Department of Mathematics
Kerchof Hall
University of Virginia
P.O. Box 400137
Charlottesville, VA 229044137
(434) 9244919
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 fulltime 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) Nonthesis 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.
Higherlevel substitutes may be approved.
Doctor of Philosophy Degree
Course Requirements 72 hours of coursework at the 500 level or above (which may include 18 hours of nontopical research: MATH 897, 898, 997, 999). A student must do satisfactory work in two semesters of analysis (MATH 731, 734), algebra (MATH 751, 752), and topology (MATH 577, 780), or the equivalent.
Examinations Passing grades on two general examinations,
chosen from analysis, algebra, and topology, and satisfactory performance on
the secondyear 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.
SecondYear Proficiency Examination Students take an informal oral examination on material from two or three selected secondyear courses. The exact content of the exam is determined by a panel of faculty members in consultation with the student. Its purpose is to gauge the student's readiness to begin carrying out research in the student's chosen area. It is normally taken in May of the second year. If any deficiencies are noted, the examining panel will make recommendations on how to fix the detected problems and meet again with the student in August.
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, although the languages required will depend on the student's research interests. These language requirements should generally be satisfied by the end of the student's third year in order to enable the student to do primary source reading in the pertinent languages.
History of Mathematics Program The requirements for the Ph.D. in the history of mathematics are the same as for the Ph.D. in mathematics, with the exception of the additional language requirements describe above. Also, students entering the program must exhibit a strong reading comprehension in two foreign languages, preferably French and German. Students interested in pursuing a Ph.D. in history of mathematics should contact Professor Karen Parshall before submitting their application.
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.
Time limit The Ph.D. may be completed in as few as three years but must be completed within seven years.
Course Descriptions
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 highlevel 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,
optionpricing 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, SturmLiouville 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 PoincaréBendixson 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; CauchyKowalewski
and Holmgren theorems; characteristics; and the HamiltonJacobi 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 singularvalue 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; RiemannStieltjes 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 553  (3) (O)
Number Theory
Prerequisite: MATH 354 or instructor permission.
Includes congruences, quadratic reciprocity, Diophantine equations, and numbertheoretic
functions, among others.
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 KrohnRhodes 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 nonEuclidean 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  (13) (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 Ktheory
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, EilenbergMaclane 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 errorcorrecting
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 KTheory
Includes projective class groups and Whitehead groups;
Milnor’s
K2 and symbols; higher Ktheory 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; GaussBonnet 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, hcobordism 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 KTheory
Studies classical cobordism theories; PontryaginThom construction;
bordism and cobordism of spaces; Ktheory 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 fixedpoint sets, orbit spaces; and local and global invariants.
MATH 896  (312) (Y)
Thesis
MATH 897  (312) (Y)
NonTopical Research, Preparation for Research
For master’s research,
taken before a thesis director has been selected.
MATH 898  (312) (Y)
NonTopical 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  (39) (Y)
Independent Research
MATH 997  (312) (Y)
NonTopical Research, Preparation for Doctoral Research
For doctoral research, taken before a dissertation director
has been selected.
MATH 999  (312) (Y)
NonTopical 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.
