Department of Mathematics
376 Fretwell Building
704-687-2580
http://www.math.uncc.edu/grad/
Mathematics
Degrees
M.S.,
Ph.D.
Mathematical
Finance Degree
The
Department of Mathematics is one of the participating departments in the
Inter-College Master of Science in Mathematical Finance program. See the
Mathematical Finance entry in the Inter-College Graduate Programs section of
this Catalog for complete information
and program requirements.
Coordinator
for Mathematics
Dr. Joel
D. Avrin
Mathematics
Education Degree
M.A.,
Ph.D. in C&I: Math Ed Specialization
Coordinator
for Mathematics Education
Dr. Victor
V. Cifarelli
Graduate
Faculty
Robert Anderson
Joel Avrin
Animikh Biswas
Charles Burnap
Wei Cai
Zongwu Cai
Victor V. Cifarelli
Ming Dai
Xingde Dai
Yuanan Diao
Jacek Dmochowski
Alan Dow
Yuri Godin
Mary Kim Harris
Gabor Hetyei
Evan G. Houston
Phillip Johnson
Janusz Kawczak
Mohammad A. Kazemi
Michael V. Klibanov
Alan L. Lambert
Thomas G. Lucas
Thomas R. Lucas
Stanislav Molchanov
Wanda Nabors
Hae-Soo Oh
Alex S. Papadopoulos
Joseph E.
Quinn
Franz Rothe
David C. Royster
Adalira Sáenz-Ludlow
Douglas S. Shafer
Isaac M. Sonin
Nicholas M. Stavrakas
Yanqing Sun
Rajeshwari Sundaram
Boris R. Vainberg
Barnet Weinstock
Volker Wihstutz
Mingxin
Xu
Alexander Yushkevich
Zhi Yi Zhang
You Lan Zhu
MASTER OF SCIENCE IN MATHEMATICS
The Master
of Science Degree in Mathematics is organized into three concentrations: the
concentration in General Mathematics, the concentration in Applied Mathematics,
and the concentration in Applied Statistics. The concentration in General
Mathematics is a robust but flexible program that allows a student to develop a
broad background in Mathematics ranging over a variety of courses chosen from
both pure and applied areas, or to tailor a program toward a particular focus
that may not be as closely covered by our other degree concentrations, e.g. one
that is interdisciplinary in nature. The concentration in Applied Mathematics
develops analytical and computational skills focused toward applications of
mathematics in the physical sciences as encountered in industry, government,
and academia. The concentration in Applied Statistics provides theoretical
understanding of, and training in, statistical methods applicable to particular
areas of business, industry, government, and academia.
All
candidates, regardless of which concentration is chosen, are required to take
MATH 5143-5144 or STAT 5126-5127; MATH 7691 (or in the case of the General
Mathematics concentration, a suitable/approved 7000 level course); and a
comprehensive exam. Students may also choose a thesis option for 3-6 credit
hours towards the required semester hour
total.
Concentration In
General Mathematics
The Master
of Science degree concentration in General Mathematics is designed both to
provide advanced skills and knowledge for persons seeking either positions in
industry or in government, or teaching positions at the community college
level, and to provide professional development to persons currently in such
positions. Graduates are also prepared to enter directly into at least the
second year of a Ph.D. program in mathematics, applied mathematics or
statistics, depending on the particular course of study.
Additional
Admission Requirements
In addition
to the general requirements for admission to the
1) Applicants must present evidence of
the satisfactory completion of at least 27 semester hours of mathematics approved
by the department Graduate Committee.
2) A satisfactory score is required on
at least the Quantitative portion of the Graduate Record Examination.
3) It is recommended that the student
have a basic knowledge of at least two of the areas of algebra, real analysis
and topology.
Concentration
Requirements
The Master
of Science degree concentration in General Mathematics requires successful
completion of at least 30 semester hours of graduate work approved by the
department Graduate Committee including: MATH 5143 and 5144 or their
equivalents, at least one course each from two of the groups I, II, III, and V,
and at least 15 hours in 7000-level courses. No credit shall be given for
6000-level math courses. With the approval of the department Graduate Committee,
one 3-hour, non-thesis 6000-level course in computer science of a theoretical
nature may be applied toward the 15 hours. Candidates for the degree
concentration must demonstrate, to the satisfaction of the department Graduate
Committee, competence on general knowledge in at least three of five groupings
of courses listed below. This may be accomplished by (a) successful performance
on a written comprehensive examination or (b) successful completion of courses
in these areas.
Group I Applied Mathematics
OPRS5111 Linear Programming (3)
OPRS5112 Non-Linear Programming (3)
OPRS5113 Game Theory (3)
OPRS5114 Dynamic Programming (3)
MATH5165 Numerical Linear Algebra (3)
MATH5172 The Finite Element Method (3)
MATH5173 Ordinary Differential Equations (3)
MATH5174 Partial Differential Equations (3)
MATH5176 Numerical Methods for Partial Differential Equations (3)
MATH7172 Partial Differential Equations (3)
MATH7176 Advanced Numerical Analysis (3)
MATH7177 Applied Optimal Control (3)
MATH7178 Comp. Methods for Fluid Dynamics (3)
MATH7273 Advanced Finite Element Analysis (3)
Group
II
Probability-Statistics
STAT5123 Applied Statistics I (3)
STAT5124 Applied Statistics II (3)
STAT5126 Theory of Statistics I (3)
STAT5127 Theory of Statistics II (3)
STAT7027 Topics in Statistics (3)
STAT7122 Advanced Statistics I (3)
STAT7123 Advanced Statistics II (3)
STAT7127 Linear Statistical Models (3)
STAT7133 Multivariate Analysis (3)
MATH5128 Applied Probability I (3)
MATH5129 Applied Probability II (3)
MATH7120 Probability Theory I (3)
MATH7121 Probability Theory II (3)
MATH7125 Stochastic Processes (3)
Group III Algebra-Topology
MATH5163 Modern Algebra (3)
MATH5164 Abstract Linear Algebra (3)
MATH5181 Introduction to Topology (3)
MATH7163 Modern Algebra I (3)
Group IV Analysis
MATH5143 Analysis I (3)
MATH5144 Analysis II (3)
MATH7141 Complex Analysis I (3)
MATH7143 Real Analysis I (3)
MATH7144 Real Analysis II (3)
Group V Computer Science
All 5000- and 6000-level Computer
Science courses.
Assistantships
A number
of graduate assistantships are available each year (with nationally-competitive
stipends) for qualified applicants. A limited number of fellowship awards can
be applied to supplement these stipends for especially qualified students.
Thesis
Completion
of a thesis is optional. With the approval of the department Graduate
Committee, a candidate may receive up to six of the 15 hours required at the
7000 level for the writing of a master's thesis on an approved topic. This
thesis may be original work, work of an expository nature, or the mathematical
formulation and solution of a particular industrial or business problem
suggested by the career interests of the student. A candidate may receive no
more than six of the hours required at the 7000 level for course and thesis work
in computer science. If the thesis option is elected, the candidate will be
required to defend his/her thesis in an oral examination.
Comprehensive
Examination
A
candidate must perform satisfactorily on an oral comprehensive examination over
his/her program of study.
Concentration In Applied Mathematics
The Master of Science degree concentration in Applied Mathematics
is designed to develop critical thinking and intuition, and to provide advanced
work in the techniques of mathematical analysis and their application to the
problems of industry and technology. Skills are developed to deal with problems
encountered in industry, business, and governmental work; to hold leadership
positions in industry or government work; to teach Applied Mathematics at the
undergraduate or community college level; and to study Applied Mathematics
leading to the Ph.D. degree.
Concentration
Requirements
A
candidate for the Master of Science degree concentration in Applied Mathematics
must complete at least 30 semester hours of graduate work approved by the
department Graduate Committee to include:
Core Requirements (21 semester hours)
1) MATH5143 Analysis I (3)
MATH5144 Analysis II (3)
MATH5165 Numerical Linear
Algebra (3)
2) One elective in Numerical Analysis
selected from:
MATH5172 The Finite Element Method (3)
MATH5176 Numerical Methods for Partial Differential Equations (3)
3) One elective in Advanced Analysis
selected from:
MATH7141 Complex Analysis I (3)
MATH7143 Real Analysis I (3)
MATH7144 Real Analysis II (3)
4) Two electives in Advanced Applied
Mathematics selected from:
MATH7172 Partial Differential Equations (3)
MATH7176 Advanced Numerical Analysis (3)
MATH7177 Applied Optimal Control (3)
MATH7178 Computational Methods for Fluid Dynamics (3)
MATH7273 Adv. Finite Element Analysis. (3)
Electives (6 semester hours)
1) One advanced elective from:
MATH7141 Complex Analysis I (3)
MATH7143 Real Analysis I (3)
MATH7144 Real Analysis II (3)
MATH7172 Partial Differential Equations (3)
MATH7176 Advanced Numerical Analysis (3)
MATH7177 Applied Optimal Control (3)
MATH7178 Computational Methods for Fluid Dynamics (3)
MATH7273 Adv. Finite Element Analysis (3)
MATH7893 Thesis (0-3)
2) One elective in Mathematics or a
suitable area of application to be selected with the approval of the student's
adviser. Suggested electives include:
OPRS5113 Game Theory (3)
STAT5123 Applied Statistics I (3)
MEGR4111 Heat Transfer (3)
MEGR4112 Intermediate Fluid Mechanics (3)
MEGR6113 Adv. Conductive Heat Transfer (3)
MEGR6141 Theory of Elasticity II (3)
Research Seminar (3 hours)
All
candidates for the degree concentration must complete three hours of MATH 7691
(Research Seminar) in which they carry out an independent project under the
supervision of a member of the graduate faculty. The project could involve a specific
application to a concrete problem of techniques identified in the literature or
studied in other courses. All projects are subject to prior approval of the
department Graduate Committee and must be successfully defended before a
committee of three graduate faculty members appointed by the department
Graduate Committee.
Assistantships
A number
of graduate assistantships are available each year (with nationally-competitive
stipends) for qualified applicants. A limited number of fellowship awards can be
applied to supplement these stipends for especially qualified students.
Thesis
A student
may choose to expand the work begun in MATH 7691 into a master's thesis by
registering for three hours of MATH 7893 to fulfill the advanced elective
requirement (1) described above. This thesis option affords the student the
opportunity to do professional/scholarly work demonstrating proficiency in the
area of Applied Mathematics.
Comprehensive Examination
Each
candidate for the degree concentration in Applied Mathematics must perform
satisfactorily on a final comprehensive examination. This examination will be
set and administered by a committee appointed by the department Graduate
Committee. It may be either in written or oral form, and it will cover those
areas of study and/or research deemed appropriate by the committee.
Concentration In
Applied Statistics
The Master
of Science degree concentration in Applied Statistics is designed to provide
advanced skills and knowledge in the planning, design, testing, and implementation
of statistical methods. Skills are developed to deal with problems encountered
in statistical applications in business, industry and government; to hold
administrative positions requiring planning and implementation of statistical
analysis; to teach statistics at the undergraduate or community college level;
and to study statistics leading to the Ph.D. degree.
Additional Admission Requirements
In
addition to the general requirements for admission to the
1) An overall GPA of at least 3.0 on
all previous college work including a GPA of at least 3.0 in courses
prerequisite to the area of applied statistics.
2) Evidence of undergraduate
preparation in mathematics and computer science including: 12 semester hours of
calculus at the level of MATH 1241/1242/2241/2242; 3 semester hours of linear
algebra at the level of MATH 2164; 3 semester hours of differential equations
at the level of MATH 2171; 6 semester hours of probability and statistics at
the level of MATH 3122/3123; and 3 semester hours of computer programming at
the level of CSCI 1100 or 1214 and its lab.
Degree Requirements
A
candidate for the Master of Science degree concentration in Applied Statistics
must complete a minimum of 33 semester hours of graduate work approved by the
department Graduate Committee including:
Core Requirements (24 semester hours)
STAT5123 Applied Statistics I (3)
STAT5124 Applied Statistics II (3)
STAT5126 Theory of Statistics I (3)
STAT5127 Theory of Statistics II (3)
STAT7027 Topics in Statistics (3)
STAT7127 Linear Statistical Models (3)
STAT7133 Multivariate Analysis (3)
MATH7691 Research Seminar (1-3)
Electives
(9 semester hours)
1) Two course selected from among:
STAT7027 Topics in Statistics (3)
MATH5128 Applied Probability I (3)
MATH5129 Applied Probability II (3)
MATH5143 Analysis I (3)
MATH5165 Numerical Linear Algebra (3)
MATH7120 Probability Theory I (3)
MATH7121 Probability Theory II (3)
MATH7143 Real Analysis I (3)
MATH7692 Research Seminar (3)
OPRS5111 Linear Programming (3)
OPRS5112 Non-linear Programming (3)
OPRS5113 Game Theory (3)
OPRS5114 Dynamic Programming (3)
2) Any MATH/STAT/OPRS course at the
7000 level.
Students
who, because of their undergraduate work or other experience, can demonstrate
sufficient knowledge of the material in one or more of the core courses may be
exempted from taking the course or courses. Exemption from a course carries no
credit towards the degree concentration.
Research Seminar and Thesis Option (3 semester hours)
All
candidates for the Master of Science degree concentration in Applied Statistics
are required to complete 3 hours of MATH 7691 (Research Seminar) in which they
carry out an independent project under the supervision of a member of the
graduate faculty. The project could involve a specific application of
techniques identified in the literature or studied in other courses. All
projects are subject to the prior approval of the department Graduate Committee
and must be successfully defended before a committee of three graduate faculty
members appointed by the department Graduate Committee.
A student
may choose to expand the work begun in MATH 7691 (Research Seminar) into a
Master's Thesis by registering for 3 hours of MATH 7893 (Thesis) to fulfill the
elective requirement under (2) above. This thesis option affords the student
the opportunity to do professional and scholarly work demonstrating proficiency
in the area of applied statistics.
Assistantships
A number
of graduate assistantships are available each year (with nationally-competitive
stipends) for qualified applicants. A limited number of fellowship awards can
be applied to supplement these stipends for especially qualified students.
Comprehensive Examination
Each
candidate for the Master of Science degree concentration in Applied Statistics
must perform satisfactorily on an oral comprehensive examination over the
candidate's program of study.
The Ph.D.
degree program in Applied Mathematics is designed to enable its students to
master a significant body of mathematics, including a specialty in applied
mathematics; to relate this knowledge to a coherent area of science or
engineering; and to carry on fundamental research in applied mathematics at a
nationally competitive level. Recipients of this degree will, according to
their abilities and choice of sub-specialty, be able to work effectively in a
research and development environment involving mathematical or statistical
analysis and modeling in business, government or industry; to teach mathematics
at the college or university level; or to carry on fundamental research in
their area of specialty.
Additional Admission Requirements
In
addition to the requirements of the
Students
are admitted to the program by the
Program of Study
The
student must complete an approved program of study, including a minor,
typically including approximately 54 credit hours. The minor is
interdisciplinary and may be satisfied by 9 hours of graduate work outside the
mathematics department, by 6 credit hours for a project in an area of
application, or by a combination of external coursework and directed project in
an area of application totaling 9 credit hours.
Each
student will have an advisory committee appointed by the department Graduate
Committee in consultation with the student and approved by the Department
Chair. It includes the prospective dissertation adviser as chair (or co-chair,
if the dissertation adviser is not a member of the Department of Mathematics).
The advisory committee should be appointed as soon as is feasible, usually
within a year after passing the Preliminary Examination. Once formed, it will
have the responsibility of constructing and approving the program of study
which includes the minor. Prior to the appointment of the advisory committee
the student will be advised by a graduate faculty member appointed by the
department Graduate Committee.
Grades
A student
is expected to achieve A's or B's in all courses included in the program of
study and must have at least a 3.0 GPA to graduate. The dissertation is graded
on a pass/unsatisfactory basis and, therefore, will not be included in the
cumulative average. An accumulation of more than two marginal (C) grades will
result in suspension of the student's enrollment in the program. If a student
makes a grade of U on any course, enrollment will be suspended and the student
cannot take further graduate work without being readmitted to the program.
Readmission to the program requires approval of the Dean of the
Transfer Credit
Only
courses with grades of A or B may be accepted for transfer credit. Transfer
credit must be recommended by the department Graduate Committee and approved by
the Dean of the
Preliminary Examination
The
student is expected to take the preliminary examination within three semesters
of being admitted to the Ph.D. program. The examination consists of two parts:
a written examination based on Real Analysis I and II (8143-8144) and a written
examination based on two other related courses chosen by the student and
approved by the department Graduate Committee. At the discretion of the
department Graduate Committee, the student may be allowed to retake a portion
of the preliminary examination a second time if the student does not pass that
portion on the first attempt. A student who fails the preliminary examination
twice is terminated from the Ph.D. program.
Qualifying Examination and Admission to
Candidacy
Each
student must pass a comprehensive oral examination covering her/his chosen
field of research and related advanced course work. The exam is conducted by
the student's Advisory Committee and may include an additional written
examination. The exam is open to the graduate faculty of the department. The
student is expected to take the qualifying examination within two years of the
appointment of the student's Advisory Committee. A student who fails the
qualifying examination twice is terminated from the Ph.D. program. The
dissertation topic may be proposed after the student has passed the qualifying
examination. A doctoral student advances to candidacy after the dissertation
topic has been approved by the student's advisory committee and the Dean of the
Assistantships
A number
of graduate assistantships are available each year (with nationally-competitive
stipends) for qualified applicants. A limited number of fellowship awards can
be applied to supplement these stipends or provide stand-alone stipends for
especially qualified students, including one award of $25,000.
Dissertation
The student must complete and defend a dissertation based on a
research program approved by the student's dissertation adviser which results
in a high quality, original and substantial piece of research. The student must
orally present and successfully defend the dissertation before the student's
Advisory Committee in a defense that is open to the public. A copy of the
dissertation must be made available to the graduate faculty of the department
at least two weeks prior to the public defense. The dissertation will be graded
on a pass/unsatisfactory basis by the Advisory Committee and must be approved
by the Department Chair and the Dean of the
Residency
Requirement
The full-time Ph.D. student must enroll for one continuous
full-time year (i.e. two consecutive semesters of at least nine graduate credit
hours in each semester) following admission to the program.
Language and Research Tool Requirements
Each
student must demonstrate a reading knowledge of French, German or Russian by
passing a written translation exam in one of these languages conducted by the
Mathematics Department. In addition, the student must demonstrate significant
computer expertise applicable to research or teaching in his or her major field
as approved by the student's Advisory Committee. The computer expertise
requirement may include course work or work on a project and may overlap with
the minor requirement.
Time Limit for Degree Completion
The
student must achieve admission to candidacy within six years after admission to
the program and complete all requirements within six years after admission to
candidacy for the Ph.D. degree. All requirements for the degree must be
completed within eight years after first registration as a doctoral student.
MASTER OF ARTS IN MATHEMATICS EDUCATION
The Master of Arts in Mathematics Education degree program is designed
primarily for secondary mathematics school teachers interested in professional
growth and graduate certification in mathematics teaching. Emphasis in this
program is given to developing depth and breadth in mathematics teaching and
learning, appropriate to the role of the secondary school teacher.
By the end of his/her first semester in the program, each student
will select a member of the Mathematics Education faculty who will serve as
his/her Graduate Advisor throughout the program. Approval of the program of
each student and provision of advice regarding progress toward the degree are
the responsibility of the Graduate Advisor.
Additional Admission Requirements
In addition to the general requirements for admission to the
1)
Twenty-seven hours of undergraduate
coursework in Mathematics beyond the freshman level, or evidence of equivalent
academic preparation.
2)
Possession of a
3)
Two years of full-time experience teaching
mathematics in a secondary school or other acceptable teaching experience.
4)
A satisfactory score is required on the
Aptitude Portion of the Graduate Record Examination.
Degree Requirements
The Master of Arts in Mathematics Education degree requires
successful completion of a minimum of 36 semester hours of graduate credit or
the equivalent. Of these, 18 hours must be in courses numbered 6000 or above.
Programs of study beyond these 36 hours may be required to remove deficiencies
in undergraduate programs or to develop areas of need, interest, or desired
experience.
Core Courses
Each candidate must complete:
1)
18 hours of graduate-level Mathematics
courses selected in consultation with the program Coordinator, with at least 9
hours of courses at the 6000-level. A recommended plan of study includes:
MATH6100 Foundations of Mathematics (3)
MATH6101 Foundations of Real Analysis (3)
MATH6102 Calculus from an Advanced Viewpoint (3)
MATH6106 Modern Algebra (3)
MATH6107 Linear Algebra (3)
MATH6118 Non-Euclidean Geometry (3)
2)
12 hours of graduate-level courses
covering mathematics education learning theory, research, and contemporary
topics in secondary mathematics teaching. These courses include:
MAED6122 Theoretical Foundations of Learning
Mathematics (3)
MAED6123 Research in Mathematics Education (3)
MAED6124 Issues in the Teaching of Secondary School
Mathematics (3)
RSCH6101 Educational Research Methods (3)
3)
6 hours of graduate-level professional
education courses including:
MDSK6260 Principles of Teacher Leadership (3)
An additional three hours of graduate-level Mathematics,
Mathematics Education, or Education courses selected in consultation with the
student's adviser.
4)
A Basic Portfolio consisting of documents
and artifacts that provides evidence of the student's professional growth
during the program.
Comprehensive Exam
Upon successful completion of all
coursework, each candidate for the degree in Mathematics Education must pass a
comprehensive final exam consisting of two parts. The student must pass an oral
exam on the mathematics content courses. The second part of the exam involves
the student presenting documentation that demonstrates their professional
growth as teachers and educational researchers. The student has the option of
presenting either a research-based project or a comprehensive portfolio. The
Graduate Advisor will advise and assist the student in planning his/her
Comprehensive Portfolio or Final Research Report.
PH.D. IN CURRICULUM AND
INSTRUCTION: MATHEMATICS EDUCATION SPECIALIZATION
In addition to the Masters of Arts in Mathematics Education
program, the department offers a Mathematics Education specialization to
students enrolled in the Ph.D. program in Curriculum and Instruction in the
Required (3):
EDCI8160
Additional MAED 8000-level courses
(21):
EDCI8004 Topics in Analysis. (3)
EDCI8008 Topics in Geometry and Topology. (3)
EDCI8100 Foundations of Mathematics. (3)
EDCI8101 Foundations of Real Analysis. (3)
EDCI8102 Calculus from an Advanced Viewpoint. (3)
EDCI8103 Computer Techniques and Numerical Methods. (3)
EDCI8105 Problem Solving in Discrete Mathematics. (3)
EDCI8106 Modern Algebra. (3)
EDCI8107 Linear Algebra. (3)
EDCI8118 Non-Euclidean Geometry. (3)
EDCI8609 Seminar. (3)
EDCI8122 Theoretical Foundations of Learning Mathematics. (3)
EDCI8123 Research in Mathematics Education. (3)
EDCI8124 Advanced Topics in Mathematics Education. (3)
EDCI8125 Issues in the Teaching of Secondary School Mathematics. (3)
EDCI8160
Courses In Mathematics, Mathematics Education And
Statistics
Mathematics
MATH 5000. Topics in Foundations or History of Mathematics. (2-3) Prerequisite: consent of the
department. Topics in the foundations or the history of mathematics selected to
supplement regular course offerings in this area of mathematics. May be repeated for credit with approval of the department.
Credit for the M.A. degree in mathematics requires approval of the department. (On
demand)
MATH 5040. Topics in Analysis. (2-3) Prerequisite: consent of the
department. Topics in the foundations or the history of mathematics selected to
supplement regular course offerings in this area of mathematics. May be repeated for credit with the approval of the department.
Credit for the M.A. degree in mathematics requires approval of the department. (On
demand)
MATH 5060. Topics in Algebra. (2-3) Prerequisite: consent of the
department. Topics in algebra selected to supplement regular course offerings
in this area of mathematics. May be repeated for credit with
the approval of the department. Credit for the M.A. degree in mathematics
requires approval of the department. (On demand)
MATH 5080. Topics in Geometry and Topology. (3) Prerequisite: consent of the
department. Topics in geometry or topology selected as to supplement regular
course offerings in this area of mathematics. May be repeated
for credit with approval of the department. Credit for M.A. degree in
mathematics requires approval of the department. (On demand)
MATH 5109. History of Mathematical Thought. (3) Prerequisite: MATH 1241 or consent
of the department. A study of the development of mathematics
in its historical setting from the earliest beginnings to modern times.
Not approved for the M.A. in mathematics degree. (Fall) (Evenings)
MATH 5128. Applied
MATH 5129. Applied Probability II. (3) Prerequisite: MATH 5128 or consent
of the department. Continuation of MATH 5128. (Spring)(Alternate
years)
MATH 5143.
MATH 5144. Analysis II. (3) Prerequisite: MATH 5143 with a grade of B or better or consent of
the department. Continuation of MATH 5143. (Spring)
MATH 5161. Number Theory. (3) Prerequisite: MATH 3163 with a grade of C or better or consent of
the department. A study of the elements of classical number theory including
divisibility, congruences, diophantine
equations, prime numbers and their distribution, quadratic reciprocity,
number-theoretic functions, and famous unsolved problems. Not approved for the
M.A. in mathematics degree. (Spring) (Alternate years)
MATH 5163. Modern Algebra. (3) Prerequisite: MATH 3163 or consent of the department. Groups, rings, integral domains, fields. (Fall)
(Alternate years)
MATH 5164. Abstract
Linear Algebra. (3)
Prerequisite: MATH 3163 and 2164 or consent of the department. Vector spaces over arbitrary fields, linear transformations,
canonical forms, multilinear algebra. (Spring)
(Alternate years)
MATH 5165. Numerical Linear Algebra. (3) Prerequisites: CSCI 1100 or 1201
and 1201L, MATH 2164 and 2171, all with a grade of C or better, or consent of
the Department. Gaussian elimination and LU decomposition
methods for linear systems. Vector and matrix norms,
condition numbers and accuracy of solutions. Solutions
of large sparse matrix systems using skyline solvers, and Jacobi,
Gauss-Seidel, and SOR iterative methods. Solution of
nonlinear systems. Least squares methods using the QR factorization.
Selected problems will be programmed for computer solution. (Fall)
(Alternate years)
MATH 5171. Numerical Solution of Ordinary Differential Equations. (3) Prerequisites: CSCI 1100 or 1201
and 1201L, MATH 2241, 2164, and 2171, all with a grade of C or better, or
consent of the Department. Numerical solution techniques for
ordinary differential equations such as Runga-kutta, multistep and extrapolation methods. Stiff solvers and stability criteria. Comparative
work with modern robust codes and visualization methods. (On demand)
MATH 5172. The Finite Element Method. (3) Prerequisites: CSCI 1100 or 1201
and 1201L, MATH 2241, 2164, and 2171, all with a grade of C or better, or
consent of the department. Boundary value problems and their variational form. Finite element basis functions,
computational techniques, isoparametric elements and
curved boundaries, alternate methods, singular problems, eigenvalue
problems. Some practical experience with an F.E.M. program
and graphical output. (Spring) (Alternate years)
MATH 5173. Ordinary Differential Equations. (3) Prerequisites: MATH 2171 and MATH
3142, or consent of the department. Existence and uniqueness theorems for
initial value problems; continuous dependence of solutions on initial values
and right hand sides; linear differential equations in R2 and Rn; non-linear differential equations in R2 and Rn: phase portraits, singularities, cycles; invariant
manifolds; linearization; singularities of planar systems; Lyapunov
stability; examples: van der Pol
oscillator, Liénard systems, Volterra-Lotka equations. (Spring)
MATH 5174. Partial Differential Equations. (3) Prerequisites: MATH 2164 and MATH
3141, or consent of department. Classification of types of
partial differential equations. Separation of variables, Sturm-Liouville problems, boundary and eigenvalue
problems, fundamental solutions and Green's theorem, Fourier series and
integrals,
MATH 5176. Numerical Methods for Partial Differential Equations. (3) Prerequisite: CSCI 1100 or 1201
and 1201L, MATH 2241, 2164, and 2171 all with a grade of C or better, or
consent of the department. Basic finite difference schemes for the solutions of
elliptic, parabolic and hyperbolic equations. Van Neuman analysis, characteristics, boundary conditions.
(Fall) (Alternate years)
MATH 5181. Introduction to Topology. (3) Prerequisite: MATH 2164 with a
grade of C or better. Topics from set theory and point set topology such as
cardinality, order, topological spaces, metric spaces, separation axioms,
compactness and connectedness. (Fall) (Alternate years)
MATH 5691. Seminar. (1-6) Prerequisite: consent of the department. Individual
or group investigation and exposition of selected topics in mathematics.
(On demand)
MATH 5692. Seminar. (1-6) Prerequisite: consent of the department. A
continuation of MATH 5691. (On demand)
MATH 6004. Topics in Analysis. (3) Prerequisite: MATH 6101 or consent of department. Topics in
analysis selected so as to complement regular course offerings in this area of
mathematics. May be repeated for credit with the consent of
department. (On demand)
MATH 6008. Topics in Geometry and Topology. (3) Prerequisite: consent of
department. Topics selected from Euclidean geometry, non-Euclidean geometry,
projective geometry, differential geometry, point-set topology, algebraic
topology. May be repeated for credit with approval of
department. (On demand)
MATH 6050. Topics in Mathematics. (3) Prerequisite: consent of the department. Topics chosen from applied mathematics applicable to other disciplines.
MATH 6100. Foundations of Mathematics. (3) Prerequisite: consent of
department. Logic, sets and axiomatic systems. (Fall,
Summer) (Alternate years)
MATH 6101. Foundations of Real Analysis. (3) Prerequisite: MATH 6100 or consent
of department. Axiomatic and historical development of the
real and complex numbers; rigorous development of limits and continuity of
functions, intermediate and extreme value theorems. (Fall) (Alternate
years)
MATH 6102. Calculus from an Advanced Viewpoint. (3) Prerequisite: MATH 6101 or its
equivalent. A continuation of MATH 6101. A rigorous approach to differentiation and integration of functions
of one real variable. (Spring) (Alternate years)
MATH 6103. Computer Techniques and Numerical Methods. (3) Prerequisite: MATH 6101 or consent
of department. Computer systems, programming, and the
computer solution of numerical problems. (Summer) (Alternate years)
MATH 6105. Problem Solving in Discrete Mathematics. (3) Prerequisite: consent of
department. Propositional and predicate calculus, counting techniques,
partially ordered sets, lattices, graphs and trees. (Alternate years)
MATH 6106. Modern Algebra. (3) Prerequisite: MATH 3163 or its equivalent or consent of
department. Topics chosen from group theory, rings and ideals, integral
domains, fields and elementary Galois theory. (Summer) (Alternate years)
MATH 6107. Linear Algebra. (3) Prerequisite: MATH 2164 or its equivalent or consent of
department. Systems of linear equations, matrices, vector
spaces, linear transformations, determinants, canonical forms of matrices,
inner products. (Summer) (Alternate years)
MATH 6118. Non-Euclidean Geometry. (3) Prerequisite: consent of
department. History of Euclid's Fifth Postulate and attempts to prove it; work
of Gauss, Bolyai, Lobachevsky
and others; systematic development of hyperbolic geometry; relative consistency
of hyperbolic geometry; relative consistency of hyperbolic and Euclidean
geometries. (Alternate years)
MATH 6171. Advanced
MATH 6172. Advanced
Applied Mathematics II. (3) Prerequisites: MATH 2241 and 2171 with grades of C or better or
consent of department. Complex analysis; probability and
statistics. (Spring) (Evenings)
MATH 6201. Statistical Techniques in Finance.
(3) This course reviews basic concepts
and introduces more advanced techniques from Probability and Statistics which
are commonly utilized in mathematical finance. Topics covered include random
variables, distributions, conditional expectations, confidence intervals and
hypothesis testing, simple and multiple regression,
multivariate analysis including factor and canonical correlation analysis, and
time series models including ARMA, ARIMA, ARCH, and GARCH.
MATH 6202. Derivatives II: Partial Differential Equations for Finance.
(3) This course deals with those partial
differential equations which are associated with financial derivatives based on
factors such as equities and spot interest rates.
MATH 6203. Stochastic Calculus for Finance.
(3) An introduction to those aspects of
partial differential equations and diffusion processes most relevant to
finance, Random walk and first-step analysis, Markov property, martingales and
semi-martingales, Brownian motion. Stochastic differential equations: Ito’s
lemma, backward and forward Kolmogorov equations, the
Feynman-Kac formula, stopping times,
MATH 6204. Numerical Methods for Financial
Derivatives. (3)
This course will
introduce students to numerical and computational techniques for solving both
European- and American-style financial derivatives. The approach will be the
finite difference method and the basic theoretical concepts will be introduced.
Final projects will involve implementing the techniques on computers. Some
spectral and Monte Carlo methods will also be discussed.
MATH 6205. Financial Computing. (3) This lab oriented course introduces
the numerical methods needed for quantitative work in finance, focusing on
derivative pricing and fixed income applications. Topics include binomial and
trinomial methods, Crank-Nicholson methods for various exotic options,
treatment of discrete dividends, numerical methods for stochastic differential
equations, random number generators, Monte-Carlo
methods for European and American options. The computing class teaches theory
and practice of numerical finance as well as the programming skills needed to
build software systems in C/C++, Java, Javascript,
and Mathematica/Matlab.
MATH 6609. Seminar. (1-3) P