Graduate Faculty
701-231-8171 or 8561
Maria Angeles Alfonseca, Ph.D.
Universidad Autonoma de Madrid, Spain, 2003
Research Interests:
Fourier Analysis, Partial Differential Equations
Nikita Barabanov, Ph.D.
University of Kiev, 1979
Research Interests:
Differential Equations, Control Theory, Optimization, Neural Networks
Marian Bocea, Ph.D.
Carnegie Mellon University, 2004
Research Interests:
Partial Differential Equations, Calculus of Variations, Mechanics of Deformable Solids
Catalin Ciuperca, Ph.D.
University of Kansas, 2001
Research Interests:
Commutative Algebras, Algebraic Geometry
Dogan Cömez, Ph.D.
University of Toronto, 1983
Research Interest:
Ergodic Theory, Measure Theory, Lp-spaces, Operator Theory, Topology
Davis Cope, Ph.D.
Vanderbilt University, 1980
Research Interests:
Partial Differential Equations, Numerical Methods, Applied Mathematics
James B. Coykendall, Ph.D.
Cornell University, 1995
Research Interests:
Algebraic Number Theory, Commutative Algebra, Ideal Theory, Dimension
Theory, Factorization Theory and K-theory
Benton Duncan, Ph.D.
University of Nebraska, 2004
Research Interests:
Operator Algebras, Noncommutative Functional Analysis, K-theory
Angela Hodge, Ph.D.
Purdue University, 2007
Research Interests:
Mathematics Education
Friedrich Littmann, Ph.D.
University of Illinois, Urbana, 2003
Research Interests:
Approximation theory, Number theory
William Martin, Ph.D.
University of Wisconsin, 1993
Research Interests:
Mathematics Education
James H. Olsen, Ph.D.
University of Minnesota, 1968
Research Interests:
Ergodic Theory, Probability Theory and Related Areas
Cristina Popovici, Ph.D.
Carnegie Mellon University, 2005
Research Interests:
Calculus of Variations, Partial Differential Equations, Mechanics of Deformable Solids
Sean Sather-Wagstaff, Ph.D.
University of Utah, 2000
Research Interests:
Commutative Algebra, Homological Algebra
Warren Shreve, Ph.D.
University of Nebraska, 1967
Research Interests:
Graph Theory, Combinatorics, Matrix Theory
Abraham Ungar, Ph.D.
Tel-Aviv University, 1973
Research Interests:
Differential Equations, Integral Transforms, Wave Propagation,
Special Relativity
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Program Description
The Department of Mathematics offers graduate
study leading to the degrees of Master of Science (M.S.) and Doctor
of Philosophy (Ph.D.). Advanced work may be specialized among
the following areas: algebra, applied mathematics, approximation theory, calculus of variations, combinatorics,
differential equations, dynamical systems, ergodic theory, graph
theory, harmonic analysis, number theory, operator theory and topology.
Beginning with their first year in residence, students are strongly
urged to attend research seminars and discuss research opportunities
with faculty members. By the end of their second semester, students
select an advisory committee and develop a plan of study specifying
how all degree requirements are to be met. One philosophical tenet of the Department of Mathematics graduate
program is that each mathematics graduate student will be well
grounded in the two very basic areas of mathematics: algebra and
analysis. To this end, each student's background will be
assessed, and the student will be directed to the appropriate
level of study in these areas.
Admissions Requirements
The Department of Mathematics graduate
program is open to all qualified graduates of universities and
colleges of recognized standing. To be admitted with full status
to the program, the applicant must:
- Hold a baccalaureate degree (or
equivalent) from an institution of higher education of recognized
standing.
- Have adequate preparation in higher
mathematics, showing potential to successfully undertake advanced
study and research as evidenced by academic performance and
experience.
- Have earned a cumulative grade
point average (GPA) of at least 3.0 or equivalent in all advanced
mathematics courses at the baccalaureate level. Students with
a GPA of at least 3.0 or equivalent in a previous graduate degree
program may be admitted in full standing.
In some of the requirements are not
met, admission on a conditional status is possible in certain
cases.
Applications for admission should be sent to The Graduate School
rather than the Department of Mathematics. The Graduate School
may be contacted for application materials. Applications will
be considered at any time. However, opportunities are improved
for those received by March 1 preceding the fall semester of intended
enrollment.
Official transcripts (transcripts having an appropriate seal or
stamp) of all previous undergraduate and graduate records must
be received by The Graduate School before the application is complete.
When a transcript is submitted in advance of completion of undergraduate
or graduate studies, an updated transcript showing all course
credits and grades must be provided prior to initial registration
at North Dakota State University.
Three letters of recommendation are
required before action is taken on any application. Personal reference
report forms are available from The Graduate School.
The Test of English as a Foreign Language (TOEFL)
examination is required of international applicants. A minimum
TOEFL score of 525 (paper test) or 193 (computer test) must be
achieved.
Financial Assistance
Teaching assistantships and a small number
of research assistantships are available. Graduate tuition is
waived for research and teaching assistants.
All students in full standing and, in certain situations, students
in conditional status are eligible for assistantships.
International students must show proficiency in reading, writing,
and speaking English. In particular, they must pass an oral proficiency
interview, which is a Test of Spoken English (TSE) prior to receiving
a teaching assistantship. This interview is the culmination of
the five-week Intensive English Language Program (IELP) available
each summer. An indication, but not a guarantee, of being able
to pass this interview is a TOEFL score of at least 600 (paper
test) or 247 (computer test). All international students applying
from outside the United States for a teaching assistantship must
expect to take the IELP.
Assistantship applications will be considered at any time. However,
opportunities are improved for those received by March 1 preceding
the fall semester of intended enrollment.
Degree Requirements
At least one year of academic work must
be spent in residence at NDSU in fulfilling graduate requirements
for each graduate degree earned. The M.S. customarily takes two
years to complete: the Ph.D. usually last three years beyond the
master's. Students must maintain a cumulative GPA of at
least 3.0 throughout their graduate career.
Two options are available: the Thesis Option
and the Comprehensive Study Option. The Thesis Option emphasizes
research and preparation of a scholarly thesis, whereas the Comprehensive
Study Option emphasizes a broader understanding of a major area
of Mathematics. Degree requirements include:
- A total of 30 credit hours in approved
graduate-level course work, depending on the degree option (see
below). Subject to the approval of the supervisory committee,
at most 6 of these 30 credits may be earned in 600-level mathematics
courses (but NOT Math 620, 621, 650, or 651) or in courses in
fields other than mathematics.
Thesis option: A total of 6 to 10 credit hours
of Math 798 (Master's Thesis), in addition to 18 credit
hours in courses numbered 700-789. These must include the two-semester
sequences in Algebra (Math 720, 721) and Real Analysis (Math
750, 751).
Comprehensive Study Option: A total of 2 to
4 credit hours of Math 797 (Master's Paper), in addition
to 24 credit hours in courses numbered 700-789. These must include
the two-semester sequences in Algebra (Math 720, 721) and Real
Analysis (Math 750, 751).
- A passing grade in two written preliminary
examinations chosen from Algebra, Applied Mathematics and Real Analysis.
- Demonstrated proficiency in a computer
programming language.
- A thesis or expository paper written
under the supervision of a faculty member and defended at an
oral examination administered by the student's supervisory
committee.
Degree requirements include:
- A total of 90 credit hours in
approved graduate-level course work, including:
a. At least 42 credit hours in courses numbered
700-789. These must include the two-semester sequences in Algebra
(Math 720, 721) and Real Analysis (Math 750, 751), and four
courses form the following list, which must be passed with a
grade of B or higher: Math 728 (Linear Algebra), Math 746 (Topology),
Math 752 (Complex Analysis), Math 754 (Functional Analysis),
Math 756 (Dynamical Systems), Math 772 (Number Theory), Math
788 (Numerical Analysis).
b. At least 3 credit hours of Math 790 (Graduate
Seminar).
c. At least 6 credit hours of Math 799 (Doctoral
Dissertation).
d. Subject to the approval of the advisory
committee, at most 12 credits may be earned in 600-level mathematics
courses (but NOT Math 620, 621, 650, or 651) or in courses in
fields other than mathematics.
- A passing grade in two written preliminary
examinations choosen from Algebra, Applied Mathematics and Real Analysis.
- Demonstrated proficiency in one foreign
language commonly used in the mathematical literature, normally
French, German, or Russian, as well as a demonstrated proficiency
in a computer programming language. A student's advisory
committee may require a second foreign language.
- A passing grade in an oral preliminary
examination administered by the student's advisory committee
after the written examinations, language requirements and all
didactic coursework have been completed. Upon passing the oral
examination, the student advances to candidacy for the Ph.D.
- A dissertation which must embody original
work constituting a definite contribution to mathematical knowledge
and demonstrate capacity for independent research, defended
at a final oral examination administered by the candidate's
advisory committee.
Credits used to satisfy the requirements for a Master's
degree at NDSU may be included in the total 90 credits required
for the Ph.D.
Students entering the doctoral program
with a Master's degree from another institution need only
complete 60 credit hours, including:
- At least 30 credit hours in courses
number 700-789 (but NOT Math 720, 721, 750, or 751).
- Subject to the approval of the advisory
committee, at most 6 credits may be earned in 600-level mathematics
courses (but NOT Math 620, 621, 650, or 651) or in courses in
fields other than mathematics.
All other requirements must be satisfied
as above.
Courses Offered
- 620 Abstract Algebra I 3
- Groups, permutations, quotient groups,
homomorphisms, rings, ideals, integers. Prereq: Math 270 or
departmental approval.
621 Abstract Algebra II 3- Division rings, integral domains, fields,
field extensions, Galois Theory. Prereq: Math 620 or departmental approval.
629 Linear Algebra 3- Vector spaces, linear transformations
eigenvalues and eigenvectors, canonical forms, inner product
spaces, and selected applications. Prereq: Math 270 or departmental approval.
630 Graph Theory 3- Graphs and directed graphs, graph models,
subgraphs, isomorphisms, paths, connectivity, trees, networks,
cycles, circuits, planarity, Euler's formula, matchings,
bipartite graphs, colorings, and selected advanced topics. Prereq:
Math 270 or departmental approval.
636 Combinatorics 3- Recurrence relations, formal power series,
generating functions, exponential generating functions, enumeration,
binomial coefficients and identities, hypergeometric functions,
Ramsey theory, Sterling and Eulerian numbers. Prereq: Math 270
or departmental approval.
640 Axiomatic Geometry 3- Hilbert's axioms for Euclidean
geometry, projective geometry, history of parallel axiom, hyperbolic
geometry, elliptic geometry. Prereq: Math 270 or departmental approval.
645 Differential Geometry 3- Basic properties of curves and surfaces,
Frenet equations, the Gauss Map, intrinsic geometry of surfaces,
geodesics, Gauss-Bonnet Theorem, and applications. Prereq: Math
270 or departmental approval.
646 Introduction to Topology 3- Topology of Euclidean space, metric
spaces, topological spaces, bases and neighborhoods, Hausdorff
property, continuity, homeomorphisms and embeddings, connectivity,
and compactness. Prereq: Math 270 or departmental approval.
647 Molecular Topology 3- Applications of topological techniques
to stereochemistry. Topics include three-dimensional manifolds,
knots, embedded graphs, chirality, topological rubber gloves,
Möbius ladders, topology of DNA, tangles, and the Ernst-Sumners
theorem. Prereq: Math 270 or departmental approval.
650 Real Analysis I 3- Sequences and convergence in R, continuity,
uniform convergence, spaces of continuous functions, compactness,
fixed point theorems, differentiability, inverse and implicit
function theorems, applications. Prereq: Math 266, and 270 or
departmental approval.
651 Real Analysis II 3- Riemann and Riemann-Stieltjes integration,
convergence theorems, multiple integration and Fubini's
Theorem, elements of Fourier analysis, applications. Prereq:
Math 650 or departmental approval.
652 Complex Analysis 3- Complex number systems, analytic and
harmonic functions, elementary conformal mapping, integral theorems,
power series, Laurent series, residue theorem, and contour integration.
Prereq: Math 265 or departmental approval.
660 Intensive MATHEMATICA 1- Thorough overview of the general purpose
mathematical software MATHEMATICA: numerical and symbolic calculations
for algebra and linear algebra, single and multivariable calculus,
ordinary and partial differential equations, 2D- and 3D-graphics,
animation, word processing. Satisfies computer programming proficiency
requirement. Prereq: Math 259 or departmental approval.
672 Number Theory 3- Properties of integers, number theoretic
functions, quadratic residues, continued fractions, prime numbers
and their distribution, primitive roots. Prereq: Math 270 or
departmental approval.
678 History of Mathemactics 3- Historical consideration emphasizing
the source of mathematical ideas, growth of mathematical knowledge,
and contributions of some outstanding mathematicians. Prereq:
Math 270 or departmental approval.
680 Applied Differential Equations 3- Power series expansions and the method
of Frobenius, special functions, and their use (Bessel functions,
Legendre polynomials); phase plane analysis. Prereq: Math 266
or departmental approval.
681 Fourier Analysis 3- Discrete and continuous Fourier transforms,
Fourier series, convergence and inversion theorems, mean square
approximation and completeness, Poisson summation, Fast-Fourier
transform. Prereq: Math 265 or departmental approval.
682 Survey of Mathematical Models 3- Lagrangian and Hamiltonian dynamics,
potential theory, diffusion, hydrodynamics, elasticity; dimensional
analysis, tensors; emphasis on how physical concepts are formulated
mathematically rather than solution methods. Prereq: Math 266
or departmental approval.
683 Partial Differential Equations 3- Solution methods for potential, diffusion,
and wave equations; treatments of homogeneous and nonhomogeneous
equations; boundary conditions; separation of variables, Green's
functions, transform techniques. Prereq: Math 680 or departmental approval.
688 Numerical Analysis I 3- Numerical solution of nonlinear equations,
interpolation, numerical integration and differentiation, numerical
solution of initial value problems for ordinary differential
equations. Prereq: Math 266 or departmental approval.
689 Numerical Analysis II 3- Numerical solutions of linear and nonlinear
systems, eigenvalue problems for matrices, boundary value problems
for ordinary differential equations, selected topics. Prereq:
Math 629 and 688 or departmental approval.
720, 721 Algebra I, II 3 each- Graduate level survey of algebra: groups,
rings, fields, Galois theory, and selected advanced topics.
Prereq: Math 621 or departmental approval .
724
Topics in Commutative Algebra- Can be repeated for credit
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726 Homological Algebra 3- An overview of the techniques of homological
algebra. Topics covered will include categories and functors,
exact sequences, (co)chain complexes, Mayer-Vietoris sequences,
TOR, and EXT. Applications to other fields will be stressed.
Prereq: Math 621 or departmental approval.
728, 729 Linear Algebra I, II 3 each- Theory of linear transformations and
matrices, canonical forms, inner product spaces, unitary spaces,
symmetric forms, generalized inverses, and selected advanced
topics. Prereq: Math 629 or departmental approval.
730, 731 Graph Theory I, II 3 each- Graduate-level survey of graph theory:
paths, connectivity, trees, cycles, planarity; genus, Eulerian
graphs, Hamiltonian graphs, factorizations, tournaments, embedding,
isomorphism, subgraphs, colorings, Ramsey theory, girth. Prereq:
Math 630 or departmental approval.
732 Introduction to Bioinformatics 3- An introduction to the principles of
bioinformatics including information relating to the determination
of DNA sequencing. Prereq: Stat 661or departmental approval.
736, 737 Discrete Mathematics I, II 3 each- Combinatorial reasoning, generating
functions, inversion formulae. Topics may include design theory,
finite geometry, Ramsey theory, and coding theory. Advanced
topics may include cryptography; combinatorial group theory;
combinatorial number theory, algebraic combinatorics, (0,l)-matrices,
and finite geometry. Prereq: Math 636 or departmental approval.
746, 747 Topology I, II 3 each- Topological spaces, convergence and
continuity, separation axioms, compactness, connectedness, metrizability,
fundamental group, and homotopy theory. Advanced topics may
include homology theory, differential topology, three-manifold
theory, and knot theory. Prereq: Math 646 or departmental approval.
749 Topics in Geometry and Topology - Can be repeated for credit.
750, 751 Analysis I, II 3 each- Lebesgue and general measure and integration
theory, differentiation, product spaces, metric spaces, elements
of classical Banach spaces, Hilbert spaces, and selected advanced
topics. Prereq: Math 651or departmental approval.
752, 753 Complex Analysis I, II 3 each- Analytic and harmonic functions, power
series, conformal mapping, contour integration and the calculus
of residues, analytic continuation, meromorphic and entire functions,
and selected topics. Prereq: Math 651or departmental approval.
754, 755 Functional Analysis I, II 3 each- Normed spaces, linear maps, Hahn-Banach
Theorem and other fundamental theorems, conjugate spaces and
weak topology; adjoint operators, Hilbert spaces, spectral theory,
and selected topics. Prereq: Math 751or departmental approval.
756 Dynamical Systems 3- A study of basic notions of topological
and symbolic dynamics. Introduction to measurable dynamics and
ergodic theory. Ergodicity, mixing and entropy of dynamical
systems. Prereq: Math 750 or departmental approval.
757 Topics in Functional Analysis - Can be repeated for credit.
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760, 761 Ordinary Differential Equations I, II 3 each- Existence, uniqueness, and extendibility
of solutions to initial value problems, linear systems, stability;
oscillation, boundary value problems, difference equations,
and selected advanced topics. Prereq: Math 751or departmental approval.
762, 763 Integral Equations I, II 3 each- Existence and uniqueness of solutions
of Fredholm and Volterra integral equations, Fredholm Theory,
singular integral equations, and selected advanced topics. Prereq:
Math 751or departmental approval.
764 Calculus of Variations 3- Variational techniques of optimization
of functionals, conditions of Euler, Weierstrass, Legendre,
Jacobi, and Erdmann, Pontryagin Maximal Principle, applications,
and selected advanced topics. Prereq: Math 651or departmental approval.
767 Topics in Applied Mathematics - Can be repeated for credit.
772, 773 Number Theory I, II 3 each- Number theoretic functions, algebraic
number fields, prime numbers and their distribution, the Prime
Number Theorem and related results, Fermat's Theorem.
Prereq: Math 672 or departmental approval.
778 Modern Probability Theory 3- See Statistics for description.
782, 783 Mathematical Methods in Physics I, II 3 each- Tensor analysis, matrices and group
theory, special relativity, integral equations and transforms,
and selected advanced topics. Prereq: Math 629and 652 or departmental approval. Cross-listed with Phys 752, 753.
784, 785 Partial Differential Equations I, II 3 each- Classification in elliptic, parabolic,
hyperbolic type; existence and uniqueness for second-order equations;
Green's functions and integral representations; characteristics,
nonlinear phenomena. Prereq: Math 751or departmental approval.
786, 787 Mixed Boundary Value Problems I, II 3 each- Methods for transient and steady-state
solutions of diffusion problems with mixed boundary conditions;
integral transforms; Green's function and integral equation
formulations, asymptotics. Prereq: Math 652 or 752 or departmental approval.
788, 789 Numerical Analysis I, II 3 each- Numerical solutions to partial differential
and integral equations, error analysis, stability, acceleration
of convergence, numerical approximation, and selected advanced
topics. Prereq: Math 689 or departmental approval.
790 Graduate Seminar 1-3
793 Individual Study/Tutorial 1-5
797 Master's Paper 1-5
798 Master's Thesis 1-10
799 Doctoral Dissertation 1-15
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