear Algebra, Multivariable Calculus, and Modern Applications, ACE

Additional problem solving session for Math 51 guided by a course assistant. Concurrent enrollment in Math 51 required. Application required: https://engineering.stanford.edu/students-academics/equity-and-inclusion-initiatives/undergraduate-programs/additional-calculus
Terms: Aut, Win, Spr | 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: Math 21 and Math 51 or equivalents.
Terms: Win, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 53: Differential Equations with Linear Algebra, Fourier Methods, and Modern Applications

Ordinary differential equations and initial value problems, linear systems of such equations with an emphasis on second-order constant-coefficient equations, stability analysis for non-linear systems (including phase portraits and the role of eigenvalues), and numerical methods. Partial differential equations and boundary-value problems, Fourier series and initial conditions, and Fourier transform for non-periodic phenomena. Throughout the development we harness insights from linear algebra, and software widgets are used to explore course topics on a computer (no coding background is needed). The free e-text provides motivation from applications across a wide array of fields (biology, chemistry, computer science, economics, engineering, and physics) described in a manner not requiring any area-specific expertise, and it has an appendix on Laplace transforms with many worked examples as a complement to the Fourier transform in the main text. Prerequisite: Math 21 and Math 51, or equivalents.
Terms: Aut, Win, Spr | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 56: Proofs and Modern Mathematics

How do mathematicians think? Why are the mathematical facts learned in school true? In this course students will explore higher-level mathematical thinking and will gain familiarity with a crucial aspect of mathematics: achieving certainty via mathematical proofs, a creative activity of figuring out what should be true and why. This course is ideal for students who would like to learn about the reasoning underlying mathematical results, but at a pace and level of abstraction not as intense as Math 61CM/DM, as a consequence benefiting from additional opportunity to explore the reasoning. Familiarity with one-variable calculus is strongly recommended at least at the AB level of AP Calculus since a significant part of the course develops some of the main results in that material systematically from a small list of axioms. We also address linear algebra from the viewpoint of a mathematician, illuminating notions such as fields and abstract vector spaces. This course may be paired with Math 51; though that course is not a pre- or co-requisite.
Terms: Aut, Win | Units: 4 | UG Reqs: WAY-FR

MATH 62CM: Modern Mathematics: Continuous Methods

A proof-based introduction to manifolds and the general Stokes' theorem. This includes a treatment of multilinear algebra, further study of submanifolds of Euclidean space (with many examples), differential forms and their geometric interpretations, integration of differential forms, Stokes' theorem, and some applications to topology. Prerequisites: Math 61CM.
Terms: Win | Units: 5 | UG Reqs: GER:DB-Math, WAY-FR

MATH 62DM: Modern Mathematics: Discrete Methods

This is the second part of a theoretical (proof-based) sequence with a focus on discrete mathematics. The central objects discussed in this course are finite fields. These are beautiful structures in themselves, and very useful in large areas of modern mathematics, and beyond. Our goal will be to construct these, understand their structure, and along the way discuss unexpected applications in combinatorics and number theory. Highlights of the course include a complete proof of a polynomial time algorithm for primality testing, Sidon sets and finite projective planes, and an understanding of a lovely magic trick due to Persi Diaconis. Prerequisite: Math 61DM or 61CM.
Terms: Win | Units: 5 | UG Reqs: WAY-FR

MATH 75SI: Learn to Give a Math Talk

This class focuses on preparing and presenting math talks. The first few weeks introduce the main skills, learning from panels, guest lectures, and discussions. In the remaining weeks, participants practice presenting math topics to classmates that are selected through class votes, and feedback is provided for improving presentations. The material in the topics is new to the students, but not too advanced, so participants learn some math also. The course is aimed at students who have taken at least one proof-based math class, and preference is given to students who have less experience presenting math to others. Application to enroll is necessary: https://forms.gle/7GSA2Vh7xZhm5AUaA
Terms: Win | Units: 2
Instructors: ; Conrad, B. (PI)

MATH 77Q: Probability and gambling

One of the earliest probabilistic discussions was in 1654 between two French mathematicians, Pascal and Fermat, on the following question: 'If a pair of six-sided dice is thrown 24 times, should you bet even money on the occurrence of at least one `double six'?' Shortly after the discussion, Huygens, a Dutch scientist, published De Ratiociniis in Ludo Aleae (The Value of all Chances in Games of Fortune) in 1657; this is considered to be the first treatise on probability. Due to the inherent appeal of games of chance, probability theory soon became popular, and the subject underwent rapid development in the 18th century with contributions from mathematical giants, such as Bernoulli, de Moivre, and Laplace. There are two fairly different lines of thought associated with applications of probability: the solution of betting/gambling and the analysis of statistical data related to quantitative subjects such as mortality tables and insurance rates. In this Introsem, we will discuss poker and other games of chance, such as daily fantasy sports, from the perspective of risk analysis. This Introsem does not require any programming knowledge, but some experience with Excel, MATLAB, R, and/or Python will enhance your experience in our discussion of daily fantasy sports. Students should be familiar with all material from Math 51. No prior knowledge of sports and games of chance is required. Students must apply through the IntroSem application process.
Terms: Win, Spr | Units: 3 | UG Reqs: WAY-FR
Instructors: ; Kim, G. (PI)

MATH 101: Math Discovery Lab

MDL is a discovery-based project course in mathematics. Students work independently in small groups to explore open-ended mathematical problems and discover original mathematics. Students formulate conjectures and hypotheses; test predictions by computation, simulation, or pure thought; and present their results to classmates. WIM. Admission is by application. Motivated students with a mathematical background of at least Math 51 or 61CM or 61DM (or equivalent) are encouraged to apply. Please visit https://mathematics.stanford.edu/math-101 for more information about the course and application.
Terms: Win | Units: 4 | UG Reqs: WAY-FR

MATH 104: Applied Matrix Theory

Linear algebra for applications in science and engineering. The course introduces the key mathematical ideas in matrix theory, which are used in modern methods of data analysis, scientific computing, optimization, and nearly all quantitative fields of science and engineering. While the choice of topics is motivated by their use in various disciplines, the course will emphasize the theoretical and conceptual underpinnings of this subject. Topics include orthogonality, projections, spectral theory for symmetric matrices, the singular value decomposition, the QR decomposition, least-squares methods, and algorithms for solving systems of linear equations; applications include clustering, principal component analysis and dimensionality reduction, regression. MATH 113 offers a more theoretical treatment of linear algebra. MATH 104 and ENGR 108 cover complementary topics in applied linear algebra. The focus of MATH 104 is on algorithms and concepts; the focus of ENGR 108 is on a few linear algebra concepts, and many applications. Prerequisites: MATH 51 and programming experience on par with CS 106A.
Terms: Aut, Win, Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 106: Functions of a Complex Variable

Complex numbers, analytic functions, Cauchy-Riemann equations, complex integration, Cauchy integral formula, residues, elementary conformal mappings. (Math 116 offers a more theoretical treatment.) Prerequisite: 52.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 107: Graph Theory

An introductory course in graph theory establishing fundamental concepts and results in variety of topics. Topics include: basic notions, connectivity, cycles, matchings, planar graphs, graph coloring, matrix-tree theorem, conditions for hamiltonicity, Kuratowski's theorem, Ramsey and Turan-type theorem. Prerequisites: 51 or equivalent and some familiarity with proofs is required.
Terms: Win | Units: 4 | UG Reqs: WAY-FR
Instructors: ; McKenzie, T. (PI); Li, Z. (TA)

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; dual space and dual basis; eigenvectors and eigenvalues; diagonalization. Includes an introduction to proof-writing. (Math 104 offers a more application-oriented treatment.) Prerequisites: Math 51
Terms: Aut, Win, Spr | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 117: Advanced Complex Analysis

Review of holomorphic and meromorphic 1-forms, product development, Gamma-function and Riemann zeta-function, Fourier series and integrals, Fourier and Laplace transforms, differential geometric and analytic approaches to Riemann surfaces and conformal mappings, introduction to hyperbolic geometry, Laplace and d-bar equations and their solvability, the Uniformization Theorem, divisors and line bundles, Riemann-Roch theorem, Abel Jacobi theory. Prerequisite: Math 116.
Terms: Win | Units: 4 | UG Reqs: WAY-FR

MATH 121: Galois Theory

Field of fractions, splitting fields, separability, finite fields. Galois groups, Galois correspondence, examples and applications. Prerequisite: Math 120 and (also recommended) 113.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Bump, D. (PI); Lopez, A. (TA)

MATH 131P: Partial Differential Equations

An introduction to techniques for solving PDE's. Topics include physical examples (such as the heat equation, wave equation, and Laplace's equation in 2 and 3 dimensions) and separation of variables with various coordinate systems to relate them to Sturm-Liouville problems using Fourier, Bessel, and Legendre series. Prerequisite: Math 53.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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. http://statweb.stanford.edu/~adembo/math-136/
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 144: Introduction to Topology and Geometry

Point set topology, including connectedness, compactness, countability and separation axioms. The inverse and implicit function theorems. Smooth manifolds, immersions and submersions, embedding theorems. Prerequisites: Math 61CM or both Math 113 and Math 171.
Terms: Win | Units: 4 | UG Reqs: WAY-FR

MATH 148: Algebraic Topology

Fundamental group, covering spaces, Euler characteristic, homology, classification of surfaces, knots. Prerequisite: 109 or 120.
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 151: Introduction to Probability Theory

A proof-oriented development of basic 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. CS majors can petition to use Math 151 in place of CS 109, provided they expect to take either CS 228 or CS 229 as well. Prerequisite: Math 61CM, or Math 52 and either Math 56 or Math 115 (or equivalent).
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR
Instructors: ; Borga, J. (PI); Serio, C. (TA)

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. Prerequisite: Math 51 and proof-writing experiences (e.g., Math 56).
Terms: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

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: Win | Units: 4 | UG Reqs: GER:DB-Math, WAY-FR

MATH 197: Senior Honors Thesis

Honors math major working on senior honors thesis under an approved advisor carries out research and reading. Satisfactory written account of progress achieved during term must be submitted to advisor before term ends. May be repeated 3 times for a max of 9 units. Contact department student services specialist to enroll.
Terms: Aut, Win, Spr, Sum | Units: 1-6 | Repeatable 3 times (up to 9 units total)

MATH 199: Reading Topics

For Math majors only. Undergraduates pursue a reading program under the direction of a Math faculty member; topics limited to those not in regular department course offerings. Credit can fulfill the elective requirement for Math majors. Departmental approval required; please contact the Student Services Specialist for the enrollment proposal form at least 2 weeks before the final study list deadline. May be repeated for credit. Enrollment beyond a third section requires additional approval.
Terms: Aut, Win, Spr, Sum | Units: 3 | Repeatable for credit

MATH 205B: Real Analysis

Point set topology, basic functional analysis, Fourier series, and Fourier transform. Prerequisites: 171 and 205A or equivalent. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win | Units: 3

MATH 210B: Modern Algebra II

Continuation of 210A. Topics in field theory, commutative algebra, algebraic geometry, and finite group representations. Prerequisites: 210A, and 121 or equivalent.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win | Units: 3
Instructors: ; Zhu, X. (PI); Iwasaki, H. (TA)

MATH 215B: Differential Topology

Topics: Basics of differentiable manifolds (tangent spaces, vector fields, tensor fields, differential forms), embeddings, tubular neighborhoods, integration and Stokes' Theorem, deRham cohomology, intersection theory via Poincare duality, Morse theory. Prerequisite: 215ANOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win | Units: 3

MATH 216B: Introduction to Algebraic Geometry

Continuation of 216A. May be repeated for credit.nnNOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win | Units: 3 | Repeatable for credit

MATH 220B: Computational Methods of Applied Mathematics (CME 306)

Numerical methods for solving elliptic, parabolic, and hyperbolic partial differential equations. Algorithms for gradient and Hamiltonian systems. Algorithms for stochastic differential equations and Monte Carlo methods. Algorithms for computational harmonic analysis. Prerequisites: advanced undergraduate level PDE and advanced undergraduate level numerical analysis. Note: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win | Units: 3

MATH 230B: Theory of Probability II (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,n(v) ergodic theory. http://statweb.stanford.edu/~adembo/stat-310b. Prerequisite: 310A or MATH 230A.
Terms: Win | Units: 3

MATH 235: Modern Markov Chains (STATS 318)

Tools for understanding Markov chains as they arise in applications. Random walk on graphs, reversible Markov chains, Metropolis algorithm, Gibbs sampler, hybrid Monte Carlo, auxiliary variables, hit and run, Swedson-Wong algorithms, geometric theory, Poincare-Nash-Cheeger-Log-Sobolov inequalities. Comparison techniques, coupling, stationary times, Harris recurrence, central limit theorems, and large deviations.
Terms: Win | Units: 3

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. Introduction to stochastic control and Bayesian filtering. Prerequisite: Math 136 or equivalent and basic familiarity with parabolic partial differential equations. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such other courses taken.
Terms: Win | Units: 3
Instructors: ; Papanicolaou, G. (PI)

MATH 238: Mathematical Finance (STATS 250)

Stochastic models of financial markets. Risk neutral pricing for derivatives, hedging strategies and management of risk. Multidimensional portfolio theory and introduction to statistical arbitrage. Prerequisite: Math 136 or equivalent. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as other courses taken.
Terms: Win | Units: 3
Instructors: ; Papanicolaou, G. (PI)

MATH 249B: Topics in Number Theory

Topics of contemporary interest in number theory. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win | Units: 3 | Repeatable for credit
Instructors: ; Zhu, X. (PI)

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
Instructors: ; Eliashberg, Y. (PI)

MATH 272: Topics in Partial Differential Equations

NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win | Units: 3 | Repeatable for credit
Instructors: ; Malinnikova, E. (PI)

MATH 275B: Topics in Applied Math II

Topics in Applied Mathematics II. May be repeated for credit. NOTE: Undergraduates require instructor permission to enroll. Undergraduates interested in taking the course should contact the instructor for permission, providing information about relevant background such as performance in prior coursework, reading, etc.
Terms: Win | Units: 3 | Repeatable for credit
Instructors: ; Ying, L. (PI)

MATH 298: Graduate Practical Training

Only for mathematics graduate students. Students obtain employment in a relevant industrial or research activity to enhance their professional experience. Students submit a concise report detailing work activities, problems worked on, and key results. May be repeated for credit up to 3 units. Prerequisite: qualified offer of employment and consent of department. Prior approval by Math Department is required; you must contact the Math Department's Student Services staff for instructions before being granted permission to enroll.
Terms: Aut, Win, Spr, Sum | Units: 1 | Repeatable for credit
Instructors: ; Luk, J. (PI); Vondrak, J. (PI)

MATH 382: Qualifying Examination Seminar

Terms: Win, Sum | Units: 1-3 | Repeatable for credit
Instructors: ; Luk, J. (PI)
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