Course descriptions • (undergraduate) • (graduate)

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


MATH 1110    Probability/Finite Mathematics (3.00)
Studies finite probability theory including combinatorics, equiprobable models, conditional probability and Bayes' theorem, expectation and variance, and Markov chains.
MATH 1140    Financial Mathematics (3.00)
Offered Fall 2017
(Offered Fall 2017)
The study of the mathematics needed to understand and answer a variety of questions that arise in everyday financial dealings. The emphasis is on applications, including simple and compound interest, valuation of bonds, amortization, sinking funds, and rates of return on investments. A solid understanding of algebra is assumed.
MATH 1150    The Shape of Space (3.00)
Offered Fall 2017
(Offered Fall 2017)
Provides an activity and project-based exploration of informal geometry in two and three dimensions. Emphasizes visualization skill, fundamental geometric concepts, and the analysis of shapes and patterns. Topics include concepts of measurement, geometric analysis, transformations, similarity, tessellations, flat and curved spaces, and topology.
MATH 1160    Algebra, Number Systems, and Number Theory (3.00)
Studies basic concepts, operations, and structures occurring in number systems, number theory, and algebra. Inquiry-based student investigations explore historical developments and conceptual transitions in the development of number and algebraic systems.
MATH 1190    A Survey of Calculus I with Algebra (4.00)
Offered Fall 2017
(Offered Fall 2017)
A first calculus course for business/biology/social-science students. Topics include college algebra/limits and continuity/differentiation and integration of algebraic and elementary transcendental functions/applications to related-rates & optimization problems as well as to curve sketching & exponential growth. At most one of MATH 1190, MATH 1210, and 1310 may be taken for credit. Prerequisite: No previous exposure to Calculus.
MATH 1210    A survey of Calculus I (3.00)
Offered Fall 2017
(Offered Fall 2017)
A first calculus course for business/biology/social-science students. Topics include limits and continuity/differentiation & integration of algebraic & elementary transcendental functions/applications to related-rates & optimization problems as well as to curve sketching & exponential growth. At most one of Math 1190, MATH 1210, and 1310 ma1y be taken for credit.
MATH 1220    A Survey of Calculus II (3.00)
Offered Fall 2017
(Offered Fall 2017)
A second calculus course for business/biology/and social-science students. Topics include differential equations/infinite series/analysis of functions of several variables/analysis of probability density functions of continuous random variables. The course begins with a review of basic single-variable calculus. Prerequisite: MATH 1210 or equivalent; at most one of MATH 1220 and MATH 1320 may be taken for credit.
MATH 1310    Calculus I (4.00)
Offered Fall 2017
(Offered Fall 2017)
A first calculus course for natural-science majors/students planning further work in mathematics/students intending to pursue graduate work in applied social sciences. Introduces differential & integral calculus for single-variable functions, emphasizing techniques/applications & major theorems, like the fundamental theorem of calculus. Prerequisite: Background in algebra/trigonometry/exponentials/logarithms/analytic geometry.
MATH 1320    Calculus II (4.00)
Offered Fall 2017
(Offered Fall 2017)
A second calculus course for natural-science majors, students planning additional work in mathematics, and students intending to pursue graduate work in the applied social sciences. Topics include applications of the integral, techniques of integration, differential equations, infinite series, parametric equations, and polar coordinates. Prerequisite: MATH 1310 or equivalent; at most one of MATH 1220 and MATH 1320 may be taken for credit.
MATH 1330    Calculus Workshop I (2.00)
Intensive calculus problem-solving workshop with topics drawn from MATH 1310. Prerequisite: Instructor permission; corequisite: MATH 1310.
MATH 1340    Calculus Workshop II (2.00)
Intensive calculus problem-solving workshop with topics drawn from MATH 1320. Prerequisite: Instructor permission; corequisite: MATH 1320.
MATH 1559    New Course in Mathematics (1.00 - 4.00)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 2310    Calculus III (4.00)
Offered Fall 2017
(Offered Fall 2017)
A continuation of Calc I and II, this course is about functions of several variables. Topics include finding maxima and minima of functions of several variables/surfaces and curves in three-dimensional space/integration over these surfaces and curves. Additional topics: conservative vector fields/Stokes' and the divergence theorems/how these concepts relate to real world applications. Prerequisite: MATH 1320 or the equivalent.
MATH 2315    Advanced Calculus and Linear Algebra I (4.00)
Offered Fall 2017
(Offered Fall 2017)
Covers the material from Math 2310 (multivariable calculus) plus topics from complex numbers, set theory and linear algebra. Prepares students for taking advanced mathematics classes at an early stage.
MATH 2559    New Course in Mathematics (1.00 - 4.00)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 2700    Euclidean and Non-Euclidean Geometry (3.00)
Examines assumptions and methods in the original text of Euclid's Elements. Covers selected geometric topics such as symmetries, spherical geometry, curvature, the dissection theory of area, constructible numbers, and the discovery of non-Euclidean geometry. Prerequisite: Some familiarity with calculus.
MATH 3000    Transition to Higher Mathematics (4.00)
Covers basic concepts with an emphasis on writing mathematical proofs. Topics include logic, sets, functions and relations, equivalence relations and partitions, induction, and cardinality. Prerequisite: Math 1320; and students with a grade of B or better in Math 3310, 3354, or any 5000-level Math course are not eligible to enroll in Math 3000.
MATH 3100    Introduction to Probability (3.00)
Offered Fall 2017
(Offered Fall 2017)
Introduces fundamental concepts/techniques of probability/the theory of randomness. Focuses on problem solving/understanding key theoretical ideas. Topics include sample spaces combinatorial analysis/discrete and continuous random variables/classical distributions/expectation/Chebyshev's inequality/independence/central limit theorem/conditional probability/generating functions. Prerequisite: MATH 1320. Recommended: knowledge of double integrals.
MATH 3120    Introduction to Mathematical Statistics (3.00)
Includes sampling theory, point estimation, interval estimation, testing hypotheses (including the Neyman-Pearson lemma and likelihood ratio tests), and regression and correlation. Prerequisite: MATH 3100.
MATH 3250    Ordinary Differential Equations (4.00)
Offered Fall 2017
(Offered Fall 2017)
Introduces the methods, theory, and applications of differential equations. Includes first-order, second and higher-order linear equations, series solutions, linear systems of first-order differential equations, and the associated matrix theory. May include numerical methods, non-linear systems, boundary value problems, and additional applications. Prerequisite: MATH 1320 or its equivalent.
MATH 3310    Basic Real Analysis (3.00)
Offered Fall 2017
(Offered Fall 2017)
A rigorous development of the properties of the real numbers and the ideas of calculus including theorems on limits/ continuity/differentiability/convergence of infinite series/the construction of the Riemann integral. The focus of students' work will be on getting experience in constructing proofs and developing examples. Prerequisite: MATH 1320.
MATH 3315    Advanced Calculus and Linear Algebra II (4.00)
This course is a continuation of MATH 2315. Covers topics from linear algebra/differential equations/real analysis. Success in this course and MATH 2315 (grades of B- or higher) exempts the student from the math major requirement of taking MATH 3351 and MATH 3250. Students are encouraged to take more advanced courses in these areas. Prerequisite: MATH 2315.
MATH 3340    Complex Variables with Applications (3.00)
Offered Fall 2017
(Offered Fall 2017)
Covers functions of a complex variable that are complex differentiable and the unusual and useful properties of such functions. Some topics: Cauchy's integral formula/power series/the residue theorem/Rouché's theorem. Applications include doing real integrals using complex methods and applications to fluid flow in two dimensions. Prerequisite: MATH 2310.
MATH 3350    Applied Linear Algebra (3.00)
Topics will include systems of linear equations, matrix operations and inverses, vector spaces and subspaces, determinants, eigenvalues and eigenvectors, matrix factorizations, inner products and orthogonality, and linear transformations. Emphasis will be on applications, with computer software integrated throughout the course. The target audience for MATH 3350 is non-math majors from disciplines that apply tools from linear algebra. Credit is not given for both MATH 3350 and 3351.
MATH 3351    Elementary Linear Algebra (3.00)
Offered Fall 2017
(Offered Fall 2017)
Includes matrices, elementary row operations, inverses, vector spaces and bases, inner products and Gram-Schmidt orthogonalization, orthogonal matrices, linear transformations and change of basis, eigenvalues, eigenvectors, and symmetric matrices. Credit is not given for both MATH 3350 and 3351. Prerequisite: MATH 1320.
MATH 3354    Survey of Algebra (3.00)
Offered Fall 2017
(Offered Fall 2017)
Surveys major topics of modern algebra: groups, rings, and fields. Presents applications to areas such as geometry and number theory; explores rational, real, and complex number systems, and the algebra of polynomials. Prerequisite: MATH 1320 or equivalent.
MATH 3559    New Course in Mathematics (1.00 - 4.00)
Offered Fall 2017
(Offered Fall 2017)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 4040    Discrete Mathematics (3.00)
Offered Fall 2017
(Offered Fall 2017)
Includes combinatorial principles, the binomial and multinomial theorems, partitions, discrete probability, algebraic structures, trees, graphs, symmetry groups, Polya's enumeration formula, linear recursions, generating functions and introduction to cryptography, time permitting. Prerequisite: MATH 3354 or instructor permission.
MATH 4080    Operations Research (3.00)
Development of mathematical models and their solutions, including linear programming, the simplex algorithm, dual programming, parametric programming, integer programming, transportation models, assignment models, and network analysis. Prerequisite: MATH 1320 and 3351.
MATH 4110    Introduction to Stochastic Processes (3.00)
Offered Fall 2017
(Offered Fall 2017)
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. Prerequisite: MATH 3100 or APMA 3100; and a knowledge of matrix algebra
MATH 4140    Mathematics of Derivative Securities (3.00)
Offered Fall 2017
(Offered Fall 2017)
This class introduces students to the mathematics used in pricing derivative securities. Topics include a review of the relevant probability theory of conditional expectation and martingales/the elements of financial markets and derivatives/pricing contingent claims in the binomial & the finite market model/(time permitting) the Black-Scholes model. Prerequisites: MATH 3100 or APMA 3100. Students should have a knowledge of matrix algebra.
MATH 4210    Mathematics for Physics (3.00)
This course covers linear algebra/complex analysis/vector differential & integral calculus. Thus it is a compressed version of MATH 3351 & MATH 3340 and a review of some of the material in MATH 2310. Emphasis is on the physical interpretation. [This course does not count as a Mathematics elective for Mathematics majors if both MATH 3351 and MATH 3340 are to be counted.] Prerequisite: MATH 2310.
MATH 4220    Partial Differential Equations and Applied Mathematics (3.00)
Offered Fall 2017
(Offered Fall 2017)
This course is a beginning course in partial differential equations/Fourier analysis/special functions (such as spherical harmonics and Bessel functions). The discussion of partial differential equations will include the Laplace and Poisson equations and the heat and wave equations. Prerequisites: MATH 3250 and either MATH 3351 or MATH 4210.
MATH 4250    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 or APMA 3080 and MATH 3310 or MATH 4310.
MATH 4300    Elementary Numerical Analysis (3.00)
Includes Taylor's theorem, solution of nonlinear equations, interpolation and approximation by polynomials, numerical quadrature. May also cover numerical solutions of ordinary differential equations, Fourier series, or least-square approximation. Prerequisite: MATH 3250 and computer proficiency.
MATH 4310    Introduction to Real Analysis (3.00)
Offered Fall 2017
(Offered Fall 2017)
This course covers the basic topology of metric spaces/continuity and differentiation of functions of a single variable/Riemann-Stieltjes integration/convergence of sequences and series. Prerequisite: MATH 3310 or permission of instructor.
MATH 4330    Advanced Multivariate Calculus (3.00)
Differential and Integral Calculus in Euclidean spaces; implicit and inverse function theorems, differential forms and Stokes' Theorem. Prerequisite: MATH 2310 or MATH 2315
MATH 4452    Algebraic Coding Theory (3.00)
Introduces algebraic techniques for communicating information in the presence of noise. Includes linear codes, bounds for codes, BCH codes and their decoding algorithms. May also include quadratic residue codes, Reed-Muller codes, algebraic geometry codes, and connections with groups, designs, and lattices. Prerequisite: MATH 3351 and 3354, or instructor permission.
MATH 4559    New Course in Mathematics (1.00 - 4.00)
This course provides the opportunity to offer a new topic in the subject of mathematics.
MATH 4595    Undergraduate Research Seminar (3.00)
Emphasizes direct contact with advanced mathematical ideas, communication of these ideas, the discovery of new results and connections among them, and the experience of mathematics as a collaborative venture among researchers at all levels. Students work collaboratively and individually on research projects, and present their results to the class. Prerequisite: Instructor permission.
MATH 4651    Advanced Linear Algebra (3.00)
Offered Fall 2017
(Offered Fall 2017)
Review of topics from Math 3351 including vector spaces, bases, dimension, matrices and linear transformations, diagonalization; however, the material is covered in greater depth with emphasis on theoretical aspects. The course continues with more advanced topics including Jordan and rational canonical forms of matrices and introduction to bilinear forms. Additional topics such as modules and tensor products may be included. Prerequisite: MATH 3351
MATH 4652    Introduction to Abstract Algebra (3.00)
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. Prerequisite: MATH 3351 or 5651
MATH 4653    Number Theory (3.00)
The study of the integers and related number systems. Includes polynomial congruences, rings of congruence classes and their groups of units, quadratic reciprocity, diophantine equations, and number-theoretic functions. Additional topics such as the distribution of prime numbers may be included. Prerequisite: MATH 3354.
MATH 4657    Bilinear Forms and Group Representations (3.00)
Offered Fall 2017
(Offered Fall 2017)
Covers the representation theory of finite groups/other interactions between linear & abstract algebra. Topics include: bilinear & sesquilinear forms & inner product spaces/important classes of linear operators on inner product spaces/the notion of group representation/complete reducibility of complex representations of finite groups/character theory/some applications of representation theory. Prerequisite: MATH 3351 (or 4651)/MATH 3354 (or 4652)
MATH 4658    Galois Theory (3.00)
This course studies the symmetries of solutions of polynomials. Topics include algebraic field extensions/field automorphisms/the fundamental theorem of Galois theory. Applications include the unsolvability of the quintic, as well as ruler & compass constructions. Prerequisites: MATH 3351 (or 4651) and MATH 4652.
MATH 4660    Algebraic Combinatorics (3.00)
Combinatorics of counting using basic tools from calculus, linear algebra, and occasionally group theory. Topics include: tableaux, symmetric polynomials, Catalan numbers, quantum binomial theorem, q-exponentials, partition and q-series identities. Bijective proofs will be emphasized when appropriate.
MATH 4720    Introduction to Differential Geometry (3.00)
Geometric study of curves/surfaces/their higher-dimensional analogues. Topics vary and may include curvature/vector fields and the Euler characteristic/the Frenet theory of curves in 3-space/geodesics/the Gauss-Bonnet theorem/and/or an introduction to Riemannian geometry on manifolds. Prerequisites: MATH 2310 and MATH 3351 or instructor permission.
MATH 4750    Introduction to Knot Theory (3.00)
Examines the knotting and linking of curves in space. Studies equivalence of knots via knot diagrams and Reidemeister moves in order to define certain invariants for distinguishing among knots. Also considers knots as boundaries of surfaces and via algebraic structures arising from knots. Prerequisite: MATH 3354 or instructor permission.
MATH 4770    General Topology (3.00)
Offered Fall 2017
(Offered Fall 2017)
Topics include abstract topological spaces & continuous functions/connectedness/compactness/countability/separation axioms. Rigorous proofs emphasized. Covers myriad examples, i.e., function spaces/projective spaces/quotient spaces/Cantor sets/compactifications. May include intro to aspects of algebraic topology, i.e., the fundamental group. Prerequisites: MATH 2310, MATH 3351, MATH 3310, or higher level versions of these courses.
MATH 4830    Seminar (3.00)
Presentation of selected topics in mathematics. Prerequisite: MATH 5310; co-requisite: MATH 5652
MATH 4840    Introduction to Mathematical Research (3.00)
This course will introduce students to the techniques and methods of mathematical research. Students will independently work with mathematical literature on a topic assigned by the instructor and present their findings in various formats (presentation, paper etc.).
MATH 4900    Distinguished Major Thesis (3.00)
This course provides a framework for the completion of a Distinguished Major Thesis, a treatise containing an exposition of a chosen mathematical topic. A faculty advisor guides a student through the beginning phases of the process of research and writing. Prerequisite: Acceptance into the Distinguished Major Program.
MATH 4901    Distinguished Major Thesis (3.00)
This is the second semester of a two semester sequence for the purpose of the completion of a Distinguished Major Thesis. A faculty member guides the student through all phases of the process which culminates in an open presentation of the thesis to an audience including a faculty evaluation committee. Prerequisite: MATH 4900.
MATH 4993    Independent Study (1.00 - 3.00)
Reading and study programs in areas of interest to individual students. For third- and fourth-years interested in topics not covered in regular courses. Students must obtain a faculty advisor to approve and direct the program.
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
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.