phd math stanford

For potential Ph.D. students

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Stanford bulletin, archive 2008-09.

Bulletin Archive

This archived information is dated to the 2008-09 academic year only and may no longer be current.

For currently applicable policies and information, see the current Stanford Bulletin .

Graduate courses in Mathematics

Primarily for graduate students; undergraduates may enroll with consent of instructor.

MATH 205A. Real Analysis

Basic measure theory and the theory of Lebesgue integration. Prerequisite: 171 or equivalent.

3 units, Aut (Ryzhik, L)

MATH 205B. Real Analysis

Point set topology, basic functional analysis, Fourier series, and Fourier transform. Prerequisites: 171 and 205A or equivalent.

3 units, Win (Vasy, A)

MATH 205C. Real Analysis

Continuation of 205B.

3 units, Spr (Katznelson, Y)

MATH 210A. Modern Algebra

Groups, rings, and fields; introduction to Galois theory. Prerequisite: 120 or equivalent.

3 units, Aut (Milgram, R)

MATH 210B. Modern Algebra

Galois theory. Ideal theory, introduction to algebraic geometry and algebraic number theory. Prerequisite: 210A.

3 units, Win (Brumfiel, G)

MATH 210C. Modern Algebra

Continuation of 210B. Representations of groups and noncommutative algebras, multilinear algebra.

3 units, Spr (Bump, D)

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.

3 units, Aut (Li, J)

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.

3 units, Win (Galatius, S)

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.

3 units, Spr (Cohen, R)

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.

3 units, not given this year

MATH 216B. Introduction to Algebraic Geometry

Continuation of 216A. May be repeated 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.

3 units, Win (Schoen, R)

MATH 217B. Differential Geometry

Riemannian manifolds, Levi-Civita connection, Riemann curvature tensor, Riemannian exponential map and geodesic normal coordinates, Jacobi fields, completeness, spaces of constant curvature, bi-invariant metrics on compact Lie groups, symmetric and locally symmetric spaces, equations for Riemannian submanifolds and Riemannian submersions. Prerequisite: 217A.

3 units, Spr (Brendle, S)

MATH 220. Partial Differential Equations of Applied Mathematics

(Same as 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.

3 units, Aut (Nolen, J)

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.

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.

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.

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.

3 units, Win (Ryzhik, L)

MATH 228A. Ergodic Theory

Measure preserving transformations and flows, ergodic theorems, mixing properties, spectrum, Kolmogorov automorphisms, entropy theory. Examples. Classical dynamical systems, mostly geodesic and horocycle forms on homogeneous spaces of SL(2,R). May be repeated for credit. Prerequisites: 205A,B.

MATH 230A. Theory of Probability

(Same as STATS 310A.) Mathematical tools: asymptotics, metric spaces; measure and integration; Lp spaces; some Hilbert spaces theory. Probability: independence, Borel-Cantelli lemmas, almost sure and Lp convergence, weak and strong laws of large numbers. Weak convergence and characteristic functions; central limit theorems; local limit theorems; Poisson convergence. Prerequisites: 116, MATH 171.

2-4 units, Aut (Diaconis, P)

MATH 230B. Theory of Probability

(Same as STATS 310B.) Stopping times, 0-1 laws, Kolmogorov consistency theorem. Uniform integrability. Radon-Nikodym theorem, branching processes, conditional expectation, discrete time martingales. Exchangeability. Large deviations. Laws of the iterated logarithm. Birkhoff's and Kingman's ergodic theorems. Recurrence, entropy. Prerequisite: 310A or MATH 230A.

2-4 units, Win (Dembo, A)

MATH 230C. Theory of Probability

(Same as STATS 310C.) Infinitely divisible laws. Continuous time martingales, random walks and Brownian motion. Invariance principle. Markov and strong Markov property. Processes with stationary independent increments. Prerequisite: 310B or MATH 230B.

2-4 units, Spr (Dembo, A)

MATH 231A. An Introduction to Random Matrix Theory

(Same as 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.

3 units, Aut (Diaconis, P)

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.

3 units, Win (Dembo, A)

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.

3 units, Spr (Staff)

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.

3 units, Win (Staff)

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.

3 units, Win (Katznelson, Y)

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.

3 units, Win (Papanicolaou, G)

MATH 238. Mathematical Finance

(Same as 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.

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.

3 units, Spr (Toussaint, A)

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.

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.

3 units, Spr (Kerckhoff, S)

MATH 245A. Topics in Algebraic Geometry: Moduli Theory

Intersection theory on the moduli spaces of stable curves, stable maps, and stable vector bundles. May be repeated for credit.

MATH 245B. Topics in Algebraic Geometry: Dessin d'Enfants

Grothendieck's theory of dessin d'enfants, a study of graphs on surfaces and their connection with the absolute Galois group of the rational numbers. Belyi's theorem, representations of the absolute Galois group as automorphisms of profinite groups, Grothendieck-Teichmuller theory, quadratic differentials, and the combinatorics of moduli spaces of surfaces. May be repeated 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.

3 units, Win (Diaconis, P)

MATH 248. Algebraic Number Theory

Introduction to modular forms and L-functions. May be repeated for credit.

1-3 units, not given this year

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.

3 units, Aut (Conrad, B)

MATH 249A. Introduction to Modular Forms

The analytic theory of holomorphic and non-holomorphic modular forms and associated L-functions. Topics include Hecke operators, L-functions, Weil's converse theorem, trace formulas, sub-convexity for L-functions and applications, and Selberg's eigenvalue conjecture. May be repeated for credit. Prerequisites: 205A,B,C, or comparable knowledge of analysis.

3 units, Aut (Soundararajan, K)

MATH 249B. Topics in Number Theory: Class Field Theory

Classification of abelian extensions of local and global fields; classical, adelic, and cohomological formulations; applications to L-functions. May be repeated for credit.

3 units, Win (Conrad, B)

MATH 249C. Topics in Number Theory: Class Field Theory and the Langlands Conjectures

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.

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.

3 units, Spr (Vasy, A)

MATH 256B. Partial Differential Equations

Continuation of 256A.

3 units, Win (Liu, T)

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.

3 units, Aut (Ionel, E)

MATH 257B. Symplectic Geometry and Topology

Continuation of 257A. May be repeated for credit.

3 units, Win (Ionel, E)

MATH 258. Topics in Geometric Analysis

May be repeated for credit.

3 units, Win (White, B)

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.

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.

3 units, Win (Bump, D)

MATH 263B. Lie Groups and Lie Algebras

Continuation of 263A. May be repeated for credit.

MATH 264. Matrix Valued Spherical Functions and Orthogonal Polynomials

Theory of spherical functions on locally compact groups and on Lie groups. Families of orthogonal polynomials with respect to a weight matrix function on the real line, and the corresponding algebra of differential operators. Spherical functions associated to the complex projective space as orthogonal polynomials. Topics may include some applications to quasi birth and death processes. My be repeated for credit. Prerequisities: 114, 205A, and 217A.

3 units, Aut (Staff)

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.

MATH 269A. Affine Complex Manifolds and Symplectic Geometry

Plurisubharmonic functions and pseudoconvexity: geometric theory. Construction of pseudoconvex shapes. Complex analysis on Stein manifolds. Symplectic geometry of Stein manifolds. Existence theorem for Stein complex manifolds. May be repeated for credit.

3 units, Aut (Eliashberg, Y)

MATH 269B. Affine Complex Manifolds and Symplectic Geometry

Symplectic convexity and Weinstein manifolds. Symplectic topology of subcritical Weinstein manifolds. From Weinstein to Stein structure. Morse-Smale theory for plurisubharmonic functions on Stein manifolds. Deformation theory for Stein complex structures. Symplectic field theory of Weinstein manifolds. May be repeated for credit.

3 units, Win (Eliashberg, Y)

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.

3 units, Win (Li, J)

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.

3 units, Spr (Eliashberg, Y)

MATH 272A. Topics in Partial Differential Equations

3 units, Aut (Tzou, L)

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.

3 units, Aut (Kerckhoff, S)

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.

3 units, Win (Carlsson, G)

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.

3 units, Spr (Milgram, R)

MATH 284A. Geometry and Topology in Dimension 3

The Poincare conjecture and the uniformization of 3-manifolds. May be repeated for credit.

MATH 284B. Geometry and Topology in Dimension 3

MATH 286. Topics in Differential Geometry

3 units, Win (Mazzeo, R), Spr (Schoen, R)

MATH 290B. Finite Model Theory

(Same as PHIL 350B.) Classical model theory deals with the relationship between formal languages and their interpretation in finite or infinite structures; its applications to mathematics using first-order languages. The recent development of the model theory of finite structures in connection with complexity classes as measures of computational difficulty; how these classes are defined within certain languages that go beyond first-order logic in expressiveness, such as fragments of higher order or infinitary languages, rather than in terms of models of computation.

MATH 292A. Set Theory

(Same as PHIL 352A.) The basics of axiomatic set theory; the systems of Zermelo-Fraenkel and Bernays-G�del. 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: PHIL160A,B, and MATH 161, or equivalents.

MATH 292B. Set Theory

(Same as PHIL 352B.) The basics of axiomatic set theory; the systems of Zermelo-Fraenkel and Bernays-G�del. 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: PHIL160A,B, and MATH 161, or equivalents.

MATH 293A. Proof Theory

(Same as 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.

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.

MATH 355. Graduate Teaching Seminar

Required of and limited to first-year Mathematics graduate students.

1 unit, Spr (Staff)

MATH 360. Advanced Reading and Research

1-9 units, Aut (Staff), Win (Staff), Spr (Staff), Sum (Staff)

MATH 361. Research Seminar Participation

Participation in a faculty-led seminar which has no specific course number.

1-3 units, Aut (Staff), Win (White, B), Spr (Kerckhoff, S), Sum (Staff)

MATH 380. Seminar in Applied Mathematics

Guest speakers on recent advances in applied mathematics.May be repeated for credit.

1 unit, Aut (Staff), Win (Staff), Spr (Staff)

MATH 381. Seminar in Analysis

1-3 units, by arrangement

MATH 384. Seminar in Geometry

1 unit, by arrangement

MATH 385. Seminar in Topology

MATH 386. Mathematics Colloquium

Guest speakers on recent advances in mathematics. May be repeated for credit.

1 unit, Aut (Staff), Win (Bump, D), Spr (Staff)

MATH 387. Seminar in Number Theory

MATH 388. Seminar in Probability and Stochastic Processes

MATH 389. Seminar in Mathematical Biology

MATH 391. Research Seminar in Logic and the Foundations of Mathematics

(Same as PHIL 391.) Contemporary work. May be repeated a total of three times for credit.

1-3 units, Spr (Mints, G; Feferman, S)

MATH 395. Classics in Geometry and Topology

Original papers in geometry and in algebraic and geometric topology. May be repeated for credit.

3 units, Aut (Brumfiel, G), Win (Staff), Spr (Cohen, R)

MATH 396. Graduate Progress

Results and current research of graduate and postdoctoral students. May be repeated for credit.

MATH 397. Physics for Mathematicians

Topics from physics essential for students studying geometry and topology. Topics may include quantum mechanics, quantum field theory, path integral approach and renormalization, statistical mechanics, and string theory. May be repeated for credit.

1 unit, Win (Staff)

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Doctoral Program

Program summary.

Students are required to

• master the material in the prerequisite courses
• pass the first-year core program
• attempt all three parts of the qualifying examinations and show acceptable performance in at least two of them (end of 1st year)
• confirm a principal dissertation research advisor and file for candidacy (early spring quarter of 2nd year)
• satisfy the depth and breadth requirements (2nd/3rd/4th year)
• successfully complete the thesis proposal meeting and submit the Dissertation Reading Committee form (winter quarter of the 3rd year)
• present a draft of their dissertation and pass the university oral examination (4th/5th year)

The PhD requires a minimum of 135 units. Students are required to take a minimum of nine units of advanced topics courses (for depth) offered by the department (not including literature, research, consulting or Year 1 coursework), and a minimum of nine units outside of the Statistics Department (for breadth). Courses for the depth and breadth requirements must equal a combined minimum of 24 units. In addition, students must enroll in STATS 390 Statistical Consulting, taking it at least twice.

All students who have passed the qualifying exams but have not yet passed the Thesis Proposal Meeting must take STATS 319 at least once each year. For example, a student taking the qualifying exams in the summer after Year 1 and having the dissertation proposal meeting in Year 3, would take 319 in Years 2 and 3. Students in their second year are strongly encouraged to take STATS 399 with at least one faculty member. All details of program requirements can be found in the Department of Statistics PhD Student Handbook (available to Stanford affiliates only, using Stanford authentication. Requests for access from non-affiliates will not be approved).

Statistics Department PhD Handbook

All students are expected to abide by the Honor Code and the Fundamental Standard .

Doctoral and Research Advisors

During the first two years of the program, students' academic progress is monitored by the department's Director of Graduate Studies (DGS). Each student should meet at least once a quarter with the DGS to discuss their academic plans and their progress towards choosing a thesis advisor (before the final study list deadline of spring of the second year). From the third year onward students are advised by their selected advisor.

Qualifying Examinations and Candidacy

Qualifying examinations are part of most PhD programs in the United States. At Stanford these exams are intended to test the student's level of knowledge when the first-year program, common to all students, has been completed. There are separate examinations in the three core subjects of statistical theory and methods, applied statistics, and probability theory, which are typically taken during the summer at the end of the student's first year. Students are expected to attempt all three examinations and show acceptable performance in at least two of them. Letter grades are not given. Qualifying exams may be taken only once. After passing the qualifying exams, students must file for PhD Candidacy, a university milestone, by early spring quarter of their second year.

While nearly all students pass the qualifying examinations, those who do not can arrange to have their financial support continued for up to three quarters while alternative plans are made. Usually students are able to complete the requirements for the M.S. degree in Statistics in two years or less, whether or not they have passed the PhD qualifying exams.

Thesis Proposal Meeting and Dissertation Reading Committee 

The thesis proposal meeting is intended to demonstrate a student's depth in some areas of statistics, and to examine the general plan for their research. In the meeting the student gives a 60-minute presentation involving ideas developed to date and plans for completing a PhD dissertation, and for another 60 minutes answers questions posed by the committee. which consists of their advisor and two other members. The meeting must be successfully completed by the end of winter quarter of the third year. If a student does not pass, the exam must be repeated. Repeated failure can lead to a loss of financial support.

The Dissertation Reading Committee consists of the student’s advisor plus two faculty readers, all of whom are responsible for reading the full dissertation. Of these three, at least two must be members of the Statistics Department (faculty with a full or joint appointment in Statistics but excluding for this purpose those with only a courtesy or adjunct appointment). Normally, all committee members are members of the Stanford University Academic Council or are emeritus Academic Council members; the principal dissertation advisor must be an Academic Council member. 

The Doctoral Dissertation Reading Committee form should be completed and signed at the Dissertation Proposal Meeting. The form must be submitted before approval of TGR status or before scheduling a University Oral Examination.

 For further information on the Dissertation Reading Committee, please see the Graduate Academic Policies and Procedures (GAP) Handbook section 4.8.

University Oral Examinations

The oral examination consists of a public, approximately 60-minute, presentation on the thesis topic, followed by a 60 minute question and answer period attended only by members of the examining committee. The questions relate to the student's presentation and also explore the student's familiarity with broader statistical topics related to the thesis research. The oral examination is normally completed during the last few months of the student's PhD period. The examining committee typically consists of four faculty members from the Statistics Department and a fifth faculty member from outside the department serving as the committee chair. Four out of five passing votes are required and no grades are given. Nearly all students can expect to pass this examination, although it is common for specific recommendations to be made regarding completion of the thesis.

The Dissertation Reading Committee must also read and approve the thesis.

For further information on university oral examinations and committees, please see the Graduate Academic Policies and Procedures (GAP) Handbook section 4.7 .

Dissertation

The dissertation is the capstone of the PhD degree. It is expected to be an original piece of work of publishable quality. The research advisor and two additional faculty members constitute the student's Dissertation Reading Committee. Normally, all committee members are members of the Stanford University Academic Council or are emeritus Academic Council members.

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For those interested in mathematics education, there are opportunities to work with several faculty who are studying mathematics teaching and learning, within and outside CTE and Stanford GSE. Current research projects are addressing issues of equity, interactions between teaching and student learning, the impact of different mathematics teaching and curricular approaches, and lesson study (teacher professional development). Students may also make mathematics education the focus of their inquiries in different courses in Stanford GSE. Students can choose to take mathematics and mathematics-related courses from the department of mathematics, engineering and other departments outside Stanford GSE, as well as work with professors and students in those departments. For those interested in teacher education and teacher professional development, there are opportunities to develop materials for pre-service and in-service mathematics teachers, and to work in the Stanford Teacher Education Program (single-subject mathematics and multiple-subject).

Students applying to this specialization will be expected to have worked in mathematics education, as a teacher or another education professional, and to have an undergraduate degree in mathematics or another subject that will inform analyses of mathematics teaching and learning. Admission depends on a combination of factors, including evidence of academic achievement, professional accomplishments, GRE scores, and fit between students' interests and program offerings.

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• Graduate Record Examination (GRE) General Test
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If you are applying to a graduate program for which the GRE is required or optional, all GRE scores we receive will be made available to the admission committee. The admission committee may decide how they wish to evaluate the scores provided. If your graduate program does not consider GRE scores, any GRE scores we receive will not be displayed to the admission committee.

Self-Reported Scores

To self-report your GRE scores, list all GRE tests you took within the past five years that you wish to be considered by the admission committee. Do not enter “superscores” (a single entry that includes your highest section scores from multiple test dates). The application system will automatically display to the admission committee the highest score you earned in each section as well as all reported scores.

Any scores you self-report on the application are considered unofficial but sufficient for the initial review process.

Official Scores

If GRE scores are required by your program, you must report your official scores directly to Stanford from ETS. Stanford’s ETS institutional code is 4704 . Individual department code numbers are not necessary. As long as you select the institutional code of 4704 , the score is electronically delivered to Stanford.

Upon successful receipt of your official scores, they will appear on the Test Scores page of your application with a status of “Verified.”

Scores expire after five years and will not be available from ETS. See below for the earliest test date Stanford considers to be valid.

Graduate programs may enforce a stricter validity period (e.g., based on their application deadline). Refer to the program’s website.

Test of English as a Foreign Language (TOEFL)

If your first language is not English, you are required to submit an official test score from the  Test of English as a Foreign Language Internet-Based Test (TOEFL iBT) .

• We accept the TOEFL iBT Home Edition and TOEFL iBT Paper Edition if you are unable to take the traditional TOEFL iBT test in a test center. If you take the Home Edition or Paper Edition, you may be required to complete additional  English placement testing prior to enrollment.
• We do not accept TOEFL Essentials scores or any other English proficiency test (e.g., IELTS, PTE).

Minimum TOEFL Requirements

Stanford’s minimum TOEFL score requirement varies depending on your field of study and planned degree:

If you score below 109 on the TOEFL and you are admitted, you will likely be required to complete additional  English placement testing prior to the start of classes.

Reporting TOEFL Scores

We accept MyBest scores , which combine your highest section scores from all test dates within the last two years. All TOEFL scores we receive, including MyBest scores, will be made available to the admission committee. The admission committee may decide how they wish to evaluate the scores provided.

You may use either of the following methods to self-report your MyBest scores on the application:

• List all TOEFL tests you took within the past two years where you earned a section score that is included in your MyBest scores. - or -
• List a single TOEFL entry with your MyBest scores. For the test date, enter the “as of” date listed on your most recent score report.

If TOEFL scores are required for your application, you must report your official scores directly to Stanford from ETS. Stanford’s ETS institutional code is 4704 . Individual department code numbers are not necessary. As long as you select the institutional code of 4704 , the score is electronically delivered to Stanford.

When you arrange for your official TOEFL scores to be sent to Stanford, the report will include both your traditional scores from your selected test date and your MyBest scores. Upon successful receipt, both sets of scores will appear on the Test Scores page of your application with a status of “Verified.” You do not need to have official scores from previous tests sent to Stanford as long as the most recent official score report includes the MyBest scores you wish to use.

Scores expire after two years and will not be available from ETS. See below for the earliest test date Stanford considers to be valid.

TOEFL Exemptions

You are exempt from submitting a TOEFL score if you meet one of the following criteria:

• You (will) have a bachelor’s, master’s, or doctoral degree from a regionally-accredited college or university in the United States (excluding territories and possessions).
• You (will) have an equivalent degree from an English-language university in Australia, Canada, Ireland, New Zealand, Singapore, or the United Kingdom.

The online application will not require you to submit a TOEFL score if you meet one of the criteria listed above for an exemption.

U.S. citizenship does not automatically exempt you from taking the TOEFL if your first language is not English.

TOEFL Waivers

You may request a waiver in the online application if you (will) have a bachelor’s, master’s, or doctoral degree from a recognized institution in a country other than Australia, Canada, Ireland, New Zealand, Singapore, and the United Kingdom in which English was the language of instruction.

You will be asked to provide the following:

• Upload an official statement certifying that your program was taught exclusively in English
• You may also link to your institution’s official website stating the language of instruction

Your waiver request will be routed to Graduate Admissions after you submit your application and pay the application fee . Allow up to 15 business days after submitting your application for a response. This will not delay the receipt of your application by your graduate program.

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Finance Requirements

I. preparation.

The study of financial economics requires a grasp of several types of basic mathematics. Students must enter with or very quickly acquire knowledge of the concepts and techniques of:

It is strongly advised that students without a strong and recent background in calculus, linear algebra, or statistics come to Stanford in June to take courses to strengthen any weak areas.

Computer programming skills are necessary in coursework (as early as the first quarter of the first year) and in research. If students do not have adequate computer programming skills, they may wish to take a computer programming course before they arrive at Stanford, or take an appropriate Stanford computer science course while here.

II. Course Requirements

All required courses must be taken for a grade (not pass/fail or credit/no credit). Exceptions are made if the required course is offered pass/fail or credit/no credit only. Each course must be passed with a grade of P or B- or better. Substitutions of required courses require approval from the faculty liaison. Waiving a course requirement based on similar doctoral level course completed elsewhere requires the approval of the course instructor, faculty liaison, and the PhD Program Office.

III. Practicum

Students are required to sign up for either a research or teaching practicum each quarter of enrollment. Below is a description of the practicum requirements for Finance students.

During the student’s first year, the student will be assigned each quarter to work with a different faculty member. This assignment will involve mentoring and advising from the faculty member and RA work from the student. The purpose of new assignments each quarter is to give the student exposure to a number of different faculty members.

In subsequent years, the practicum will take the form of a research or teaching mentorship, where the student is expected to provide research or teaching support under the guidance and advice of a faculty member. Faculty assignments here will be made through informal discussions between faculty and students, and may be quarterly, or for the entire year.

For students of all years, one requirement to satisfy the practicum is that students regularly attend the Finance seminar. The only exception to this will be if there is a direct and unavoidable conflict between the seminar and necessary coursework.

IV. Summer Research Papers

All students in all years are expected to complete a research paper over the summer, and present this paper in the Fall quarter. A draft of this research paper should be submitted by the end of September to the field liaison. Students can continue to work on and improve their paper up to their presentation. Presentations of summer research will always be viewed as research in progress.

For students completing their first year, the summer paper should demonstrate the mastery of a specific area in the literature. This can be accomplished by either (i) presenting the preliminary development of a research idea or (ii) presenting work co-authored with faculty. The student will be expected to present this paper to a gathering of three Finance faculty members of the student’s choosing in October.

For students completing their second year, the summer paper should develop a research idea that was approved during the oral exam at the beginning of the summer (see below). “Develop” does not mean complete - students will be evaluated based on whether they have made reasonable progress on their research topic and on whether they have identified an appropriate research question. A passing grade on the second-year paper is one requirement for admission to candidacy.

In all years after the second year, the summer research paper should be a well-developed research paper. (Well-developed does not mean completed – research is always presented as work in progress. Rather, it means that the work shows enough progress and development to merit a seminar presentation.) Students will then present their papers to the overall Finance faculty and PhD student body in scheduled talks over the Fall quarter. Student presentations will typically be 45 minutes, save for job market paper presentations, which will be a full hour and a half.

More generally, these presentations throughout all years will be a primary manner that faculty who are not advising the student become familiar with the student’s work, and will play a crucial role in the assessment of the student’s academic progress.

V. Field Exam

Students take the field exam in the summer after the first year. Material from the field exam will be based on required first year coursework. This includes required finance courses, as well as the required microeconomic and econometric classes. The primary purpose of the exam is to ascertain that students have learned the introductory material that is a necessary foundation for understanding and undertaking research in the field. Additionally, studying for the field exam will give students the opportunity to review and synthesize material across all their different first year courses. Students may be asked to leave the program if they fail the field exam, or may be allowed to retake the exam at the Faculty’s discretion. Students who fail the field exam two times will be required to leave the program.

VI. Teaching Requirement

One quarter of course assistantship or teaching practicum. This requirement must be completed prior to graduation.

VII. Finance Oral Exam

The finance oral exam takes place at the end of the spring quarter of the second year, in early June.

At the beginning of the spring quarter of the second year, the student meets with the liaison to determine three finance faculty members who will administer the exam. The student then meets with the selected faculty examiners to discuss a set of topics that will be covered in the finance oral exam. These topics will generally be chosen from coverage in the Finance PhD classes. An important component of the exam involves the student identifying a particular research area to discuss at the exam. The student will be expected to discuss major results in the literature related to this area and to identify important unresolved questions that need to be addressed. In addition the student will be expected to discuss how one or more of these questions might be addressed either theoretically or empirically. During the exam, the student should agree with the faculty members on a topic for the second-year paper (see above).

The results from the finance oral exam plus the result from the second-year summer research paper (presented in the fall of 3rd year) and overall performance in the program are weighed in the decision to admit to candidacy.

VIII. Candidacy

Admission to candidacy for the doctoral degree is a judgment by the faculty of the student’s potential to successfully complete the requirements of the degree program. Students are required to advance to candidacy by September 1 before the start of their fourth year in the program.

IX. University Oral Exam

The university oral examination is a defense of the dissertation work in progress. The student orally presents and defends the thesis work in progress at a stage when it is one-half to two-thirds complete. The oral examination committee tests the student on the theory and methodology underlying the research, the areas of application and portions of the major field to which the research is relevant, and the significance of the dissertation research. Students are required to successfully complete the oral exams by September 1 before the start of their fifth year in the program.

X. Doctoral Dissertation

The doctoral dissertation is expected to be an original contribution to scholarship or scientific knowledge, to exemplify the highest standards of the discipline, and to be of lasting value to the intellectual community. The Finance faculty defer to the student’s Dissertation Reading Committee to provide general guidelines (e.g., number of chapters, length of dissertation) on the dissertation.

Typical Timeline

Years one & two.

• Field Requirements
• Directed Reading & Research
• Advancement to Candidacy
• Formulation of Research Topic
• Annual Evaluation
• Continued Research

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105-year-old Stanford graduate finally gets her diploma after 80+ year wait

Virginia hislop had waited 83 years for this day, by garvin thomas • published june 18, 2024 • updated on june 22, 2024 at 9:21 pm.

On Sunday morning, commencement exercises for Stanford University's Graduate School of Education started late. The school's undergraduate ceremony had run behind schedule, so the smaller ceremonies for graduate students ended up being pushed back half an hour.

It was a minor inconvenience for most of the 160 students getting their master's degrees and doctorates in education. For one student, though, it was downright inconsequential. Virginia Hislop had waited 83 years for this day, so what were another 30 minutes?

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"My goodness, I've waited a long time," were Hislop's exact words when she accepted her diploma.

The 105-year-old Hislop, who grew up in Southern California and now lives in Yakima, Washington, said she always wanted to go to Stanford. Her mother had attended the school in the 1920s.

"There was a desire to come to Stanford and take advantage of everything I could," Hislop said.

Hislop earned her undergraduate degree in 1940 and by the summer of 1941, Hislop had earned enough credits to qualify for a master's degree in education and only needed to write a thesis to finish meeting the degree requirements. But then, on the eve of the Second World War, her husband George, a second lieutenant in the U.S. Army, was called up to active duty. He was ordered to report to Fort Sill, Oklahoma.

“Not my idea of a place for a honeymoon,” Hislop said. “But I had no choice in the matter.”

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Hislop was soon the mother of two small children, so returning to her studies was not a possibility.

Still, the lack of a master's in education did not stop Hislop from spending a lifetime being involved in education.

"No, it had absolutely no effect," Hislop said.

For decades, Hislop served on boards and committees overseeing every level of schooling, from kindergarten to college in Yakima.

“I gave it a great deal of thought and tried to improve the education where I lived,” Hislop said. 

One thing Hislop did not give a great deal of thought to, however, was that nearly-finished degree. It was such a non-issue her son-in-law had never heard the story until recently. He contacted Stanford to inquire about it and learned something revelatory: sometime after Hislop left Stanford, the thesis requirement for a master's had been dropped. She had earned the degree, after all.

"I was surprised and pleased," Hislop said.

So, by her grand and great-grandchildren, Hislop joined the class of 2024 on the commencement stage and received a well-deserved standing ovation. 

She viewed it as a recognition, not just for her diploma, but for all the work in education she has done in the past 80 years. 

"I feel like I've made a difference in my community," she said.

Hundreds of students walk out of Stanford commencement in protest

Stanford University costs over $92,000 a year—how much students actually pay, according to income level

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