Department of Mathematics
216 Kerchof Hall
University of Virginia
P.O. Box 400137
Charlottesville, VA 229044137
(434) 9244919 Fax: (434) 9823084
www.math.virginia.edu
Overview In a world of increasing technological complexity,
knowledge of mathematics is the gateway to the pursuit of many fields. Mathematics
has long been the language of choice for expressing complex relationships and
describing complicated patterns and processes. It is now true that many fields,
in addition to mathematics and the sciences, rely on this in a fundamental way.
What was formerly "abstract" mathematics to many
has become the concrete stuff of everyday life. "The unreasonable effectiveness
of mathematics" manifests itself today in such familiar things as CAT and
MRI scans, compact discs, satellite communications, and computer animation.
These were all rendered possible by new discoveries made by mathematicians within
the last fifty years. Even the efficient operation of our financial markets
is based, in part, on relatively recent theorems of mathematical analysis and
probability theory.
Mathematics research today is a vibrant and dynamic enterprise.
Thousands of mathematicians worldwide are at work on an unimaginably broad range
of questions. Exciting recent advances include the proof of Fermat’s Last
Theorem, the classification of the finite simple groups, the proof of the Bieberbach
conjecture, and the computerassisted proof of the fourcolor theorem. The discipline
and creativity required by the study of mathematics can be a formidable preparation
for later life. Past students of mathematics have had successful careers in
almost every sphere, including all the professions. The scope of mathematics
courses offered at the University of Virginia allows majors to tailor their
own programs. Students electing to major in mathematics should consult carefully
with a faculty advisor to ensure the selection of a program of courses that
provides a solid grounding in the fundamentals of higher mathematics and is
appropriate to future goals.
Faculty The faculty of the Department of Mathematics
is committed to excellence in teaching and research. Its members carry out highlevel
research on diverse problems in algebra, analysis, topology, probability, and
statistics, mathematical physics, and the history of the discipline. Their research
has been widely published in prestigious research journals and is recognized
internationally. Members of the department have won Sloan fellowships, Humboldt
fellowships, and other scholarly honors, as well as numerous research grants.
Many are currently supported by grants from the National Science Foundation
and other federal agencies. Most have held visiting professorships abroad. In
addition, the department offerings and ambiance are enhanced each year by the
presence of several internationally recognized visiting faculty.
Students There are currently about 100 students majoring
in mathematics. Class sizes vary from a few large introductory classes to an
average class size of twenty students for upperlevel courses. This small class
size affords students the opportunity to get individual attention.
Students who graduate with degrees in mathematics successfully
pursue a variety of different careers. Many go directly into jobs in industry,
insurance (as actuaries), government, finance, and other fields. Employers in
the past have included Morgan Stanley, General Motors, MITRE Corp., the Census
Bureau, the National Security Agency, and various consulting firms. Many find
themselves well equipped to go on to professional schools in law, medicine,
and business. Some go directly into teaching. Others have gone on to graduate
programs in mathematics, applied mathematics, statistics, engineering, systems
engineering, economics, and computer science. Students who have combined the
mathematics major with courses in computer programming, economics, and business
have done exceptionally well in the job market.
Requirements for Major Normally, the calculus sequence
MATH 131, 132, and 231 or its equivalent must be completed before a student
can declare a major in mathematics. At least a 2.200 average in the calculus
sequence and a minimum grade of C in MATH 231 or its equivalent are required.
However, the department may grant special permission to declare a major to a
student who has only completed MATH 131 and 132, and at least one mathematics
course (other than MATH 231 or its equivalent) which could be counted toward
the major in mathematics, provided the student completes MATH 231 or its equivalent
in the semester following the declaration of a mathematics major.
To graduate with a major in mathematics the student must show computer proficiency by completing CS 101, CS 120 or PHY 254 or an approved equivalent course with a grade of C or higher. This should be done as early as possible.
To help guide the student through the major, the mathematics
department offers five concentrations. Completion of one of these concentrations
is required. Each concentration contains a set of nine required mathematics
courses (approximately 28 credits). To graduate, a student must obtain minimum
grades of C in seven of these courses and C in the other two.
Up to two courses that are being counted for another College major or another College minor can also be counted for the major in mathematics.
Up to two courses that are taken from outside the University
and which are equivalent to College Mathematics courses may be offered for
the
College mathematics major.
Certain substitutions are allowed in all options, for example, MATH 531 for MATH 331, MATH 551 for MATH 351, MATH 552 for MATH 354, and PHYS 553 for MATH 430. PHIL 542 Symbolic Logic is an approved elective for both the major and minor in Mathematics.
A. The Basic Concentration
This traditional program for the mathematics major provides
an overview of key areas:
MATH 325

Ordinary Differential Eq.

4

MATH 351

Elementary Linear Algebra

3

MATH 354

Survey of Algebra

3

Two from the following three:


MATH 310

Introduction to



Mathematical Probability

3

MATH 331

Basic Real Analysis

3

MATH 334

Complex Variables with Applications

3

Four electives at the 300 level or higher, of which at least two are MATH courses

12

Students fulfilling the requirements for this option have a
wide range of career opportunities, from law to business to any field that requires
deductive, logical reasoning skills.
B. The Graduate Preparatory Concentration
This concentration is for the student who plans to attend graduate
school in mathematics or an allied field. The program emphasizes the fundamental
ideas of mathematics with substantial work in proving and understanding the
basic theorems. It consists of:
MATH 325

Ordinary Differential Eq.

4

MATH 334

Complex Variables with Applications

3

MATH 531

Intro. to Real Analysis I

3

MATH 551

Advanced Linear Algebra

3

MATH 552

Intro. to Abstract Algebra

3

Four electives at the 300 level or higher, of which at least two are MATH courses. (You may wish to take MATH 331 in preparation for MATH 531, MATH 351 in preparation for MATH 551, and MATH 354 in preparation for MATH 552).
This constitutes the minimum expected of an incoming graduate student in most programs nationwide. The department strongly recommends MATH 533 (Advanced Multivariate Calculus), as well as courses in differential geometry (MATH 572) or topology (MATH 577). Many of our graduate school bound students take additional courses, including 700level graduate courses.
C. The Probability and Statistics Concentration
This concentration is designed to give the student a good theoretical
underpinning in probability and statistics, as well as the opportunity to go
deeper in these fields. The program can lead to a Master of Science in Statistics
with one additional year of course work, if additional courses in statistics
are taken in the fourth year. (Those interested in the M.S. in Statistics should
contact the graduate advisor in the Department of Statistics prior to the beginning
of their fourth year.) The requirements for the concentration are the following:
MATH 325

Ordinary Differential Eq.

4

MATH 310

Intro. to Mathematical Probability

3

MATH 312

Intro. to Mathematical Statistics

3

MATH 331

Basic Real Analysis or


MATH 334

Complex Variables with Applications

3

MATH 351

Elementary Linear Algebra

3

MATH 354

Survey of Algebra

3

MATH 511

Stochastic Processes

3

STAT 512

Applied Linear Models

3

One additional course chosen from:



MATH 430

Elementary Numerical Analysis

3

MATH 531

Intro. to Real Analysis I

3

STAT 313

Design and Analysis of Sample Surveys

3

STAT 513

Applied Multivariate Statistics

3

STAT 516

Experimental Design

3

STAT 517

Applied Time Series

3

STAT 519

Intro. to Mathematical Statistics

3

D. The Financial Mathematics Concentration
This program provides the student with a broad background of basic mathematics, which is essential for an understanding of the mathematical models used in the financial markets. The mathematics of modern finance includes probability, statistics, regression, time series, partial differential equations, stochastic processes, stochastic calculus, numerical methods, and analysis. The program consists of:
MATH 325

Ordinary Differential Eq.

4

MATH 310

Intro. to Mathematical Probability

3

MATH 312

Intro. to Mathematical Statistics

3

MATH 331

Basic Real Analysis or


MATH 334

Complex Variables with Applications

3

MATH 351

Elementary Linear Algebra

3

MATH 354

Survey of Algebra

3

MATH 514

Mathematics of Derivative Securities

3

Two additional courses chosen from:



MATH 408

Operations Research

3

MATH 430

Elementary Numerical Analysis

3

MATH 511

Stochastic Processes

3

STAT 512

Applied Linear Models

3

STAT 517

Applied Time Series

3

SYS 360

Stochastic Decision Models

3 
Two courses^{(1)} chosen from:



ECON 201

Microeconomics

3

ECON 202

Macroeconomics

3

COMM 201

Introduction to Financial Accounting

3

COMM 202

Intro. to Mgmt. Accounting

3

^{(1)} Completing all four courses is recommended.
E. Fiveyear Teacher Education Program
This option leads to both Bachelor of Arts and Master of Teaching
degrees after five years. The program is for both elementary and secondary teachers
and is administered by the Curry School of Education. Required courses include:
MATH 325

Ordinary Differential Equations

4

MATH 310

Intro. to Mathematical Probability

3

MATH 312

Intro. to Mathematical Statistics

3

MATH 331

Basic Real Analysis or


MATH 334

Complex Variables with Applications

3

MATH 351

Elementary Linear Algebra

3

MATH 354

Survey of Algebra

3

MATH 404

Discrete Mathematics

3 
MATH 501

History of Calculus or


MATH 503

History of Mathematics

3

MATH 570

Introduction to Geometry

3

The Curry School has additional requirements for this program.
Distinguished Majors Program in Mathematics The department
offers a Distinguished Majors Program (DMP) to qualified majors in mathematics.
The departmental committee for the DMP grants admission to the program, usually
at the end of the student’s fourth semester. Criteria for acceptance into
the program are based on the GPA in mathematics, letters of recommendation from
mathematics instructors, and the cumulative GPA in the College (which should
be near 3.400 or higher).
The DMP is the same as the graduate school preparatory concentration, except that in the fourth year the students take the seminar course MATH 583 in which they give an hour lecture and prepare a written exposition of their work in the seminar under faculty guidance. As with the concentrations, the DMP must consist of at least nine courses.
Three levels of distinction are possible: distinction, high
distinction, or highest distinction. The departmental recommendation for the
level of distinction to be awarded is based on the quality of the student’s
seminar presentations, the overall work in the DMP, and the entire major program,
as well as the student’s College GPA.
Requirements for Minor in Mathematics Students who wish
to declare a minor in mathematics must complete the calculus sequence through
MATH 231 or its equivalent with at least a 2.000 average.
To graduate with a minor in mathematics a student must complete five courses approved by the department of mathematics with minimum grades of C in three of the courses and minimum grades of C in the other two. An approved course must carry at least three credits. Currently, the approved courses are those from the College department of mathematics with the MATH mnemonic numbered 300 or higher. Either Math 331 or Math 354 should be one of the five approved courses. Courses with the STAT mnemonic or from other departments or institutions can be taken if approved by the undergraduate committee.
Up to two courses that are being counted for another College major or another College minor can also be counted for the minor in mathematics.
Up to two courses that are taken from outside the University and which are equivalent to College Mathematics courses may be offered for the College mathematics minor.
Echols Mathematics Club is an undergraduate club for
mathematics students that sponsors lectures, mathematics films, problem solving
sessions for the Putnam Mathematical Competition and other similar activities.
Additional Information For more information, contact John Faulkner, Lower Division Advisor, Room 325, 9244942, or Thomas Kriete, Upper Division Advisor, Room 205, 9244932, Kerchof Hall, P.O. Box 400137, Charlottesville, VA 229044137; www.math.virginia.edu.
Course Descriptions
Mathematics
The entering College student has a variety of courses in mathematics
from which to choose. Among those that may be counted toward the College area
requirement in natural science and mathematics, are several options in calculus,
elementary (noncalculus based) courses in probability and in statistics, and
courses dealing with computer techniques in mathematics.
MATH 103 (precalculus) is available for students who need to
improve basic skills that are required in other courses such as calculus, chemistry,
psychology, economics, and statistics. However, it may not be counted toward
the area requirement in natural science and mathematics. Students planning to
major in the social sciences, arts, or humanities who wish to take a mathematics
course but omit the study of calculus may choose from MATH 111 (Elementary Probability
Theory) and MATH 114 (Financial Mathematics). Even though it is not a prerequisite
for STAT 112, MATH 111 is frequently taken prior to STAT 112. MATH 115 and 116
are introductory courses that investigate familiar areas of elementary mathematics
at a profound level and are intended for first and secondyear nonmajors,
especially those preparing to teach in elementary and middle schools.
In MATH 114, the students study the mathematics needed to understand
and answer a variety of questions that arise in everyday financial dealings.
The emphasis in this course will be on applications, including simple and compound
interest, valuation of bonds, rates of return on investments, and more. Although
the topics in this course are drawn primarily from business and economics, students
of all majors are welcome and should find the applications interesting and relevant.
The study of calculus is the foundation of college mathematics
for students planning to major in mathematics or the physical sciences or anticipating
a career or graduate study in any of the natural sciences, engineering, or applied
social sciences (such as economics). There are essentially two programs of study
available in calculus:
1. MATH 121, 122 is a terminal oneyear sequence intended for business,
biology, and social science majors;
2. MATH 131, 132, 231 is the traditional calculus sequence intended for
students of mathematics and the natural sciences, as well as for students
intending to pursue graduate work in the applied social sciences;
The MATH 121, 122 sequence is unacceptable as a prerequisite
for mathematics courses numbered 231 and above. Students anticipating the need
for higher mathematics courses such as MATH 325 (Differential Equations) or
MATH 310, 312 (Probability and Statistics) should instead elect the MATH 131,
132, 231 sequence. Credit is not allowed for both MATH 121 and 131 (or its equivalent).
Students who have previously passed a calculus course in high
school may elect MATH 122, 131, 132, or 231 as their first course, depending
on placement, preparation, and interest. A strong high school calculus course
is generally adequate preparation for MATH 132 as a first calculus course, even
if advanced placement credit has not been awarded for MATH 131. Students planning
to take any advanced course in mathematics should not take MATH 122, because
credit for that course must be forfeited if the student takes MATH 132 (or its
equivalent).
MATH 133 and 134 is a twosemester calculus workshop sequence
taken in conjunction with specific sections of MATH 131 and 132. Participants
in the calculus workshop meet for six hours per week to work in small groups
on challenging problem sets related to material covered in MATH 131 and 132.
They typically enjoy getting to work closely with fellow calculus students,
and find that their performance in MATH 131 and 132 is significantly improved.
Permission is required to sign up for the calculus workshop. For more information,
contact Professor Jeffrey Holt, Calculus Workshop Coordinator; 9244927; jjh2b@
virginia.edu.
Exceptionally wellprepared students (who place out of both
MATH 131 and 132) may choose either MATH 231 or 325 (Differential Equations)
as their first course.
Advanced placement credit in the calculus sequence is granted
on the basis of the College Entrance Examination Board Advanced Placement Test
(either AB or BC). A score of 4 or 5 on the AB test or on the AB subscore of
the BC test gives the student credit for MATH 131. A score of 4 or 5 on the
BC test gives the student credit for both MATH 131 and 132. The Department of
Mathematics offers short advisory online placement tests during fall orientation.
Precommerce students are required to take a statistics course,
usually STAT 112, and one other mathematics course, usually MATH 111, 121, 122,
or MATH 131.
Substitutions There are numerous instances
of equivalent courses offered by
the
Department
of Mathematics as well as by the Department of Applied Mathematics in the School
of Engineering and Applied Science. A student may not offer for degree credit
two equivalent courses (e.g., MATH 131 and APMA 111, or MATH 131 and MATH 121).
Up to two courses, taken from outside of the University and which are equivalent
to College Mathematics courses, may be offered for the College Mathematics
major or minor. The following are equivalent courses from the School of Engineering
and Applied
Sciences:
 APMA 213 and MATH 325 Ordinary Differential Equations
 APMA 302 and MATH 404 Discrete Mathematics
 APMA 308 and MATH 351 Elementary Linear Algebra
 APMA 310 and MATH 310 Introduction to Mathematical Probability
 APMA 507 and MATH 430 Elementary Numerical Analysis
 SYS 321 and MATH 408 Operations Research
Course Descriptions
MATH 103  (3) (Y)
Precalculus
Prerequisite: High school algebra II and geometry.
Studies computational
skills, patterns of quantitative problem solving, and mathematical thought. Includes
linear and quadratic equations,
polynomials, inverse functions, logarithms, arithmetic and geometric sequences,
trigonometric functions, and linear systems. (Does not satisfy the College natural
science and mathematics requirement.)
MATH 111  (3) (S)
Probability/Finite Mathematics
Studies finite probability theory including
combinatorics, equiprobable models, conditional probability and Bayes’ theorem,
expectation and variance, and Markov chains.
MATH 114  (3) (Y)
Financial Mathematics
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 115  (3) (IR)
The Shape of Space
Provides an activity and projectbased 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 116  (3) (IR)
Algebra, Number Systems, and Number Theory
Studies basic concepts, operations, and structures occurring
in number systems, number theory, and algebra. Inquirybased student investigations
explore historical developments and conceptual transitions in the development
of number and algebraic systems.
MATH 121  (3) (S)
Applied Calculus I
Topics include limits and continuity; differentiation and integration
of algebraic and elementary transcendental functions; and applications to maximumminimum
problems, curve sketching and exponential growth. Credit is not given for both
MATH 121 and 131.
MATH 122  (3) (S)
Applied Calculus II
Prerequisite: MATH 121 or equivalent.
A second calculus course for
business, biology, and social science students. Analyzes functions of several
variables, their graphs, partial
derivatives and optimization; multiple integrals. Reviews basic single variable
calculus and introduces differential equations and infinite series. Credit is
not given for both MATH 122 and 132.
MATH 131  (4) (S)
Calculus I
Prerequisite: Background in algebra, trigonometry, exponentials,
logarithms, and analytic geometry.
Introduces calculus with emphasis on techniques
and applications. Recommended for natural science majors and students planning
additional work
in mathematics. The differential and integral calculus for functions of a single
variable is developed through the fundamental theorem of calculus. Credit is
not given for both MATH 121 and 131.
MATH 132  (4) (S)
Calculus II
Prerequisite: MATH 131 or equivalent, or instructor
permission.
Continuation of 131. Applications of the integral, techniques
of integration, infinite series, vectors. Credit is not given for both MATH
122 and 132.
MATH 133  (2) (Y)
Calculus Workshop I
Prerequisite: Instructor permission; corequisite:
MATH 131.
Intensive calculus problemsolving workshop with topics drawn
from MATH 131.
MATH 134  (2) (Y)
Calculus Workshop II
Prerequisite: Instructor permission; corequisite:
MATH 132.
Intensive calculus problemsolving workshop with topics drawn
from MATH 132.
MATH 231  (4) (S)
Calculus III
Prerequisite: MATH 132 or its equivalent.
Studies functions of several
variables including lines and planes in space, differentiation of functions of
several variables, maxima and
minima, multiple integration, line integrals, and volume.
MATH 310  (3) (Y)
Introduction to Mathematical Probability
Prerequisite: MATH 132. A knowledge of double integrals
is recommended.
Includes sample spaces, combinatorial analysis, discrete and
continuous random variables, classical distributions, expectation, Chebyshev
theorem, independence, central limit theorem, conditional probability, and
generating functions.
MATH 312  (3) (Y)
Introduction to Mathematical Statistics
Prerequisite: MATH 310.
Includes sampling theory, point estimation,
interval estimation, testing hypotheses (including the NeymanPearson lemma and
likelihood ratio
tests), and regression and correlation.
MATH 325  (4) (S)
Ordinary Differential Equations
Prerequisite: MATH 132 or its equivalent.
Introduces the methods, theory,
and applications of differential equations. Includes firstorder, second and
higherorder linear equations, series
solutions, linear systems of firstorder differential equations, and the associated
matrix theory. May include numerical methods, nonlinear systems, boundary value
problems, and additional applications.
MATH 325P  (4) (S)
Ordinary Differential Equations
Prerequisite: MATH 132 or its equivalent.
Usually offered in the spring,
this course covers the same material as MATH 325 with some additional topics,
including an introduction
to SturmLiouville theory, Fourier series and boundary value problems, and
their connection with partial differential equations. Physics majors should enroll
in MATH 325P, although no knowledge of physics is assumed.
MATH 331  (3) (S)
Basic Real Analysis
Prerequisite: MATH 132.
Concentrates on proving the basic theorems
of calculus, with due attention to the beginner with little or no experience
in the techniques
of proof. Includes limits, continuity, differentiability, the BolzanoWeierstrass
theorem, Taylor’s theorem, integrability of continuous functions, and uniform
convergence.
MATH 334  (3) (Y)
Complex Variables with Applications
Prerequisite: MATH 231.
Topics include analytic functions, Cauchy formulas,
power series, residue theorem, conformal mapping, and Laplace transforms.
MATH 351  (3) (S)
Elementary Linear Algebra
Prerequisite: MATH 132.
Includes matrices, elementary row operations,
inverses, vector spaces and bases, inner products and GramSchmidt orthogonalization,
orthogonal
matrices, linear transformations and change of basis, eigenvalues, eigenvectors,
and symmetric matrices.
MATH 354  (3) (Y)
Survey of Algebra
Prerequisite: MATH 132 or equivalent.
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.
MATH 404  (3) (E)
Discrete Mathematics
Prerequisite: MATH 354 or instructor permission.
Includes combinatorial
principles, the binomial and multinomial theorems, partitions, discrete probability,
algebraic structures, trees, graphs,
symmetry groups, Polya’s enumeration formula, linear recursions, and generating
functions.
MATH 408  (3) (Y)
Operations Research
Prerequisite: MATH 132 and 351.
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.
MATH 430  (3) (IR)
Elementary Numerical Analysis
Prerequisite: MATH 325 and computer proficiency.
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 leastsquare approximation.
MATH 452  (3) (IR)
Algebraic Coding Theory
Prerequisite: MATH 351 and 354, or instructor permission.
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, ReedMuller
codes, algebraic geometry codes, and connections with groups, designs, and lattices.
MATH 475  (3) (IR)
Introduction to Knot Theory
Prerequisite: MATH 354, or instructor permission.
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.
MATH 493  (3) (IR)
Independent Study
Reading and study programs in areas of interest to individual
students. For third and fourthyears interested in topics not covered in regular
courses. Students must obtain a faculty advisor to approve and direct the program.
MATH 495  (3) (IR)
Undergraduate Research Seminar
Prerequisite: Instructor permission.
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.
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
calculus in the 17th century, and how those ideas were
perfected and extended by succeeding generations of mathematicians. Emphasizes
primary source materials when possible.
MATH 503  (3) (O)
The History of Mathematics
Prerequisite: MATH 231 and 351, or instructor permission.
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 when possible.
MATH 504  (3) (E)
Discrete Mathematics
Prerequisite: graduate standing.
Includes combinatorial principles,
the binomial and multinomial theorems, partitions, discrete probability, algebraic
structures, trees, graphs,
symmetry groups, Polya’s enumeration formula, linear recursions, and generating
functions.
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.
Includes sorting problems,
recursive algorithms, and dynamic data structures.
MATH 510  (3) (Y)
Mathematical Probability
Prerequisite: Graduate standing and MATH 132, or equivalent.
Those who have received credit for MATH 310 may not take 510 for credit.
Studies
the development and analysis of probability models through the basic concepts
of sample spaces, random variables, probability distributions,
expectations, and conditional probability. Also includes 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
theory selected from Random walks, Markov processes, Brownian motion, Poisson
processes, branching processes, stationary
time series, linear filtering and prediction, queuing process, and renewal
theory.
MATH 512  (3) (Y)
Mathematical Statistics
Prerequisite: MATH 510 and graduate standing.
Topics include methods
of estimation, general concepts of hypothesis testing, linear models and estimation
by least squares, categorical data, and
nonparametric statistics. Those who have received credit for MATH 312 may not
take 512 for credit.
MATH 514  (3) (Y)
Mathematics of Derivative Securities
Prerequisite: MATH 231 or 122 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 521  (3) (Y)
Advanced Calculus with Applied Mathematics
Prerequisite: MATH 231, 325; 351 recommended.
Topics include vector
analysis, Green’s, Stokes’,
divergence theorems, conservation of energy, potential energy functions. Emphasis
on physical interpretation. Also includes SturmLiouville problems, Fourier
series, special functions, orthogonal polynomials, and Green’s functions.
MATH 522  (3) (Y)
Partial Differential Equations and Applied Mathematics
Prerequisite: MATH 521.
Introduces complex variables and partial differential
equations. Topics include analytic functions, complex integration, power series,
residues,
conformal mapping; separation of variables, boundary value problems, Laplace’s
equation, wave equation, and heat equation.
MATH 525  (3) (IR)
Dynamical Systems
Prerequisite: MATH 231, 325, 351, or instructor permission.
Studies
the qualitative geometrical theory of ordinary differential equations. Topics
include basic wellposedness (existence, uniqueness, continuation
of solutions, dependence on parameters, comparison theory); linear and periodic
systems (Floquet theory); stability theory (Lyapunov’s method and invariance
theory, domain of attraction, comparison principle); perturbation of linear
systems; center manifold theorem; periodic solutions and Poincare´Bendixson
theory; Hopf bifurcation; introduction to chaotic dynamics; control theoretic
questions; and differentialgeometric methods (Lie theory).
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: graduate standing.
Topics include 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
groups, rings, and fields, and presents applications to other areas of mathematics,
such as 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,
including an introduction to the theory of 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
Topics selected from analytic geometry, affine geometry, projective
geometry, and hyperbolic and nonEuclidean geometry.
MATH 572  (3) (E)
Introduction to Differential Geometry
Prerequisite: MATH 231 and 351, or instructor permission.
Topics selected
by the instructor from 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) (IR)
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.
In exceptional
circumstances, a student may undertake 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.
