Please refer to SIS or Lou’s list for details about current instructors and current enrollment numbers.

## PhD level courses and research seminars

MATH 7000    Seminar on College Teaching (1.00 - 3.00)
Offered Fall 2017
(Offered Fall 2017)
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 degree. Prerequisite: Graduate standing in mathematics.
MATH 7010    Seminar on Research in Mathematics (1.00 - 3.00)
This seminar discusses the issues related to research in Mathematics. There are speakers from the different areas of mathematics represented at the University of Virginia. Credit may not be used towards a Master's degree. Prerequisite: Graduate standing in mathematics.
MATH 7250    Ordinary Differential Equations and Dynamical Systems (3.00)
Topics include well-posedness and stability of dynamical flows, attractors, invariant manifolds and their properties, and dissipative and Hamiltonian systems. Prerequisite: MATH 5310 and linear algebra, or the equivalent.
MATH 7305    Problems in Analysis (3.00)
Applications of the theory presented in MATH 7310, 7320, and 7340 to specific examples in real and complex analysis. The course emphasizes problem-solving and preparation for the General Examination in Analysis. Problems are based on those from past General Exams. This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
MATH 7310    Real Analysis and Linear Spaces I (3.00)
Introduces measure and integration theory. Prerequisite: MATH 5310 or equivalent.
MATH 7320    Real Analysis and Linear Spaces II (3.00)
Offered Fall 2017
(Offered Fall 2017)
Additional topics in measure theory. Banach and Hilbert spaces, and Fourier analysis. Prerequisite: MATH 7310, 7340, or equivalent.
MATH 7340    Complex Analysis I (3.00)
Offered Fall 2017
(Offered Fall 2017)
Studies the fundamental theorems of analytic function theory.
MATH 7360    Probability Theory I (3.00)
Rigorous introduction to probability, using techniques of measure theory. Includes limit theorems, martingales, and stochastic processes. Prerequisite: 7310 or equivalent.
MATH 7370    Probability Theory II (3.00)
Continuation of Probability Theory I. Elements of stochastic processes, including Brownian motion, continuous time martingales, and Markov processes.
MATH 7410    Functional Analysis I (3.00)
Studies the basic principles of linear analysis, including spectral theory of compact and selfadjoint operators. Prerequisite: MATH 7340 and 7310, or equivalent.
MATH 7420    Functional Analysis II (3.00)
Studies the spectral theory of unbounded operators, semigroups, and distribution theory. Prerequisite: MATH 7410 or equivalent.
MATH 7450    Introduction to Mathematical Physics (3.00)
An introduction to classical mechanics, with topics in statistical and quantum mechanics, as time permits. Prerequisite: MATH 5310.
MATH 7559    New Course in Mathematics (1.00 - 4.00)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 7600    Homological Algebra (3.00)
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. Prerequisite: MATH 5770.
MATH 7705    Problems In Topology (3.00)
A continuation of the theory presented in MATH 5770 and 7800 intensively training students to apply the theory to proving theorems and solving problems in topology, especially in preparation for the General Examination in Topology. Problems are based on those from past General Exams. This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
MATH 7751    Algebra I (3.00)
Offered Fall 2017
(Offered Fall 2017)
Studies groups, rings, fields, modules, tensor products, and multilinear functions. Prerequisite: MATH 5651, 5652, or equivalent.
MATH 7752    Algebra II (3.00)
Studies groups, rings, fields, modules, tensor products, and multilinear functions. Prerequisite: MATH 5651, 5652, or equivalent.
MATH 7753    Algebra III (3.00)
Offered Fall 2017
(Offered Fall 2017)
Studies the Wedderburn theory, commutative algebra, and topics in advanced algebra. Prerequisite: MATH 7751, 7752, or equivalent.
MATH 7754    Algebra IV (3.00)
Further topics in algebra.
MATH 7755    Problems in Algebra (3.00)
A continuation of the theory presented in MATH 7751 and 7752 intensively training students to apply the theory to proving theorems in algebra, especially in preparation for the General Examination in Algebra. Problems are based on those from past General Exams. This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
MATH 7800    Algebraic Topology I (3.00)
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. Prerequisite: MATH 5352, 5770, or equivalent.
MATH 7810    Algebraic Topology II (3.00)
Offered Fall 2017
(Offered Fall 2017)
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. Prerequisite: MATH 7800.
MATH 7820    Differential Topology (3.00)
Offered Fall 2017
(Offered Fall 2017)
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. Prerequisite: MATH 5310, 5770, or equivalent.
MATH 7830    Fiber Bundles (3.00)
Examines fiber bundles; induced bundles, principal bundles, classifying spaces, vector bundles, and characteristic classes, and introduces K-theory and Bott periodicity. Prerequisite: MATH 7800.
MATH 7840    Homotopy Theory (3.00)
Offered Fall 2017
(Offered Fall 2017)
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. Prerequisite: MATH 7800.
MATH 8250    Partial Differential Equations (3.00)
Theory of distributions. Sobolev spaces and their properties (trace and embedding theorems). Theory of elliptic equations. Time-dependent partial differential equations: parabolic and hyperbolic equations. Topics in nonlinear partial differential equations. Prerequisites: MATH 7410 and 7250.
MATH 8310    Operator Theory I, II (3.00)
Topics in the theory of operators on a Hilbert space and related areas of function theory.
MATH 8320    Operator Theory I, II (3.00)
Topics in the theory of operators on a Hilbert space and related areas of function theory.
MATH 8360    Stochastic Calculus and Differential Equations (3.00)
This course presents the basic theory of stochastic differential equations and provides examples of its applications. It is an essential topic for students preparing to do research in probability. Topics covered include a review of the relevant stochastic process and martingale theory; stochastic calculus including Ito's formula; existence and uniqueness for stochastic differential equations, strong Markov property; and applications. Prerequisite: MATH 7360 and 7370, or instructor permission.
MATH 8380    Random Matrices (3.00)
Discusses fundamental problems and results of the theory of random matrices, and their connections to tools of algebra and combinatorics: Wigner's semicircle law, free probability, Gaussian, circular, and beta ensembles of random matrices, bulk and edge asymptotics and universality, Dyson's Brownian motion, determinantal point processes, and discrete analogues of random matrix models. Prerequisite: MATH 7360 or instructor permission.
MATH 8410    Harmonic Analysis (3.00)
This course studies real variable methods for singular integrals and related functional spaces.
MATH 8450    Topics in Mathematical Physics (3.00)
Applies functional analysis to physical problems; scattering theory, statistical mechanics, and quantum field theory.
MATH 8559    New Course in Mathematics (1.00 - 4.00)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 8600    Commutative Algebra (3.00)
The foundations of commutative algebra, algebraic number theory, or algebraic geometry.
MATH 8620    Algebraic Geometry (3.00)
Studies the foundations of algebraic geometry.
MATH 8630    Algebraic Number Theory (3.00)
Theory of number fields and local fields, ramification theory, further topics as chosen by instructor.
MATH 8700    Lie Groups (3.00)
Studies basic results concerning Lie groups, Lie algebras, and the correspondence between them.
MATH 8710    Lie Algebras (3.00)
Studies basic structure theory of Lie algebras.
MATH 8720    Differential Geometry (3.00)
Offered Fall 2017
(Offered Fall 2017)
Studies differential geometry in the large; connections; Riemannian geometry; Gauss-Bonnet formula; and differential forms.
MATH 8750    Topology of Manifolds (3.00)
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. Prerequisite: Math 5770.
MATH 8850    Topics in Algebraic Topology (3.00)
Selected advanced topics in algebraic topology.
MATH 8851    Group Theory (3.00)
Studies the basic structure theory of groups, especially finite groups.
MATH 8852    Representation Theory (3.00)
Offered Fall 2017
(Offered Fall 2017)
Studies the foundations of representation and character theory of finite groups.
MATH 8855    Theory of Algebras (3.00)
Studies the basic structure theory of associative or nonassociative algebras.
MATH 8880    Transformation Groups (3.00)
Studies groups of transformations operating on a space; properties of fixed-point sets, orbit spaces; and local and global invariants.
MATH 8995    Thesis (3.00 - 12.00)
Offered Fall 2017
(Offered Fall 2017)
Thesis
MATH 8998    Non-Topical Research, Preparation for Research (1.00 - 12.00)
Offered Fall 2017
(Offered Fall 2017)
For master's research, taken before a thesis director has been selected.
MATH 8999    Non-Topical Research (1.00 - 12.00)
Offered Fall 2017
(Offered Fall 2017)
For master's thesis, taken under the supervision of a thesis director.
MATH 9000    Mathematics Colloquium (0.00)
Forum for invited speakers giving mathematical colloquium talks.
MATH 9010    History of Mathematics Seminar (1.00 - 3.00)
Discusses subjects from the history of mathematics.
MATH 9020    Graduate Seminar (0.00)
This is a meeting place for junior faculty members and graduate students to discuss mathematics and give talks reflecting the mathematical interests of the participants.
MATH 9250    Harmonic Analysis and PDEs (3.00)
Offered Fall 2017
(Offered Fall 2017)
Harmonic Analysis and PDEs seminar
MATH 9310    Operator Theory Seminar (3.00)
Offered Fall 2017
(Offered Fall 2017)
Operator Theory Seminar
MATH 9360    Probability Seminar (3.00)
Offered Fall 2017
(Offered Fall 2017)
Probability Seminar
MATH 9410    Galois-Grothendieck Seminar (3.00)
Offered Fall 2017
(Offered Fall 2017)
Galois-Grothendieck Seminar
MATH 9450    Mathematical Physics Seminar (3.00)
Mathematical Physics Seminar
MATH 9559    New Course in Mathematics (1.00 - 4.00)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 9800    Topology Seminar (3.00)
Offered Fall 2017
(Offered Fall 2017)
Topology Seminar
MATH 9820    Geometry Seminar (1.00 - 3.00)
Offered Fall 2017
(Offered Fall 2017)
Discusses subjects from geometry.
MATH 9950    Algebra Seminar (3.00)
Offered Fall 2017
(Offered Fall 2017)
Algebra Seminar
MATH 9995    Independent Research (3.00 - 9.00)
Offered Fall 2017
(Offered Fall 2017)
Independent Research
MATH 9998    Non-Topical Research, Preparation for Doctoral Research (1.00 - 12.00)
Offered Fall 2017
(Offered Fall 2017)
For doctoral research, taken before a dissertation director has been selected.
MATH 9999    Non-Topical Research (1.00 - 12.00)
Offered Fall 2017
(Offered Fall 2017)
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. For doctoral dissertation, taken under the supervision of a dissertation director.

## Master level and mathematical education courses

MATH 5010    The History of the Calculus (3.00)
Studies the evolution of the various mathematical ideas leading up to the development of calculus in the 17th century, and how those ideas were perfected and extended by succeeding generations of mathematicians. Emphasizes primary source materials. Prerequisite: MATH 2310 and 3351, or instructor permission.
MATH 5030    The History of Mathematics (3.00)
Studies the development of mathematics from classical antiquity to the end of the 19th century, focusing on critical periods in the evolution of geometry, number theory, algebra, probability, and set theory. Emphasizes primary source materials. Prerequisite: MATH 2310 and 3351, or instructor permission.
MATH 5100    Probability (3.00)
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. Prerequisite: MATH 1320 or equivalent, and graduate standing. Credit cannot be received for both MATH 3100 and 5100.
MATH 5250    Differential Equations and Dynamical Systems (3.00)
A second course in ordinary differential equations, from the dynamical systems point of view. Topics include: existence and uniqueness theorems; linear systems; qualitative study of equilibria and attractors; bifurcation theory; introduction to chaotic systems. Further topics as chosen by the instructor. Applications drawn from physics, biology, and engineering. Prerequisites:MATH 3351 and MATH 3310 or equivalent.
MATH 5305    Proofs in Analysis (3.00)
This course reviews the proofs of the main theorems in analysis in preparation for the advanced graduate analysis courses. This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
MATH 5330    Advanced Multivariate Calculus (3.00)
Differential and Integral Calculus in Euclidean spaces; implicit and inverse function theorems, differential forms and Stokes' Theorem. Prerequisite: Calculus III (either MATH 2310 or MATH 2315) and one of MATH 5310 or MATH 5770.
MATH 5559    New Course in Mathematics (1.00 - 4.00)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 5700    Introduction to Geometry (3.00)
Topics selected from analytic, affine, projective, hyperbolic, and non-Euclidean geometry. Prerequisite: MATH 2310, 3351, or instructor permission.
MATH 5720    Introduction to Differential Geometry (3.00)
Topics selected from the theory of curves and surfaces in Euclidean space and the theory of manifolds. Prerequisite: MATH 2310 and 3351, or instructor permission.
MATH 5770    General Topology (3.00)
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. Prerequisite: MATH 2310 and 3351, and 3310.
MATH 5855    Proofs in Algebra (3.00)
This course reviews the proofs of the main theorems in algebra in preparation for the advanced graduate algebra courses.This course is offered in the summer and restricted to Mathematics and Graduate Arts and Science students.
MATH 5896    Supervised Study in Mathematics (3.00)
Offered Fall 2017
(Offered Fall 2017)
A rigorous program of supervised study designed to expose the student to a particular area of mathematics. Prerequisite: Instructor permission and graduate standing.
MATH 6060    AFDA: Mathematical Modeling with Probability and Statistics (3.00)
Examines experimental design and probability and statistics through exploring, analyzing, and interpreting data sets. Explores the graphing calculator as a tool to display and analyze data obtained from sampling, observations, measurement, experiments, and internet sources.
MATH 6120    Measurement and Data Analysis (3.00)
Measurement and Data Analysis
MATH 6452    Functions and Algebra (3.00)
Functions and Algebra
MATH 6453    Number Systems and Number Theory for K-8 Mathematics Specialists (3.00)
Number Systems and Number Theory for K-8 Mathematics Specialists
MATH 6454    Rational Numbers and Proportional Reasoning (3.00)
Rational Numbers and Proportional Reasoning
MATH 6559    New Course in Mathematics (1.00 - 4.00)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 6600    Algebra for Middle School Specialists (3.00)
Algebra for Middle School Specialists
MATH 6630    AAO Introductory College Algebra and Trigonometry (3.00)
AAO Introductory College Algebra and Trigonometry
MATH 6650    AAO Calculus with Applications (3.00)
AAO Calculus with Applications
MATH 6660    Euclidean Geometry (3.00)
Euclidean Geometry
MATH 6670    AAO Probability and Statistics (3.00)
Explores introductory descriptive statistics, probability, and statistical inference. Develops conceptual understanding and procedural fluency in problem settings based on real data which investigate the use of visual methods from summarizing quantitative information, basic experimental design, sampling methods, and interpretation of statistical analysis.
MATH 6700    Geometry and Measurement for K-8 Math Specialists (3.00)
Geometry and Measurement for K-8 Math Specialists
MATH 6760    MM Data Analysis, Probability, and Statistics for Middle School Teachers (3.00)
Focuses on the representation of data for decision making and predictability based on data analysis as it relates to middle school mathematics and defined in the NCTM Professional Standards for School Mathematics and Virginia SOLS in Mathematics. Teachers deepen their understanding and use of the fundamental ideas in mathematics that underlie the probability and statistics strand.
MATH 6800    Teaching Mathematics to Diverse Populations (3.00)
Teaching Mathematics to Diverse Populations