Program Director

Dr. G. Jay Kerns
620 Lincoln Building
(330) 941-3310
gkerns@ysu.edu

Program Description                    

The Department of Mathematics and Statistics offers the M.S. degree in mathematics. Options for this degree include:

  • predoctoral studies,
  • applied mathematics,
  • computer science,
  • secondary mathematics,
  • statistics, and
  • actuarial science.

Graduate faculty members have a broad range of research interests in both pure and applied areas. The curriculum stresses theoretical as well as computational math­ematics and is flexible enough to key a student’s program to individual interests and abilities. Re­ceiving a well-rounded education in mathematics, graduates can pursue a Ph.D., secure a position in government or industry, or further a teaching career. The department has extensive computing facilities that include microcomputers, workstations, mainframe, and access to supercomputers.

Admission Requirements

In addition to the minimum College of Graduate Studies admission requirements, students must also have the following:

COURSETITLES.H.
A cumulative undergraduate cumulative grade point average of at least 3.0 (on a 4.0 scale) in all undergraduate mathematics an statistics courses.
A completed sequence in standard calculus comparable to:
MATH 1571Calculus 14
MATH 1572Calculus 24
MATH 2673Calculus 34
Previous courses in discrete structures and linear algebra comparable to:
MATH 3715Discrete Mathematics3
MATH 3720Linear Algebra and Matrix Theory3
Previous course in abstract algebra comparable to:
MATH 5821Topics in Abstract Algebra4
MATH 5851Topics in Analysis4
The Graduate Record Examination

Students not satisfying all of the above may be admitted with provisional status subject to the approval of the graduate program director and the graduate dean.

Graduate Faculty

Guang-Hwa (Andy) Chang, Ph.D., Professor
Biostatistics

Richard G. Goldthwait, Ph.D., Assistant Professor
Math education

Jozsi Z. Jalics, Ph.D., Associate Professor
Computational neuroscience; mathematical biology; dynamical systems; partial differential equations

G. Jay Kerns, Ph.D., Professor
Signed measures; infinite divisibility; exchangeability in probability and statistics; applications of stochastic processes

Lucy Xiaojing Kerns, Ph.D., Assistant Professor
Simultaneous confidence bands; minimum effective doses; benchmark dose methodology

Thomas L. Madsen, Ph.D., Assistant Professor
Abstract algebra; group theory; representation theory

Nguyet Thi Nguyen, Ph.D., Assistant Professor
Financial models; Monte Carlo simulation; actuarial science

Anita C. O'Mellan, Ph.D., Professor
Graph theory; combinatorics; early childhood mathematics education

Alicia Prieto Langarica, Ph.D., Associate Professor
Mathematical biology; agent-based modeling

Stephen Rodabaugh, Ph.D., Professor
Foundations of topology and fuzzy logic: point-set, lattice-theoretic, and categorical methods

Thomas Smotzer, Ph.D., Professor
Real analysis; measure theory; operator theory

Angela Spalsbury, Ph.D., Professor, Chair
Functional analysis; operator theory; measure theory

Jamal K. Tartir, Ph.D., Professor
Set-theoretic topology

Thomas P. Wakefield, Ph.D., Professor
Character theory; actuarial science

Eric J. Wingler, Ph.D., Professor
Real analysis; complex analysis; operator theory

George Yates, Ph.D., Professor
Applied mathematics; partial differential equations; mathematical biology; nonlinear waves

  • A minimum of 33 semester hours of credit excluding MATH 5821 Topics in Abstract Algebra and MATH 5851 Topics in Analysis
  • A cumulative grade point average of at least 3.0
  • Students entering without a prior course in abstract algebra must include MATH 5821 Topics in Abstract Algebra in their program, to be taken in the earliest available semester, and students entering with­out a prior course in theoretical analysis must include MATH 5851 Topics in Analysis in their program, to be taken in the earliest available semester. These courses are not included in the 33-semester-hour minimum requirement.
  • The student’s combined undergraduate and graduate programs must include a mathemat­ics core comprising the following courses or their equivalent:
    COURSETITLES.H.
    MATH 5821Topics in Abstract Algebra (taken in the earliest available semester)4
    MATH 5851Topics in Analysis (taken in the earliest available semester)4
    MATH 5825Advanced Linear Algebra3
    MATH 5852Real Analysis 23
    MATH 6996Mathematical Project1-3
  • Satisfactory performance on written and oral examinations. The subject matter for these examinations must be approved by the Graduate Executive Committee.  Additionally, the following distribution requirements apply:
    • Written exams in MATH 5852, Math 5825 and the first course in the student's chosen course sequence
    • Oral exam on thesis, or oral exam on a project and two courses
    • At least half of the hours of the courses examined must be at the 6900 level
  • At least 17 hours of the student's approved program must be at the 6900 level. In addition to completing the courses which make up the mathematics core, students must complete at least one course sequence for depth and at least fifteen additional hours of elective courses to satisfy the breadth requirement for the degree. The course groupings are described below.
  • MATH 6999 Thesis is highly recommended
  • Before completing 12 semester hours, the student must submit the entire degree program for approval and evaluation by the Graduate Executive Committee in the Department of Mathematics and Statistics. Subsequent revisions to this program must be approved by the Graduate Executive Committee. An abstract of a proposed thesis must be submitted for approval prior to registering for the course.
  • Students must participate in an exit interview during the semester in which they plan on graduating. The exit interview will be conducted with one or more members of the Graduate Executive Committee and must be scheduled by the student prior to the thesis or project presentation.

Course Sequences for Depth

The description of the recommended course sequences for depth will refer to the following list.  The sequences offered depend upon student interest. 

COURSETITLES.H.
Abstract Algebra
MATH 6922Advanced Topics in Group and Ring Theory3
MATH 6923Advanced Topics in Field Theory3
Actuarial Mathematics
STAT 6988Modeling in Financial Economics3
STAT 5802Theory of Interest3
Advanced Data Analysis
STAT 6940Advanced Data Analysis3
STAT 6948Linear Models3
Differential Equations
MATH 6955Advanced Differential Equations3
MATH 6957Partial Differential Equations3
Mathematical Statistics
STAT 6943Mathematical Statistics 13
STAT 6944Mathematical Statistics 23
Operations Research
MATH 5845Operations Research3
MATH 6942Advanced Operations Research3
Topology
MATH 6980Topology 13
MATH 6981Topology 23

Predoctoral Studies in Mathematics and Applied Mathematics

COURSETITLES.H.
MATH 6922Advanced Topics in Group and Ring Theory3
MATH 6923Advanced Topics in Field Theory3
MATH 6975Complex Analysis 13
MATH 5852Real Analysis 23
MATH 6980Topology 13
STAT 6940Advanced Data Analysis3
MATH 6955Advanced Differential Equations3
STAT 6943Mathematical Statistics 13
MATH 5861Numerical Analysis 23
MATH 5845Operations Research3
Electives
Select two or more sequences in areas of interest

Statistics

COURSETITLES.H.
Core Requirements
STAT 6940Advanced Data Analysis3
STAT 6948Linear Models3
STAT 6943Mathematical Statistics 13
STAT 6944Mathematical Statistics 23
Electives
Select 9 additional hours of statistics courses.9

Actuarial Science

COURSETITLES.H.
Core Requirements
STAT 5802Theory of Interest3
STAT 6943Mathematical Statistics 13
STAT 6944Mathematical Statistics 23
Electives
Select from statistic and actuarial science course offerings

Applied Mathematics

COURSETITLES.H.
Core Requirements
STAT 6940Advanced Data Analysis3
MATH 6955Advanced Differential Equations3
STAT 6943Mathematical Statistics 13
MATH 5861Numerical Analysis 23
MATH 5845Operations Research3
Depth Requirement
Select the second course in on e of the sequence

Secondary/Community College Mathematics

COURSETITLES.H.
STAT 6943Mathematical Statistics 13
or
STAT 6940Advanced Data Analysis3
MATH 6915Mathematical Foundations3
Select one of the following:3
MATH 6922Advanced Topics in Group and Ring Theory3
MATH 6923Advanced Topics in Field Theory3
MATH 6975Complex Analysis 13
MATH 6965Abstract Analysis 13
MATH 6980Topology 13
MATH 6922Advanced Topics in Group and Ring Theory3
MATH 6923Advanced Topics in Field Theory3
Those students seeking certification should consult an advisor in the school of Education.

Computer Science

Students in coursework in computer science in addition to mathematics should plan their graduate program in consultation with advisors in both the Department of Mathematics and Statistics and the Department of Computer Science and Information Systems.

Accelerated MS Mathematics

Undergraduate students can apply for admission into the accelerated program for the MS in Mathematics after completing 78 semester hours with a GPA of 3.3 or higher. After being admitted into the program, students can take a maximum of nine semester hours of graduate coursework that can count toward both an bachelor's and master's degree from the Department of Mathematics and Statistics. The courses chosen to count for both undergraduate and graduate coursework must be approved by the Graduate Executive Committee within the Department upon admission into the program. An additional six hours of graduate coursework can be completed as an undergraduate and used exclusively for graduate credit.

Learning Outcomes

Students will develop and demonstrate the ability to reason mathematically by constructing mathematical proofs and recognizing and analyzing accurate numerical data in appropriate core courses. Students will learn that truth in mathematics is verified by careful argument, and will demonstrate the ability to make conjectures and form hypotheses, test the accuracy of their work, and effectively solve problems.

Students will learn to identify fundamental concepts of mathematics as applied to science and other areas of mathematics, and to interconnect the roles of pure and applied mathematics.

Students will demonstrate that they can communicate mathematical ideas effectively, both orally and in writing, by completing a graduate project or thesis involving an investigative mathematical project, together with oral and written examinations.

Students in cooperative doctoral programs will demonstrate their ability to create significant, original mathematics.

Graduate Courses

MATH 5821    Topics in Abstract Algebra    4 s.h.

A course in abstract algebra aimed at developing a broad understanding of the subject. Credit will not be given for both MATH 3721 and MATH 5821.
Prereq.: MATH 3715 and MATH 3720.

MATH 5825    Advanced Linear Algebra    3 s.h.

A study of abstract vector spaces, linear transformations, duality, canonical forms, the spectral theorem, and inner product spaces.
Prereq.: MATH 3721.

MATH 5828    Number Theory    3 s.h.

A study of congruences, Diophantine equations, quadratic residues, special number theory functions, and selected applications.
Prereq.: MATH 3721.

MATH 5835    Introduction to Combinatorics and Graph Theory    3 s.h.

The pigeonhole principle; permutations, combinations, the binomial theorem; the inclusion-exclusion principle; recurrence relations; graphs and digraphs, paths and cycles, trees, bipartite graphs and matchings.
Prereq.: MATH 3715 and MATH 3720.

MATH 5845    Operations Research    3 s.h.

An introduction to operations research with emphasis on mathematical methods. Topics may include: linear programming, sensitivity analysis, duality theory, transportation problems, assignment problems, transshipment problems, and network problems.
Prereq.: MATH 3715 and MATH 3720.

MATH 5851    Topics in Analysis    4 s.h.

A course in analysis aimed at developing a broad understanding of the subject. Credit will not be given for both MATH 3751 and MATH 5851.
Prereq.: MATH 2673, MATH 3720, and MATH 3715.

MATH 5852    Real Analysis 2    3 s.h.

Uniform convergence of sequences of functions and some consequences; functions on n-space: derivatives in vector spaces, mean value theorem, Taylor's formula, inverse mapping theorem, implicit mapping theorem.
Prereq.: MATH 3720 and MATH 3751 or equivalent.

MATH 5860    Topics in Numerical Analysis    3 s.h.

A course in numerical analysis aimed at developing a broad understanding of the subject. Credit will not be given for both MATH 3760 and MATH 5860.
Prereq.: MATH 3720 and CSIS 2610.

MATH 5861    Numerical Analysis 2    3 s.h.

Numerical methods of initial-value problems, eigenvalue problems, iterative methods for linear and nonlinear systems of equations, and methods involving least squares, orthogonal polynomials, and fast Fourier transforms.
Prereq.: MATH 2673 and MATH 3760 or equivalent.

MATH 5875    Complex Variables    3 s.h.

Complex numbers and their geometric representation, analytic functions of a complex variable, contour integration, Taylor and Laurent series, residues and poles, conformal mapping.
Prereq.: MATH 3751 or equivalent.

MATH 5895    Selected Topics in Mathematics    2-3 s.h.

The study of a standard mathematical topic in depth or the development of a special area of mathematics. May be repeated twice.
Prereq.: 24 s.h. of mathematics applicable to the mathematics major including either MATH 3721 or MATH 3751.

MATH 5895W    Selected Topics in Mathematics Topology 2    2-3 s.h.

The study of a standard mathematical topic in depth or the development of a special area of mathematics. May be repeated twice.
Prereq.: 24 s.h. of mathematics applicable to the mathematics major including either MATH 3721 or MATH 3751.

MATH 6905    Teaching Practicum    1 s.h.

Intensive preparation for teaching lower-level mathematics courses, featuring formal instruction and orientation on teaching issues, evaluated presentations, mentored classroom instruction, and weekly teaching seminars. Topics include course design, policies, syllabi, grading; classroom teaching problems; orientation in Mathematics Assistance Center, specific lower-level mathematics courses, online tutorial services. Required of and limited to graduate assistants in the Department of Mathematics and Statistics. To be taken each semester student is a graduate assistant. Grading is S/U. Does not count toward credit in the program.

MATH 6910    Advanced Engineering Mathematics 1    3 s.h.

Theory and solution techniques used in engineering applications. Topics include brief review of ordinary differential equations and linear algebra; vector calculus, integral theorems, complex analysis, series, residue theory, potential theory, special functions, integral transforms, partial differential equations and applications in mathematical modeling.
Prereq.: MATH 3705.

MATH 6911    Advanced Engineering Mathematics 2    3 s.h.

Theory and solution techniques used in engineering applications. Topics include brief review of ordinary differential equations and linear algebra; vector calculus, integral theorems, complex analysis, series, residue theory, potential theory, special functions, integral transforms, partial differential equations and applications in mathematical modeling.
Prereq.: MATH 6910.

MATH 6915    Mathematical Foundations    3 s.h.

Order-theoretic and monadic foundations of mathematics: ordered structures; topologies; powerset operators of a function; applications to continuity, compactness, algebra, logic, and calculus.
Prereq.: MATH 3721 Abstract Algebra I and MATH 3751 Real Analysis I, or permission of graduate coordinator.

MATH 6922    Advanced Topics in Group and Ring Theory    3 s.h.

A continuation of MATH 5821 with special emphasis on groups acting on sets, Sylow's Theorem and its applications, ring homomorphisms, ideals, and polynomial rings. Credit will not be given for MATH 4822 and MATH 6922.
Prereq.: MATH 3721 or MATH 5821.

MATH 6923    Advanced Topics in Field Theory    3 s.h.

This course introduces the major results in advanced field theory. These results include splitting fields, algebraic extensions, finite extensions, cyclotomic polynomials, and finite fields. Credit will not be given for MATH 4823 and MATH 6923.
Prereq.: MATH 4822 or MATH 6922.

MATH 6924    Galois Theory    3 s.h.

An introduction to Galois Theory with special emphasis on the Galois group, the Fundamental Theorem of Galois Theory, and radical extensions.
Prereq.: MATH 4823 or MATH 6923.

MATH 6928    Advanced Number Theory    3 s.h.

Advanced study of number theory: theory and distribution of primes, computational number theory, and additive number theory.
Prereq.: MATH 5828.

MATH 6930    Differential Geometry    3 s.h.

Classical differential geometry of curves and surfaces, differentiable manifolds with tensors.
Prereq.: MATH 5852.

MATH 6942    Advanced Operations Research    3 s.h.

Topics may include integer programming, advanced linear programming, nonlinear programming, dynamic programming, queuing theory, Markov analysis, game theory, and forecasting models.
Prereq.: MATH 5845 and STAT 3743 Probability and Statistics.

MATH 6955    Advanced Differential Equations    3 s.h.

Proofs of existence and uniqueness of nonautonomous, nonlinear equations. Additional topics may include advanced linear systems, partial differential equations, and integral equations.
Prereq.: MATH 5852 and either MATH 3705 Differential Equations or MATH 4855, or permission of graduate coordinator.

MATH 6957    Partial Differential Equations    3 s.h.

An introduction to partial differential equations (PDE) and their applications. The classification of the basic types of linear partial differential equations, development of how boundary and initial conditions affect solutions, exploration, and application of solution techniques for PDEs and explosions in orthogonal functions will be presented.
Prereq.: MATH 3705 and MATH 3720 or equivalent .

MATH 6965    Abstract Analysis 1    3 s.h.

Lebesgue integration and measure on the real line. General measure theory and functional analysis, including the Radon-Nikodym theorem, the Fubini theorem, the Hahn-Banach theorem, the closed graph and open mapping theorems, and weak topology.
Prereq.: MATH 5852 and either MATH 4880 or MATH 6915 or permission of graduate coordinator.

MATH 6975    Complex Analysis 1    3 s.h.

Analytic and meromorphic functions of a complex variable, contour integration, the Cauchy-Goursat theorem, Taylor and Laurent series, residues and poles, conformal mapping. Credit will not be given for both MATH 5875 and MATH 6975.
Prereq.: MATH 3751 Real Analysis I, or permission of graduate coordinator.

MATH 6980    Topology 1    3 s.h.

Basic concepts of topological spaces and mappings between them, including compactness, connectedness, and continuity. Credit will not be given for both MATH 4880 and MATH 6980.
Prereq.: MATH 3721 Abstract Algebra I and MATH 3751 Real Analysis I, or permission of graduate coordinator.

MATH 6981    Topology 2    3 s.h.

Separation, metrization, compactification. Additional topics will be selected from point-set topology, fuzzy topology, algebraic topology, combinatorial topology, topological algebra.
Prereq.: MATH 4880 or MATH 6980, or permission of graduate coordinator.

MATH 6990    Independent Study    1-3 s.h.

Study under the supervision of a staff member. May be repeated.
Prereq.: Consent of graduate coordinator.

MATH 6995    Special Topics    1-3 s.h.

Specialized topics selected by the staff. May be repeated up to 12 semester hours.
Prereq.: Permission of graduate coordinator and department chair.

MATH 6996    Mathematical Project    1-3 s.h.

Individual research project culminating in a written report or paper, though not as broad in scope as a thesis. May be repeated once if the second project is in a different area of mathematics.

MATH 6999    Thesis    3 s.h.

A student may register for six semester hours in one semester or for three semester hours in each of two semesters.

MATH 7005    Advanced Topics in Categorical Topology    3 s.h.

Content varies with each offering. Implements ideas from MATH 6915, MATH 6980, MATH 6981, and studies categorical methods in topology and related concrete categories. Emphasis on current literature and open questions. May be repeated with approval of graduate coordinator.
Prereq.: MATH 6915, MATH 6980, MATH 6981, or equivalent, or permission of the graduate coordinator.

MATH 7015    Advanced Topics in Foundations of Topology    3 s.h.

Content varies with each offering, implements ideas from MATH 6915, MATH 6980, MATH 6981, and studies foundations of topology from a variety of viewpoints (algebraic, categorical, logical, order theoretic, powerset theoretic, set theoretic, etc.). Emphasis on current literature and open questions. May be repeated with approval of graduate coordinator.
Prereq.: MATH 6915, MATH 6980, MATH 6981, or equivalent, or permission of graduate coordinator.

MATH 7025    Advanced Topics in General Topology    3 s.h.

Content varies with each offering, implements ideas from MATH 6915, MATH 6980, MATH 6981, and studies various topics in point-set topology. Emphasis on current literature and open questions. May be repeated with approval of graduate coordinator.
Prereq.: MATH 6980, MATH 6981, or equivalent, or permission of graduate coordinator.

MATH 7035    Advanced Topics in Lattice-Valued Topology    3 s.h.

Content varies with each offering. Implements ideas from MATH 6915, MATH 6980, MATH 6981, and studies topology from the standpoint of lattice-valued (fuzzy) subsets. Emphasis on current literature and open questions. May be repeated with approval of graduate coordinator.
Prereq.: MATH 6915, MATH 6980, MATH 6981, or equivalent, or permission of the graduate coordinator.

MATH 7045    Advanced Topics in Topological Analysis    3 s.h.

Content varies with each offering. Implements ideas from MATH 6915, MATH 6965, MATH 6966, MATH 6980, MATH 6981, and studies the overlap between topology and abstract analysis (topological games, topological groups, separate versus joint continuity, etc.). Emphasis on current literature and open questions. May be repeated with approval of graduate coordinator.
Prereq.: MATH 6915, MATH 6965, MATH 6980, MATH 6981, or equivalent, or permission of graduate coordinator.

MATH 7055    Seminar in Topology and Abstract Analysis    3 s.h.

Content varies with each offering. Implements ideas from MATH 6915, MATH 6930, MATH 6965, MATH 6980, MATH 6981, MATH 6984, and focuses on current research activities of seminar participants. Student registrants are expected to make at least one major presentation each month of the term. May be repeated with approval of graduate coordinator.
Prereq.: Permission of graduate coordinator.