Sample PhD Course Programs
All department Ph.D. students choose specialized fields for their advanced course work and dissertation research. Although the construction of a suitable program of specialized courses is a matter for the advisor and student to work out individually, it is convenient to exhibit sample programs in each of the most popular special areas ‑‑ operations research, statistics, optimization & mathematical programming, probability & stochastic processes, discrete mathematics, and numerical mathematics. The first two years of each sample program constitute a possible Master’s program in the specialization indicated.
Regardless of an ultimate specialization, however, the first two or three semesters of a typical program are devoted primarily to "basic" courses. These are selected to provide the student with the appropriate foundation to take the Ph.D. Candidacy Examination. Students already possessing some of this basic background, e.g., in analysis or probability, would make substitutions, of course, and possibly proceed to more advanced coursework.
Below are exhibited sample programs, consistent with the above guidelines, for several areas of specialization.
Operations Research
The term Operations Research originally described investigations designed to facilitate the making of executive decisions, and carried out by broadly based multidisciplinary teams. But with the passage of time the name has been increasingly associated with certain mathematical ideas and techniques, notably those of optimization, stochastic processes, and systems modeling, which have repeatedly proved useful in such investigations.
The program at Johns Hopkins attempts to balance the study of modern quantitative techniques and the basic mathematics and statistics required for their mastery with suitable exposure to actual decision problems. Typically these may involve questions of inventory and production scheduling, ecology, managerial economics, health service research, and the design of transportation and communication networks. The student's electives should be chosen to promote such exposure.
During the first graduate year, which is ordinarily spent in satisfying the department's basic requirements, the student and advisor design a program of further coursework. This program is tailored to the objectives of the individual and reflects an appropriate balance between theory and application.
Year l
Fall Spring
110.405 Analysis
I 550.662
Optimization Algorithms
550.661 Foundations
of Optimization 550.672
Graph Theory
550.671 Combinatorial
Analysis 550.692
Matrix Analysis
Year 2
Fall Spring
550.681 Numerical
Analysis 550.453
Game Theory
550.420 Probability
Theory 550.463
Network Models
570.609 Facility
Location, or 550.426
Stochastic Processes
600.766 Combinatorial
Optimization
Years 3,4
Completion of advanced coursework and electives (for example, in operations
research, optimization, statistics, health sciences, systems analysis,
and econometrics).
Dissertation research.
Statistics
A student who wishes to construct a doctoral program concentrating in theoretical or applied statistics should obtain training in major areas of probability, statistics and data analysis. The program listed below is designed to prepare the student for independent research and to acquaint him or her with widely used statistical and analytical techniques. Faculty of the department are particularly interested in image analysis and biological applications. There are close connections with the Department of Biostatistics in the School of Hygiene and Public Health; some students may choose to write dissertations under the direction of faculty of that department on topics generated by actual problems in the biomedical field.
Year l
Fall Spring
110.405 Analysis 550.630
Statistical Theory
550.620 Probability
Theory I 550.621
Probability Theory II
550.681 Numerical
Analysis 550.692
Matrix Analysis
Year 2
Fall Spring
550.413 Applied
Statistics & Data Analysis 550.432
Linear Statistical Models
550.631 Statistical
Inference 550.730
Topics in Statistics
550.671 Combinatorial
Analysis 550.692
Matrix Analysis
Years 3,4
Completion of advanced coursework and electives (for example, in statistics,
probability, stochastic processes, operations research, numerical analysis,
econometrics, psychometrics, or biostatistics).
Dissertation research.
Optimization/Mathematical Programming
Mathematical optimization is concerned with finding the "best" answer to a problem rather than just finding "good" answers. While such a concept dates back to antiquity, the term optimum was first used by Leibnitz in his Theodicy (1710). The remarkable efficiency of the digital computer has spurred interest in optimization, which now finds application in diverse areas of engineering, economics, statistics, and the natural sciences.
The program at Johns Hopkins provides the student with opportunity to study and do research in a broad range of optimization topics. These include linear and nonlinear programming, network flows, and integer programming. Central to the study of mathematical optimization is a solid foundation in basic mathematics, especially algebra and analysis. Many students may come with adequate background in these areas, but a wide selection of courses is available for the student wishing to fortify and extend his or her knowledge of mathematics.
The course offerings in optimization are diverse; the student is encouraged to begin coursework in optimization in the first year and to work out a satisfactory advanced program with the advisor.
Year 1
Fall Spring
110.405 Analysis
I 550.662
Optimization Algorithms
550.661 Foundations
of Optimization 550.672
Graph Theory
550.681 Numerical
Analysis 550.692
Matrix Analysis
Year 2
Fall Spring
550.420 Probability
Theory 550.426
Stochastic Processes I
550.671 Combinatorial
Analysis 550.453
Game Theory
550.775 Numerical
Methods for Optimization 550.762 Advanced
Nonlinear Programming
Years 3,4
Completion of advanced coursework and electives (for example, in optimization,
mathematical programming, operations research, and computer
science).
Dissertation research.
Probability/Stochastic Processes
An intuitive notion of "chance" or "randomness" is born of everyday experience. Investigation of the notion is basic to the whole problem of rational interpretation of nature. The effort to quantify the notion mathematically and apply it systematically to various phenomena began with Pascal and Fermat (1654). Thus has arisen a branch of pure mathematics - probability theory ‑ concerned with the construction and investigation of the mathematical model of randomness. With its application to an increasing variety of phenomena and situations, the level of generality, abstractness, and mathematical sophistication of probability theory has also increased.
Stochastic processes are processes developing in time or space in a manner governed by probabilistic laws. Examples are the path of a particle in Brownian motion, the growth of a population, and the fluctuating output of gasoline in successive runs of an oil‑refining mechanism. The theory of stochastic processes represents the "dynamic" part of probabilistic model‑building, in which one studies a process from the point of view of the interdependence of its states at various times and the limiting behavior of the process.
Probability theory and stochastic process theory provide models for the random phenomena studied in science and engineering, and thus provide the foundation of the statistical methodologies for interpretation of data.
Year 1
Fall Spring
110.405 Analysis
I 550.621
Probability Theory II
550.620 Probability
Theory I 550.672
Graph Theory
550.671 Combinatorial
Analysis 550.692
Matrix Analysis
Year 2
Fall Spring
110.605 Real
Variables I 110.606
Real Variables II
550.661 Foundations
of Optimization 550.426
Stochastic Processes
550.681 Numerical
Analysis 550.630
Statistical Theory
Years 3,4
Completion of advanced coursework and electives (for example, in probability,
stochastic processes, time series and signal analysis, control theory,
statistical mechanics, and optimization).
Dissertation research.
Numerical Analysis and Algorithms
As a science, numerical analysis is concerned with the means by which mathematical problems can be solved approximately using the finite resources of a computer. Sometimes this involves the development of algorithms to compute values from solutions which have been determined mathematically. Often, however, methods must be devised to approximate quantities which cannot be expressed in closed form. Above all, the limitations of the computer --- finite storage, finite speed, and finite precision of arithmetic --- must be taken into account.
As an art, numerical analysis is concerned with choosing that method (and suitably applying it) which is "best" suited to the solution of a particular problem. Since the definition of "best" must pay attention to obtaining the highest accuracy with the greatest probability over the widest class of examples using the smallest amount of human and computer resources, numerical analysis is a particularly active area of research. Each change in computer technology, each additional class of problems, and each new mathematical technique causes a reevaluation of what is "best".
This program stresses numerical analysis and other analytical and computational methods useful in solving applied mathematics problems arising in industry, government, and scientific and engineering laboratories.
Year l
Fall Spring
110.405 Analysis
I 550.662
Optimization Algorithms
550.661 Foundations
of Optimization 550.672
Graph Theory
550.681 Numerical
Analysis 550.692
Matrix Analysis
Year 2
Fall Spring
110.416 Ordinary
Differential Equations 110.417
Partial Differential Equations
550.420 Probability
Theory 550.630
Statistical Theory
550.463 Algorithms
I 600.464
Algorithms II
Years 3,4
Completion of advanced coursework and electives (for example, in advanced
mathematical programming, approximation theory, operations research,
applied functional analysis, and appropriate fields of science and engineering).
Dissertation research.
Discrete and Combinatorial Mathematics
Discrete mathematics treats a range of problems structured around finite sets. Although problems of counting and enumeration, and of optimizing a function over a finite set, for example, date from antiquity, all but the simplest remained intractable until sufficiently powerful computers became available. Indeed, the digital computer, which calculates within a finite system of arithmetic, has been both stimulus and tool for revitalization of fields such as combinatorics and graph theory. Crystal structure, particle physics, and communication networks are illustrative sources of inherently discrete mathematical problems; other topics in mathematics (for example matrix theory and linear programming) generate fundamental questions that are combinatorial or graph‑theoretic.
The program below is designed for the student with interest in discrete mathematics and optimization. It can lead to research specialization in combinatorics or graph theory, or in optimization theory involving discrete mathematical systems.
Year 1
Fall Spring
110.405 Analysis
I 550.662
Optimization Algorithms
550.661 Foundations
of Optimization 550.672
Graph Theory
550.671 Combinatorial
Analysis 550.692
Matrix Analysis
Year 2
Fall Spring
550.420 Probability
Theory 550.426
Stochastic Processes
550.681 Numerical
Analysis 550.463
Network Models
600.463 Algorithms
I 600.464
Algorithms II
Years 3,4
Completion of advanced coursework and electives (for example, in combinatorial
algorithms, integer programming, matroid optimization problems,
matching theory, and extremal graph theory).
Dissertation research.