Department of Mathematics
P.O. Box 400137
University of Virginia
Charlottesville, VA 229044137
Phone: (434) 9244919
Fax: (434) 9823084
www.virginia.edu/~woodson
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 75 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 wellequipped
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
2.2 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 or 120, or an approved equivalent
course. This should be done as early as possible.
To help guide the student through the major, the mathematics department
offers six concentrations. Completion of one of these concentrations
is required. Each concentration contains a set of nine or ten required
courses (approximately 28 credit hours). To graduate, a student
must obtain minimum grades of C in seven of these courses and C
in the other two.
Certain substitutions are allowed in all options, for example,
MATH 531 for MATH 331 and MATH 551 for MATH 354.
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 
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
351 
Elementary
Linear Algebra 
3

MATH
531 
Intro.
to Real Analysis I 
3

MATH
551 
Intro.
to Abstract Algebra I 
3

MATH
552 
Intro.
to Abstract Algebra II 
3

Three electives at the 300 level or higher.(You may wish to take
MATH 354 in preparation for MATH 551 and MATH 331 in preparation
for MATH 531.)
This constitutes the minimum expected of an incoming graduate student
in most programs nationwide. The department strongly recommends
MATH 532 (Real Analysis in Several Variables), as well as courses
in differential geometry and topology (MATH 572 and 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, but is not limited to, probability, statistics, regression,
time series, partial differential equations, stochastic processes,
stochastic calculus, numerical methods, and analysis. Probability
and statistics and some acquaintance with numerical methods are
essential as is some knowledge of economics/accounting and some
computing experience. Additional background in statistics, optimization,
and stochastic processes is also desirable. 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

Two
courses chosen from^{(1)}: 
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. Actuarial Concentration This concentration offers some
of the basic mathematics and statistics necessary for a successful
career in actuarial science, and it provides some of the academic
background needed to pass the first few actuarial exams.
Actuaries use mathematics, statistics, and financial theory to
analyze future events, especially those related to insurance and
pension programs. They may work for insurance companies, consulting
firms, government, employee benefits departments of large organizations,
banks, investment firms, or more generally, businesses that need
to assess the financial consequences of risk.
To become an actuary, the student must pass a series of examinations
administered by the professional actuarial societies: the Society
of Actuaries (SOA) and the Casualty Actuarial Society (CAS). The
first few exams are jointly administered. Exams which correspond
to various courses are indicated below.
The program consists of nine courses as follows:
MATH 325^{(1)} 
Ordinary Differential Eq. 
4 
MATH 310^{(2)} 
Intro. to Mathematical Probability 
3 
MATH 312^{(2)} 
Intro. to Mathematical Statistics 
3 
MATH 331 
Basic Real Analysis 
3 
MATH 351 
Elementary Linear Algebra 
3 
MATH
354 
Survey
of Algebra 
3

and
either: 
both 
MATH 517/
STAT 540^{(3)} 
Actuarial Mathematics/Statistics 
and 
STAT
541^{(4)} 
Actuarial
Risk Theory 
6

and one
elective from the list below 
3

or: 
one
of either 
MATH 517/
STAT 540^{(3)} 
Actuarial Mathematics/Statistics 
or 
STAT
541^{(4)} 
Actuarial Risk Theory 
3

and two
electives from the list below 
6

Electives
(all strongly recommended) 
MATH 408^{(5)} 
Operations
Research 
3

MATH
514 
Mathematics
of Derivative Securities 
3

MATH
430^{(6)} 
Elementary
Numerical Analysis 
3

MATH
511 
Stochastic
Processes 
3

STAT
512^{(7)} 
Applied
Linear Models 
3

STAT
517^{(9)} 
Applied
Time Series 
3

(1) Part of exam 100, offered jointly by the SOA
and CAS.
(2) Exam 110, offered jointly by the SOA and CAS.
(3) Required SOA exams 140, 150/CAS exam 4A.
(4) Required SOA exam 151/CAS exam 5A.
(5) Elective SOA exam 130.
(6) Elective SOA exam 135/required CAS
exam 3C.
(7) Onethird of required SOA exam 120/CAS exam
3A.
(8) Onehalf of required SOA exam 160.
(9) Twothirds of required SOA exam 120/ CAS exam
3A.
It is highly advantageous for students interested in this concentration
to take both MATH 310 and 312 in their second year. Actuarial Mathematics
(MATH 517/STAT 540) and Actuarial Risk Theory (STAT 541) form the
core of the actuarial program. Both of these courses are offered
every year if there is sufficient student interest, and otherwise
in alternate years. With sufficient early course preparation, a
summer internship after the third year has been an integral part
of the program for those students who wished to intern.
Other courses which are recommended but not required include:
ECON
201 
Microeconomics 
3

ECON
202 
Macroeconomics 
3

ECON
301 
Intermediate
Microecon. 
3

ECON
302 
Intermediate
Macroecon. 
3

ECON
434 
Theory
of Financial Markets 
3

ECON
435^{(9)} 
Corporate
Finance 
3

STAT
514^{(10)} 
Survival
Analysis and Reliability Theory 
3

(9) Required SOA exam 230/required CAS exam 5A.
(10) SOA exam 160.
F. 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
501 
History
of Calculus or 
MATH
503 
History
of Mathematics 
3

MATH
570 
Introduction
to Geometry 
3

One
elective at the 300 level or higher 
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. Admission to the program is granted by the
departmental committee for the DMP, 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 3.4 or higher).
The DMP is the same as the graduate school preparatory concentration,
except that in the fourth year the students also 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.
Note that MATH 531 and 551 are prerequisites for the seminar. 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 2.0
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. Courses
with the STAT mnemonic or from other departments or institutions
can be taken if approved by the undergraduate committee.
Courses that are being counted for a major or another minor cannot
also be counted for the minor in mathematics.
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 Charles Dunkl, Lower Division Advisor, Room 223, 9244939,
or Thomas Kriete, Upper Division Advisor, Room 205, 9244932, Kerchof
Hall, Charlottesville, VA 229044137; www.math.virginia.edu.

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 108 (Modes of Mathematical
Thinking) or MATH 111 (Elementary Probability Theory). 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:
MATH 121, 122 is a terminal oneyear sequence intended for business,
biology, and social science majors; 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 two semester 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 well prepared 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. Students who wish to enter the calculus sequence
but who have not received advanced placement credit should consult
the Student Handbook for placement guidelines based on grades and
achievement test scores. The Department of Mathematics offers short
advisory 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.
Warning 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 101, or MATH 131 and MATH 121).
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 108  (3) (IR)
Modes of Mathematical Thinking
Studies logic, number systems, functions, analytic geometry, equations,
matrices, enumeration, computer algebra systems. Intended for liberal
arts students and emphasizes the connection between analyticalgebraic
and geometric reasoning in the understanding of mathematics. Facilitated
by the use of a modern computer algebra system, such as Maple.
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 121S  (4) (IR)
Introduction to Calculus
Prerequisite: Instructor permission.
Includes limits and continuity; differentiation and integration
of algebraic and elementary transcendental functions; and applications
to maximumminimum problems, curve sketching and exponential growth.
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 300  (3) (IR)
Foundations of Analysis
Prerequisite: MATH 132 or equivalent.
Topics from logic and the construction of mathematical proofs, basic
set theory, number systems, continuity of functions, and foundations
of analysis. Intermediate introduction of the standards of mathematical
rigor and abstraction that are encountered in advanced mathematics,
based on the material of the calculus and other basic mathematics.
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 334/534  (3) (Y)
Complex Variables With Applications
Prerequisite: MATH 231 and graduate standing for MATH 534. 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 404/504  (3) (E)
Discrete Mathematics
Prerequisite: MATH 354 or instructor permission, and graduate standing
for MATH 504.
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 453  (3) (O)
Number Theory
Prerequisite: MATH 354 or instructor permission. Includes congruences,
quadratic reciprocity, Diophantine equations, and numbertheoretic
functions, among others.
MATH 475  (3) (IR)
Introduction to Knot Theory
Prerequisite: MATH 331, 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 fourth years 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 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 517  (3) (IR)
Actuarial Mathematics
Prerequisite: MATH 312 or 512, instructor permission.
Covers the main topics required by students preparing for the examinations
in actuarial statistics, set by the American Society of Actuaries.
Topics include life tables, life insurance and annuities, survival
distributions, net premiums and premium reserves, multiple life
functions and decrement models, valuation of pension plans, insurance
models, benefits, and dividends.
MATH 521  (3) (Y)
Advanced Calculus 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, 532  (3) (Y)
Introduction to Real Analysis I, II
Prerequisite: MATH 231, 351.
Includes the basic topology of Euclidean spaces, continuity, and
differentiation of functions on Euclidean spaces; RiemannStieltjes
integration, convergence of sequences and series of functions; and
equicontinuous families of functions, Weierstrass theorem, inverse
function theorem and implicit function theorem, integration of differential
forms, and Stokes' Theorem.
MATH 551, 552  (3) (Y)
Introduction to Abstract Algebra I, II
Prerequisite: MATH 351 or instructor permission.
Introduces algebraic systems: groups, rings, fields, vector spaces
and their general properties; subsystems, quotient systems, homomorphisms.
Includes permutation groups, polynomial rings, and groups and rings
of matrices. Additional topics may include applications to linear
algebra and number theory.
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) (Y)
Introduction to Geometry
Prerequisite: MATH 231 and 351 or instructor permission.
Topics selected from analytic geometry, affine geometry, projective
geometry, hyperbolic and nonEuclidean geometry.
MATH 572  (3) (IR)
Introduction to Differential Geometry
Prerequisite: MATH 231.
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.
Includes topological spaces and continuous functions; product and
quotient topologies; compactness and connectedness; separation and
metrication; and the fundamental group and covering spaces.
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.
