GRADUATE STUDIES IN MATHEMATICS

Graduate Studies in Mathematics at Tulane University

For additional information about the Mathematics Department at Tulane University, please visit

http://www.math.tulane.edu/

HISTORY

Graduate School

Tulane is a privately endowed university located in New Orleans, Louisiana. At present it has an enrollment of about 10,000 students from almost every state and from 60 foreign countries.

The Mathematics program has, if anything, been strengthened by the reorganization of Tulane University in the aftermath of hurricane Katrina.

Tulane's mathematical tradition can be traced back to the late nineteenth century, when Florian Cajori, later an expert in the history of mathematical notation, and the most famous translator of Isaac Newton's Principia, was the recipient of the first doctorate in mathematics from Tulane University (1894). Four undergraduates from the period up to the end of World War II (as well as Cajori) later became presidents of the Mathematical Association of America (Duren, McShane, Moise, Young); one (McShane) was a member of the National Academy of Sciences. In the 1950's Tulane became a major center in mathematical research. From 1970 to 2000, 123 Ph.D.'s were awarded.

The mathematics life at Tulane is enlivened by the distinguished mathematicians who visit each year for long or short periods, and by the international symposia which are held here from time to time. The department holds the annual Clifford Lectures, a week-long series of talks by a distinguished mathematician. A mini-conference supported by the National Science Foundation is held in conjunction with each of the Clifford lecture series. The first Clifford lecturer, in 1984, was Fields Medal recipient Charles Fefferman of Princeton University. In 1985 and 1986, the Clifford lecturers were Fields Medal winners, S. T. Yau of UC San Diego and William Thurston of Princeton University. The Clifford lecturers from 1987 through 1990 were Saharon Shelah of Hebrew University in Jerusalem, Clifford Taubes of Harvard University, Charles Peskin of Courant Institute and Haim Brezis of Université de Paris and Rutgers. From 1991 to 1996 they were Sylvain Cappell of Courant Institute of Mathematical Sciences, Nigel Hitchin of the University of Warwick and Persi Diaconis of Harvard University, Peter Sarnak of Princeton University and Dan Voiculescu of UC Berkeley. In 1994, a special conference on semigroups was held in honor of Alfred H. Clifford. In 1997 the Clifford lecturers were Paul Fife of University of Utah (Spring) and Peter Kronheimer of Harvard University (Fall). The speakers from 1998 to the present were Peter Bickel and Alexander Chorin of UC Berkeley, Robert Friedman of Columbia University, Sergei N. Artemov of City University of New York, T. J. Pedley of Cambridge University and Yakov Eliashberg of Stanford University.

ADMISSION TO THE GRADUATE PROGRAMS

Commencement 2000

The Mathematics Department at Tulane University offers a Ph.D. degree in Mathematics as well as Master of Science degrees in Mathematics, Applied Mathematics and in Statistics. These programs are described below. Undergraduate students majoring in mathematics or other sciences (like engineering, physics or computer science) with a strong interest in mathematics are encouraged to apply for admission to any one of the graduate programs. People who already hold undergraduate degrees in mathematics or other sciences are also encouraged to apply.

Requirements for admission into the Tulane Graduate School include:

An undergraduate GPA of at least 3.0 on a 4.0 scale;

The Graduate Record Examination (GRE) general test;

Three letters of recommendation from people who know the applicant's academic potential. Recommendations from mathematicians are particularly desirable.

For foreign students with English as a second language, the English language proficiency exam, TOEFL, is also required.

An application form.

APPLICATION FORM

The best way to apply is to fill out and submit a Web-based application form. Alternatively, you can download a paper-based application. In both cases, visit
https://app.applyyourself.com/?id=tulane-se
If you have any problems receiving the application, you may inquire at
Graduate Coordinator
Mathematics Department
Tulane University
6823 St. Charles Ave.
New Orleans, LA 70118
phone: (504) 865-5727
fax: (504) 865-5063
gradapp@math.tulane.edu

FINANCIAL ASSISTANCE

Most graduate students receive tuition waivers and teaching assistantships, which carry a stipend of $17,000. Teaching Assistants typically teach three laboratories (each meets weekly), although more advanced students may teach one section of an undergraduate course. All Ph.D. students are required to teach an undergraduate course, or to serve as teaching assistants in problem sessions attached to undergraduate courses, for at least two semesters during their residence.

In addition, about 30% of our Ph.D. students are supported by Fellowships, which carry stipends varying from $18,000 to $22,500 per year, plus a tuition waiver. All candidates (U.S. citizens) for Teaching Assistantships in the Department of Mathematics are considered for these fellowships. Students holding a fellowship are not required to teach at all or have reduced teaching duties.

STUDENT LIFE

Newcomb pottery

The Tulane Mathematics Department is known for its friendly atmosphere and its practice of fostering close contact and cordial relations between faculty and graduate students. To us, this is a very important aspect of life here, and we strive to maintain it. The ratio of graduate students to faculty members is kept between 0.8 and 1.4. This is important to us because it allows all new graduate students to soon become familiar with everyone and feel at home. It also allows the faculty to get to know the students during their first semester.

The incoming graduate students are advised by the Graduate Studies Coordinator of the Mathematics Department. The Coordinator, in consultation with the students, determines appropriate first-year courses for each student, according to their preparation and interests. Throughout the program, the Graduate Coordinator continues to help the students plan their studies and realize their mathematical interests.

All students are given a cubicle in one of three large graduate student rooms equipped with desks, desktop PC's, bookcases, chalkboards and a phone. Students also have unlimited access to the computer rooms, lounge and the Mathematics Library, all located within the Department. The lounge, or "commons room", is a place where people gather after seminars and colloquia for refreshments and discussion.

DEGREE PROGRAMS

Sample Analysis problem

Students in the Library

For more detailed information on the graduate programs described below, including required courses, please visit http://www.math.tulane.edu/graduate/

Ph.D. in Mathematics

Completing this degree takes about 5 years, depending on the student's preparation and progress satisfying the requirements. For advanced incoming students, limited transfer credit is possible. The Ph.D. prepares the students for a research career in mathematics in industry or academia.

The course requirements are 48 hours of graduate course work at the 700-level. Up to two courses may be taken in another department with the approval of the Graduate Studies Committee. A number of second year courses and specialized seminars are offered annually.

All students in their first year of the Ph.D. degree must pass a preliminary exam on topics from advanced calculus and linear algebra. During the second year, students must take comprehensive written exams on three fundamental areas of mathematics. One of the areas must be Analysis. While the other two areas are chosen by each student, . Before advancing to candidacy, students must pass an oral exam on more specific topics of research interest. In addition, the students must pass one language examination in mathematical French, German, or Russian. Finally, students must write and defend a dissertation containing original research.

Mathematicians with a Ph.D. from Tulane University have been successful getting jobs in a variety of colleges, research universities, government research laboratories and industries, including biotechnology, e-commerce and financial institutions.

Master of Science in Mathematics

This program is designed to provide students with the opportunity to broaden and deepen their knowledge of core areas of mathematics. The course work is designed to provide both breadth of knowledge and depth in an area of interest to the student. This experience will prepare the student for further studies leading to a Ph.D. degree in mathematics.

The requirements for the Master's degree in Mathematics include 30 hours of graduate course work. These must include 4 required courses (12 credit hours), 15 credit hours chosen from a list of graduate courses, and an independent study course (3 credit hours) in which the student develops a topic in depth and writes a report. A comprehensive examination is also required.

Master of Science in Applied Mathematics

Reaction-diffusion

This program is designed to provide students with the opportunity to broaden and deepen their knowledge of mathematics with an emphasis on those areas that have been most important in science and engineering. Students will also examine, through seminars and case studies, examples of significant applications of mathematics to other areas. This expanded base of knowledge, together with extensive experience in problem solving, is excellent preparation for further studies leading to the Ph.D. degree or for immediate employment in many areas of industry and government.

The program is open to students who have a Bachelor's degree in mathematics or a related field, and have completed undergraduate courses equivalent to Linear Algebra, Numerical Methods, and Analytical Methods. Proficiency in a programming language is essential. Students who have not completed all of these courses may be admitted and are required to take them during the first year.

The requirements for the Master's degree in applied mathematics include 30 hours of graduate course work. These must include 5 required courses (15 credit hours), 12 credit hours chosen from a list of graduate courses, and an independent study course (3 credit hours) in which the student develops a topic in depth and writes a report. A maximum of 6 credit hours taken outside the department may be counted toward the degree. A comprehensive examination is also required. There is also a requirement of proficiency in one of MATLAB, Fortran, C, or C++.

Master of Science in Statistics

Binormal distribution

The Master of Science degree in Statistics combines theory and application. Students in statistics will be trained in data collection, the editing and presentation of large data sets, the analyses of these sets and the mathematical foundations upon which all of these areas are based. The training has the two-fold purpose of preparing the student to enter commercial, governmental and other work areas which extensively rely on statistical information and to prepare the student to continue in pursuit of a more advanced degree. Students with appropriate background (three semesters of Calculus and some knowledge of elementary statistics) usually complete the program in one or two academic years.

Course prerequisites include the equivalent of Math 601: Probability and Statistics and Math 609: Linear Algebra. Students without these prerequisites may take them without credit toward the M.S. degree.

The requirements for the Master's degree in Statistics is 30 hours of graduate course work. These must include 5 required courses (15 credit hours), 12 credit hours chosen from a list of graduate courses, and an independent study course (3 credit hours) in which the student develops a topic in depth and writes a report. A comprehensive examination is also required.

COURSES OFFERED

Student at work

The current programs and atmosphere at Tulane provide mathematics graduate students and faculty great flexibility in their directions of study and research. A wide selection of courses, including advanced topics seminars and elementary courses, are offered yearly. In this way, students are able to begin their studies at the appropriate level.

A general list of graduate courses is given below. Two-semester course sequences are offered starting with part I in the fall and part II in the spring. In addition to these courses, we offer special topics courses and seminars primarily in the areas in which the faculty will direct Ph.D. dissertations.

MATH 601 Probability and Statistics

MATH 603 Introduction of Stochastic Processes

MATH 605-6 Real Analysis I & II

MATH 609 Linear Algebra

MATH 611 Abstract Algebra I

MATH 620 Combinatorics

MATH 621 Differential Geometry

MATH 630 Complex Analysis

MATH 631 Scientific Computing

MATH 635 Optimization Theory

MATH 647 Analytic Methods of Applied Mathematics

MATH 701-2 Topology I & II

MATH 711-2 Algebra I & II

MATH 721-2 Analysis I & II

MATH 724 Mathematical Statistics

MATH 726 Linear Models

MATH 731-2 Applied Math I & II

Math 734 Numerical PDEs

Math 736 Data Analysis

MATH 751-2 Differential Geometry I& II

Math 753-4 Partial Differential Equations I& II

MATH 755 Probability

MATH 757-8 Scientific Computation I & II

MATH 771-9 Special Topics Courses

FACULTY AND AREAS OF RESEARCH

Professor M. Kalka

The Mathematics Department consists of 24 regular faculty members, several postdoctoral researchers and frequent visiting faculty in many areas of mathematics.

Its faculty enjoys national and international recognition in Algebra, Analysis, Differential Geometry, Mathematical Physics, Probability and Statistics, Scientific Computation, Theoretical Computer Science, and Topology. The researchers in Scientific Computation and in Statistics, and an increasing number of faculty in other areas, collaborate actively with colleagues in other units of the university such as the Schools of Engineering, Liberal Arts and Sciences, Medicine, and Public Health.

During the past five years our regular faculty have published over 100 research articles and several books. The regular faculty direct theses in very diverse areas which range through all of Pure Mathematics, Applied Mathematics, and Statistics. You can read brief descriptions of our specialties below. More detailed information can be found in

http://www.math.tulane.edu/faculty/

Algebra

Tulane has had a pioneering role in the research on semigroups, abelian groups, modules over valuation domains, and ordered algebraic systems. Some of the standard reference books in these areas are from Tulane authors. Recent work of the faculty is on problems in commutative algebra, the theory of rings and modules, and in the structure and cohomology of commutative semigroups (especially in the finite case). Weekly research seminars lead students to the frontiers of current research.

Faculty: John Dauns, Tai Huy Ha,

Visiting Faculty: Michael Joyce

Adjunct: Karl Hofmann

Domain Theory and Theoretical Computer Science

Domains are structures that are equipped with partial orders having special properties. Interest in these structures arose in the late 1960s when it was realized they could be used to produce models of the untyped lambda calculus of Church and Curry. This calculus can be viewed as a prototypical programming language without assignment. A whole area of research has sprung up since then, the focus of which is to investigate the structure of domains as well as their applications to areas ranging from theoretical computer science to pure mathematics. For example, recent work has shown that domains can be used to model fractals and to provide a novel approach to Riemann integration that extends established theorems in that area. This last work relies on a construction that introduces probability theory into domain theory. Domain theory is characterized by the relatively simply constructions that are available, but that have surprisingly general applications. The focus of the research taking place here at Tulane encompasses both the theory of domains and their applications to areas such as theoretical computer science. In recent years, the interest has been on modeling crypto-protocols used in the security community to establish secure communication between users on systems such as the Internet. Tulane enjoys collaborations with researchers at a number of other institutions in this area, including the University of Oxford, the Naval Research Laboratory in Washington, D.C., and the University of Paris VII. Collaborators from these institutions regularly visit Tulane, and we currently have an Oxford Ph.D. who holds a postdoctoral fellowship in the department.

Faculty: Michael W. Mislove

Geometry and Topology

Tulane has a sizable group working in a diverse array of topics in topology and geometry. Research in topology includes algebraic topology, 3 and 4 dimensional manifold theory, continuum theory and holomorphic dynamics. Research in geometry includes complex, differential, and algebraic geometry. The overlap and interplay among various fields of geometry and topology is apparent in this group: the researchers often use a variety of techniques from different areas to attack a problem, for example techniques from gauge theory and algebraic geometry have been used to study topological questions about 4-dimensional manifolds. This group currently has two weekly seminars (topology and geometry) that are well-attended by Tulane students and faculty as well as faculty from Loyola University.

Faculty: Morris Kalka (Chairman), Slawomir Kwasik, James T. Rogers, Jr., Albert L. Vitter III, David Yang

Applied Mathematics and Partial Differential Equations

The work of the group focuses on the qualitative behavior of solutions to partial differential equations, such as concentration, spatial and temporal patterns, stability, long time behavior, blow-up, phase separation and interfacial motion. Recently, we have studied anisotropic heat transfer. The equations are often nonlinear and are motivated by applications of mathematics to natural sciences. The goal is to understand a phenomenon and to develop mathematical methods that apply to more general situations.

Faculty: Steven I. Rosencrans, Xuefeng Wang.

Post Docs: Michael Nicholas, Xingzhou Yang

Symbolic Analysis

This area relates to mathematical problems that have their origin in Symbolic Computation. These include the closed-form evaluation of series and integrals, many of which have connections to Dynamical Systems, Combinatorics and Number Theory. The goal is to develop algorithms that will obtain such closed forms or prove their impossibility. Recent work of this group includes the development of Landen transformations that are the analogue of the classical Arithmetic Geometric mean for elliptic integrals.

Faculty: Victor Moll,

Visiting Faculty: Tewodros Amdeberhan (on leave)

Computational Mathematics

This group works on numerical methods for partial differential equations and maintains a strong interest in applications to general fluid dynamics with emphasis on biofluid dynamics. Recent work of this group includes accurate computational methods for flows in bounded domains and developing improved immersed boundary methods for aquatic animal locomotion, bacterial chemotaxis and bioremediation. Members of this group were co-founders of the Center for Computational Science (CCS) where they collaborate in research with faculty from the School of Engineering, the sciences, Health sciences, and more. Through the CCS, this group holds grants that support postdoctoral researchers, graduate students, undergraduates, and distinguished visitors.

Faculty: Ricardo Cortez, Lisa J. Fauci, Alexander Kurganov

Probability and Statistics

Group members work in applied probability, probability, statistics, and stochastic processes. Specific research interests include the bootstrap, censored data and survival analysis, dynamical systems with random perturbations, likelihood methods, linear models, Markov chains and Markov processes, and statistical inference. This group cooperates in research, applications, and seminars with the Department of Biostatistics.

Faculty: Michelle Lacey, John R. Liukkonen, Alexander D. Wentzell,

Mathematical Physics

The research in this area ranges from broad topics in Mathematical Physics such as cosmology and general relativity to geometric aspects of super-symmetric string theories and M-theory.

Faculty: Frank Tipler.

FACILITIES

Gibson Hall

The Mathematics Department is housed in the upper floors of Gibson Hall, a stone structure built in 1894. Here are located faculty, graduate students, and staff offices, as well as classrooms, seminar rooms and computers linked to Tulane's main computing system. The department also contains the A. H. Clifford Mathematics Research Library, housing some 28,000 bound volumes and subscribing to 243 journals devoted to all areas of mathematics.

The department has a network of Linux computers, Windows workstations, advanced Silicon Graphics and SUN workstations. This network links to a campus-wide RS6000 system for e-mail and software applications. Graduate students are provided with adequate computing resources, ethernet connections, and offices.

LOCATION

Kermit Ruffins

Tulane University is located in America's most exciting and most visited city. Our department is on St. Charles Avenue, across from Audubon Park, in a quiet residential area full of majestic oak trees and fine old antebellum homes. Often-photographed streetcars provide an easy ride to the picturesque French Quarter. New Orleans has a rich cultural life, with a symphony orchestra, operas, ballets, plays, a noted art museum, many art galleries, excellent jazz, a major jazz festival and many other events. During Mardi Gras (40 days before Easter) the town fills with parades and revelry. New Orleans is also famous for its cuisine; it boasts a number of great restaurants, and many more with good inexpensive meals.