### Doctoral program

The main goal of the Ph.D. program is to train mathematicians and statisticians who intend to make research in these areas their life work. Since 1967 when the University of Toledo joined the Ohio university system, the Department of Mathematics has offered a strong doctoral program and its graduates now occupy academic positions in colleges and universities around the world.

# Programs

In the first two years of doctoral study, the emphasis is on a core curriculum, designed to provide students with a broad base of knowledge in the major areas of modern mathematics or statistics. The following represent typical curricula for the first two years in each of the three tracks:

## Pure Mathematics

**First Year:**- Real Analysis, Topology, Algebra.
**Second Year:**- Complex Analysis and electives such as Differential Geometry, Functional Analysis, Algebraic or General Topology, Differential Equations.

## Applied Mathematics

**First Year:**- Real Analysis, Topology, Differential Equations.
**Second Year:**- Algebra, Complex Analysis, and electives such as Partial Differential Equations, Dynamical Systems, Functional Analysis, Convex Analysis.

## Statistics

**First Year:**- Real Analysis, Probability and Statistical Theory, Linear Statistical Models, and Multivariate Statistics.
**Second Year:**- Functional Analysis, Nonparametric Statistics, Categorical Data Analysis, Statistical Consulting, and Statistical Computing.

# Examinations

The written Ph.D qualifying examination (offered in each of the Fall and Spring semesters) must be passed by the end of the student's second year in the program. Students wishing to specialize in pure or applied mathematics choose to be examined in two of the following subject options: Real Analysis, Topology, Algebra and Differential Equations. Students interested in specializing in statistics must pass the exams in Real Analysis and in Probability and Statistical theory.

Subsequent to passing the qualifying exam, the student prepares for a specialized oral examination under the supervision of a faculty adviser. This exam must be passed before the end of the student's third year. Generally, it is expected that the oral examination topic will be closely related to the student's eventual dissertation research.

# Dissertation

The defining stage of the Ph.D. program is the writing and defence of a dissertation, demonstrating the student's ability to independently attack and solve in an original manner a significant mathematical or statistical problem. No firm timetable can be given for completion of this stage but generally, it can be expected to take two to three years. Possible areas for thesis research in the Department include group theory, non-commutative ring theory, approximation theory, harmonic analysis, partial differential equations, dynamical systems, differential geometry, relativity, scattering theory, wavelets, general topology, category theory, statistics, optimal control and dynamic games. The following list of recent dissertation titles gives some indication of the type of research undertaken by doctoral students in the past:

- Lagrangian submanifold of almost symplectic manifolds
- Finitistic dimension of monomial algebras
- Some results about empirical likelihood method
- Theory and applications of 2-D nonseparable wavelet interpolation and approximation
- The holonomy groups of four dimensional neutral metrics
- Lattice-ordered rings
- Theory and applications of multiwavelet approximation
- Geometric structures of the dynamical equations
- Wavelet interpolation and applications for partial differential equations
- On locally finite automorphism groups
- Volume constrained Douglas problem and the stability of liquid bridges between two coaxial tubes
- Topologically-algebraic categories of universal algebras
- Essentially algebraic categories of partial algebras
- Two problems in the theory of Banach algebras
- Epireflective hulls in
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