Department of Mathematics

Kerchof Hall
University of Virginia
P.O. Box 400137
Charlottesville, VA 22904-4137
(434) 924-4919
www.math.virginia.edu

Degree Requirements

Programs of Study The Department of Mathematics administers programs leading to the degree of Master of Arts, Master of Science, and Doctor of Philosophy. These programs provide diverse opportunities for advanced study and research in algebra, analysis, topology, and mathematical physics.

The Master of Arts and Master of Science degrees are normally completed within two years, though in some cases, these degrees can be completed in one calendar year (two semesters and a summer session). The M.A. and M.S. programs differ mainly in course requirements. The M.S. degree requires specific courses in algebra, analysis, and topology. In contrast, the course requirements for the M.A. degree are flexible and based on individual needs. The M.A. candidate has two options, one requiring an expository paper for a thesis, and the other substituting additional course work in place of a thesis.

The Doctor of Philosophy degree is normally completed within five years. Candidates for the Ph.D. must fulfill certain course requirements and examinations beyond the master’s level. The most important addition is the Ph.D. dissertation, which is based on original research performed under the supervision of a faculty member.

All full-time graduate students are required, as part of their program, to gain teaching experience by assisting the instruction of undergraduate courses.

Master of Arts Degree

Course Requirements (a) Thesis option: 24 credits of courses approved by the graduate committee at the 500 level or above (some courses from other departments and thesis research can count towards the 24 credits). (b) Non-thesis option: 30 credits of courses at the 500 level or above (no reading or research courses), which must include MATH 531, 533 (or replacements from among 731, 732, 734) and MATH 551, 552 (or replacements from 751, 752), and cannot include more than 9 credits from other departments.

Thesis (option (a) only): The master’s thesis is an expository paper written under the supervision of a faculty advisor.

Examinations A passing grade on the final master’s exam (or both parts of the general examination); specific content of the exam should be agreed on by the student and the examiners well in advance. The candidate must be a registered student at the time of the exam, and must finish the degree requirements within three years of passing the exam.

Language Facility in reading mathematical literature in one foreign language (French, German, Russian, Italian, or a substitute acceptable to the department) as confirmed by an examination administered by a member of the department. Two years of undergraduate credit in one of the languages will meet this requirement.

Master of Science Degree The requirements for the M.S. degree are the same as for the M.A. degree, except the program must include MATH 731, 734, MATH 751, 752, MATH 577 and a topology course at the 700 level. Higher-level substitutes may be approved.

Doctor of Philosophy Degree

Course Requirements A student must do satisfactory work in two semesters of analysis (MATH 731, 734), algebra (MATH 751, 752), and topology (MATH 577 and a 700 level topology course), or the equivalent.

Examinations Passing grades on two general examinations, chosen from analysis, algebra, and topology, and satisfactory performance on the second year proficiency examination.

General Examinations The general exams are written exams, which are set and graded by the graduate committee. They test whether the student has the inventiveness and command of basic material to pursue a Ph.D. degree, and are usually taken in the second year of graduate study.

Second Year Proficiency Examination In consultation with the Graduate Advisory Committee, students select two or three second year courses to form the basis for the second year proficiency examination. It is formatted as a conversation between the student and a panel of faculty members, and tests whether the student has mastered material relevant to the intended dissertation area. It is typically completed in May of the second year.

Language Facility in reading mathematical literature in one language (French, German, Russian, Italian, or a substitute acceptable to the department), as demonstrated by an exam administered by the department, in which students are required to translate passages from mathematical works in the given language. The language requirement should generally be satisfied by the end of the fourth year, or by the date of the Ph.D. defense, whichever comes first. Students pursing research in the history of mathematics are required to pass a written translation examination in two foreign languages, typically French and German, and this requirement should in general be satisfied by the end of the third year.

Dissertation and Defense Written under the supervision of the major advisor, the Ph.D. dissertation must contain original contributions to the field of mathematics. The main results of the dissertation are presented at a public oral defense. A committee consisting of the major advisor and three other faculty members (two from within the department and one from outside) must approve the dissertation and defense in order for the dissertation to be considered accepted by the faculty.

Course Descriptions

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MATH 501 - (3) (E)
The History of the Calculus
Prerequisite: MATH 231 and 351, or instructor permission.
Studies the evolution of the various mathematical ideas leading up to the development of the calculus in the seventeenth century, and how those ideas were perfected and extended by succeeding generations of mathematicians. Emphasizes primary source materials.

MATH 503 - (3) (O)
The History of Mathematics
Prerequisite: MATH 231 and 351, or instructor permission.
Studies the development of mathematics from classical antiquity through the end of the nineteenth century, focusing on the critical periods in the evolution of such areas as geometry, number theory, algebra, probability, and set theory. Emphasizes primary source materials.

MATH 506 - (3) (IR)
Algorithms
Prerequisite: MATH 132 and computer proficiency.
Studies abstract algorithms to solve mathematical problems and their implementation in a high-level language. Topics include sorting problems, recursive algorithms, and dynamic data structures.

MATH 510 - (3) (Y)
Mathematical Probability
Prerequisite: MATH 132 or equivalent, and graduate standing. Students may not receive credit for both MATH 310 and 510.
Studies the development and analysis of probability models through the basic concepts of sample spaces, random variables, probability distributions, expectations, and conditional probability. Additional topics include distributions of transformed variables, moment generating functions, and the central limit theorem.

MATH 511 - (3) (Y)
Stochastic Processes
Prerequisite: MATH 310 or instructor permission.
Topics in probability selected from Random walks, Markov processes, Brownian motion, Poisson processes, branching processes, stationary time series, linear filtering and prediction, queuing processes, and renewal theory.

MATH 512 - (3) (Y)
Mathematical Statistics
Prerequisite: MATH 510 or equivalent, and graduate standing.
Studies methods of estimation, general concepts of hypothesis testing, linear models and estimation by least squares, categorical data, and nonparametric statistics.

MATH 514 - (3) (Y)
Mathematics of Derivative Securities
Prerequisite: MATH 231 or 122 or its equivalent, and a knowledge of probability and statistics. MATH 310 or its equivalent is recommended.
Topics include arbitrage arguments, valuation of futures, forwards and swaps, hedging, option-pricing theory, and sensitivity analysis.

MATH 517 - (3) (IR)
Actuarial Mathematics
Prerequisite: MATH 312 or 512 or instructor permission.
Covers the main topics required by students preparing for the examinations in actuarial statistics, set by the American Society of Actuaries. Topics include life tables, life insurance and annuities, survival distributions, net premiums and premium reserves, multiple life functions and decrement models, valuation of pension plans, insurance models, benefits, and dividends.

MATH 521 - (3) (Y)
Advanced Calculus and Applied Mathematics
Prerequisite: MATH 231, 325.
Includes vector analysis, Green’s, Stokes’, divergence theorems, conservation of energy, and potential energy functions. Emphasizes physical interpretation, Sturm-Liouville problems and Fourier series, special functions, orthogonal polynomials, and Green’s functions.

MATH 522 - (3) (Y)
Partial Differential Equations and Applied Mathematics
Prerequisite: MATH 521 (351 recommended).
Introduces partial differential equations, Fourier transforms. Includes separation of variables, boundary value problems, classification of partial differential equations in two variables, Laplace and Poisson equations, and heat and wave equations.

MATH 525 - (3) (IR)
Advanced Ordinary Differential Equations
Prerequisite: MATH 231, 325, 351 or instructor permission.
Studies the qualitative geometrical theory of ordinary differential equations. Includes basic well posedness; linear systems and periodic systems; stability theory; perturbation of linear systems; center manifold theorem; periodic solutions and Poincaré-Bendixson theory; Hopf bifurcation; introduction to chaotic dynamics; control theoretic questions; differential geometric methods.

MATH 526 - (3) (IR)
Partial Differential Equations
Prerequisite: MATH 231, 325, and 351, or instructor permission.
A theoretical introduction from a classical viewpoint. Includes harmonic and subharmonic functions; wave and heat equations; Cauchy-Kowalewski and Holmgren theorems; characteristics; and the Hamilton-Jacobi theory.

MATH 530 - (3) (IR)
Computer Methods in Numerical Analysis
Prerequisite: MATH 351, 430, and computer proficiency.
Studies underlying mathematical principles and the use of sophisticated software for spline interpolation, ordinary differential equations, nonlinear equations, optimization, and singular-value decomposition of a matrix.

MATH 531 - (3) (Y)
Introduction to Real Analysis
Prerequisite: MATH 231, 351.
Includes the basic topology of Euclidean spaces; continuity, and differentiation of functions of a single variable; Riemann-Stieltjes integration; and convergence of sequences and series.

MATH 533 - (3) (Y)
Advanced Multivariate Calculus
Prerequisite: MATH 531.
Differential and Integral Calculus in Euclidean spaces; implicit and inverse function theorems, differential forms and Stokes’ Theorem.

MATH 534 - (3) (Y)
Complex Variables with Applications
Prerequisite: MATH 231 and graduate standing.
Includes analytic functions, Cauchy formulas, power series, residue theorem, conformal mapping, and Laplace transforms.

MATH 551 - (3) (Y)
Advanced Linear Algebra
Prerequisite: MATH 351 or instructor permission.
This course includes a systematic review of the material usually considered in MATH 351 such as matrices, determinants, systems of linear equations, vector spaces, and linear operators. However, these concepts will be developed over general fields and more theoretical aspects will be emphasized. The centerpiece of the course is the theory of canonical forms, including the Jordan canonical form and the rational canonical form. Another important topic is general bilinear forms on vector spaces. Time permitting, some applications of linear algebra in differential equations, probability, etc. are considered.

MATH 552 - (3) (Y)
Introduction to Abstract Algebra
Prerequisite: MATH 351 or instructor permission.
Focuses on structural properties of basic algebraic systems such as groups, rings and fields. A special emphasis is made on polynomials in one and several variables, including irreducible polynomials, unique factorization and symmetric polynomials. Time permitting, such topics as group representations or algebras over a field may be included.

MATH 554 - (3) (Y)
Survey of Algebra
Prerequisite: MATH 132 or equivalent and graduate standing.
Surveys major topics of modern algebra such as groups, rings, and fields. Presents applications to geometry and number theory. Explores the rational, real, and complex number systems, and the algebra of polynomials.

MATH 555 - (3) (IR)
Algebraic Automata Theory
Prerequisite: MATH 351.
Introduces the theory of sequential machines, finite permutation groups and transformation semigroups. Includes examples from biological and electronic systems as well as computer science, the Krohn-Rhodes decomposition of a state machine, and Mealy machines.

MATH 570 - (3) (O)
Introduction to Geometry
Prerequisite: MATH 231, 351, or instructor permission.
Selected topics from analytic, affine, projective, hyperbolic, and non-Euclidean geometry.

MATH 572 - (3) (E)
Introduction to Differential Geometry
Prerequisite: MATH 231, 351, or instructor permission.
Studies the theory of curves and surfaces in Euclidean space and the theory of manifolds.

MATH 577 - (3) (Y)
General Topology
Prerequisite: MATH 231; corequisite: MATH 551 or equivalent.
Topological spaces and continuous functions, connectedness, compactness, countability and separation axioms, and function spaces. Time permitting, more advanced examples of topological spaces, such as projectives spaces, as well as an introduction to the fundamental group will be covered.

MATH 583 - (3) (SI)
Seminar
Prerequisite: Instructor permission.
Presentation of selected topics in mathematics. Usually for DMP students.

MATH 596 - (3) (S)
Supervised Study in Mathematics
Prerequisite: Instructor permission and graduate standing.
A rigorous program of supervised study designed to expose the student to a particular area of mathematics. Regular homework assignments and scheduled examinations are required.

MATH 700 - (1-3) (Y)
Seminar on College Teaching
Prerequisite: Graduate standing in mathematics.
Discussion of issues related to the practice of teaching, pedagogical concerns in college level mathematics, and aspects of the responsibilities of a professional mathematician. Credits may not be used towards a Master’s or Ph.D. degree.

MATH 731 - (4) (Y)
Real Analysis and Linear Spaces I
Prerequisite: MATH 531 or equivalent.
Introduces measure and integration theory.

MATH 732 - (3) (O)
Real Analysis and Linear Spaces II
Prerequisite: MATH 731, 734, or equivalent.
Additional topics in measure theory. Banach and Hilbert spaces, and Fourier analysis.

MATH 734 - (4) (Y)
Complex Analysis I
Studies the fundamental theorems of analytic function theory.

MATH 735 - (3) (O)
Complex Analysis II
Prerequisite: MATH 734 or equivalent.
Studies the Riemann mapping theorem, meromorphic and entire functions, topics in analytic function theory.

MATH 736 - (3) (E)
Mathematical Theory of Probability
Prerequisite: MATH 731 or equivalent.
Rigorous introduction to probability, using techniques of measure theory. Includes limit theorems, martingales, and stochastic processes.

MATH 741 - (3) (Y)
Functional Analysis I
Prerequisite: MATH 734 and 731, or equivalent.
Studies the basic principles of linear analysis, including spectral theory of compact and selfadjoint operators.

MATH 742 - (3) (E)
Functional Analysis II
Prerequisite: MATH 741 or equivalent.
Studies the spectral theory of unbounded operators, semigroups, and distribution theory.

MATH 745 - (3) (IR)
Introduction to Mathematical Physics
Prerequisite: MATH 531.
An introduction to classical mechanics, with topics in statistical and quantum mechanics, as time permits.

MATH 751, 752 - (4) (Y)
Algebra I, II
Prerequisite: MATH 551, 552, or equivalent.
Studies groups, rings, fields, modules, tensor products, and multilinear functions.

MATH 753 - (3) (Y)
Algebra III
Prerequisite: MATH 751, 752, or equivalent.
Studies the Wedderburn theory, commutative algebra, and topics in advanced algebra.

MATH 760 - (3) (SI)
Homological Algebra
Prerequisite: MATH 577.
Examines categories, functors, abelian catqegories, limits and colimits, chain complexes, homology and cohomology, homological dimension, derived functors, Tor and Ext, group homology, Lie algebra homology, spectral sequences, and calculations.

MATH 780 - (3) (Y)
Algebraic Topology I
Prerequisite: MATH 552, 577, or equivalent.
Topics include the fundamental group, covering spaces, covering transformations, the universal covering spaces, graphs and subgroups of free groups, and the fundamental groups of surfaces. Additional topics will be from homology, including chain complexes, simplicial and singular homology, exact sequences and excision, cellular homology, and classical applications.

MATH 781 - (3) (Y)
Algebraic Topology II
Prerequisite: MATH 780.
Devoted to chomology theory: cohomology groups, the universal coefficient theorem, the Kunneth formula, cup products, the cohomology ring of manifolds, Poincare duality, and other topics if time permits.

MATH 782 - (3) (Y)
Differential Topology
Prerequisite: MATH 531, 577, or equivalent.
Topics include smooth manifolds and functions, tangent bundles and vector fields, embeddings, immersions, transversality, regular values, critical points, degree of maps, differential forms, de Rham cohomology, and connections.

MATH 783 - (3) (Y)
Fiber Bundles
Prerequisite: MATH 780.
Examines fiber bundles; induced bundles, principal bundles, classifying spaces, vector bundles, and characteristic classes, and introduces K-theory and Bott periodicity.

MATH 784 - (3) (Y)
Homotopy Theory
Prerequisite: MATH 780.
Definition of homotopy groups, homotopy theory of CW complexes, Huriewich theorem and Whitehead’s theorem, Eilenberg-Maclane spaces, fibration and cofibration sequences, Postnikov towers, and obstruction theory.

MATH 825 - (3) (O)
Differential Equations
Topics in the theory of ordinary and partial differential equations.

MATH 830 - (3) (SI)
Topics in Function Theory
Topics in real and complex function theory.

MATH 831, 832 - (3) (Y)
Operator Theory I, II
Topics in the theory of operators on a Hilbert space and related areas of function theory.

MATH 836, 837 - (3) (SI)
Topics in Probability Theory and Stochastics Processes
Topics in probability, stochastic processes and ergodic theory.

MATH 840 - (3) (SI)
Harmonic Analysis
Studies Banach and C* algebras, topological vector spaces, locally compact groups, Fourier analysis.

MATH 845 - (3) (SI)
Topics in Mathematical Physics
Applies functional analysis to physical problems; scattering theory, statistical mechanics, and quantum field theory.

MATH 851 - (3) (SI)
Group Theory
Studies the basic structure theory of groups, especially finite groups.

MATH 852 - (3) (SI)
Representation Theory
Studies the foundations of representation and character theory of finite groups.

MATH 853 - (3) (SI)
Algebraic Combinatorics
Studies geometries, generating functions, partitions, and error-correcting codes and graphs using algebraic methods involving group theory, number theory, and linear algebra.

MATH 854 - (3) (SI)
Arithmetic Groups
Prerequisite: MATH 752.
General methods of analyzing groups viewed as discrete subgroups of real algebraic subgroups. Additional topics include the congruence subgroup problem.

MATH 855 - (3) (SI)
Theory of Algebras
Studies the basic structure theory of associative or nonassociative algebras.

MATH 860 - (3) (SI)
Commutative Algebra
The foundations of commutative algebra, algebraic number theory, or algebraic geometry.

MATH 862 - (3) (SI)
Algebraic Geometry
Studies the foundations of algebraic geometry.

MATH 865 -(3) (SI)
Algebraic K-Theory
Includes projective class groups and Whitehead groups; Milnor’s K2 and symbols; higher K-theory and finite fields.

MATH 870 - (3) (SI)
Lie Groups
Studies basic results concerning Lie groups, Lie algebras, and the correspondence between them.

MATH 871 - (3) (SI)
Lie Algebras
Studies basic structure theory of Lie algebras.

MATH 872 - (3) (SI)
Differential Geometry
Studies differential geometry in the large; connections; Riemannian geometry; Gauss-Bonnet formula; and differential forms.

MATH 875 - (3) (SI)
Topology of Manifolds
Prerequisite: Math 577.
Studies regular and critical values, gradient flow, handle decompositions, Morse theory, h-cobordism theorem, Dehn’s lemma in dimension 3, and disk theorem in dimension 4.

MATH 880 - (3) (SI)
Generalized Cohomology Theory
Topics include the axiomatic generalized cohomology theory; representability and spectra; spectra and ring spectra; orientability of bundles in generalized cohomology theory; Adams spectral sequence, and stable homotopy.

MATH 883 - (3) (SI)
Cobordism and K-Theory
Studies classical cobordism theories; Pontryagin-Thom construction; bordism and cobordism of spaces; K-theory and Bott periodicity; formal groups, and cobordism.

MATH 885 - (3) (SI)
Topics in Algebraic Topology
Selected advanced topics in algebraic topology.

MATH 888 - (3) (SI)
Transformation Groups
Studies groups of transformations operating on a space; properties of fixed-point sets, orbit spaces; and local and global invariants.

MATH 896 - (3-12) (Y)
Thesis

MATH 897 - (3-12) (Y)
Non-Topical Research, Preparation for Research
For master’s research, taken before a thesis director has been selected.

MATH 898 - (3-12) (Y)
Non-Topical Research
For master’s thesis, taken under the supervision of a thesis director.

MATH 925 - (3) (Y)
Differential Equations and Dynamical Systems Seminar

MATH 931 - (3) (Y)
Operator Theory Seminar

MATH 936 - (3) (Y)
Probability Seminar

MATH 941 - (3) (Y)
Analysis Seminar

MATH 945 - (3) (Y)
Mathematical Physics Seminar

MATH 950 - (3) (Y)
Algebra Seminar

MATH 952 - (3) (IR)
Coding Theory Seminar

MATH 980 - (3) (Y)
Topology Seminar

MATH 996 - (3-9) (Y)
Independent Research

MATH 997 - (3-12) (Y)
Non-Topical Research, Preparation for Doctoral Research
For doctoral research, taken before a dissertation director has been selected.

MATH 999 - (3-12) (Y)
Non-Topical Research
For doctoral dissertation, taken under the supervision of a dissertation director.

The Mathematics Colloquium is held weekly, the sessions being devoted to research activities of students and faculty members, and to reports by visiting mathematicians on current work of interest.