HANDBOOK
FOR
MATHEMATICS
MAJORS

• Preface

INTRODUCTION

• What is Mathematics?
• Career Opportunities
• What Do Our Math Majors Really Do?

ACADEMIC PROGRAMS

• Intermediate and Advanced Courses
• Requirements
• Recommended Programs
• Honors Work

MISCELLANEOUS INFORMATION

• Student Research
• Contests and Competitions
• The Orr Prize
• The Baum Prize
• Invited Lectures
• Majors Committee
• Faculty Interests

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Pure Mathematics

Pure mathematics has as its main purpose the search for a deeper understanding of mathematics itself. As a result pure mathematics seems at first far removed from everyday life. However, many important applications have been the results of advances in pure mathematics. This subject is traditionally divided into four main areas: algebra, analysis, geometry, and logic. However, some of the most exciting developments throughout the history of mathematics have resulted from the interaction between different areas.

Algebra is the study of abstract mathematical systems. These systems, such as groups, rings, and fields, generalize properties of familiar structures such as integers, polynomials, and matrices. The general, abstract approach of algebra has been fruitful in solving many problems in both mathematics and other disciplines. For example, using algebraic techniques, one can show that it is impossible to trisect an angle using only a straight-edge and compass. Group theory has been employed liberally in quantum mechanical physics and physical chemistry. Other recent applications of algebra include cryptography and coding theory.

Analysis is the study of infinite processes. As such, it concerns itself with phenomena that are continuous as opposed to discrete. Starting with the fundamental notions of function and limit, it builds differential and integral calculus, which is the mathematics of continuous change. Analysis in turn gives rise to a deeper and more general study of functions of both real and complex variables. The area is pervasive, and it finds rich and varied applications in almost every field of pure and applied mathematics.

Geometry is the study of curves, surfaces, their higher-dimensional analogues, and the properties they possess under various types of transformations. Geometry and topology frequently make use of techniques and notions from algebra and analysis. Geometry is an important subject for our understanding of the nature and structure of spatial relations.

Logic is at the very foundation of mathematics. In this field one studies the formulation of mathematical statements, the meaning and nature of mathematical truth and proof, and what can possibly be proved in a mathematical system. For example, there is a famous theorem due to Kurt Goedel that says that in any logical system rich enough to contain arithmetic there are true statements that can neither be proved nor disproved. Logic has found many important applications in the study of computability in computer science.

Applied Mathematics

Applied mathematics is the development and use of mathematical concepts and techniques to solve problems in many other disciplines. Unlike pure mathematics, the areas of applied mathematics fall under no simple classification. Nonetheless, the following topics cover many of the important applications.

Applied analysis involves the study of techniques for analyzing continuous processes and phenomena. For example, many methods from real and complex analysis are utilized when looking at problems of a physical or computational nature. Differential equations and numerical analysis are two examples of subjects that come under this heading.

Combinatorics, the study and enumeration of patterns and configurations, is one technique for analyzing phenomena that do not behave in a smooth or continuous fashion. Techniques from algebra and other areas are applied to study a wide variety of problems in such areas as graph theory, scheduling, and game theory. There are also many important applications of combinatorics to computer science.

Probability and statistics are among the most fundamental tools for mathematical modeling. The importance of probability lies in its formulation of chance (or stochastic) processes and its applicability to the analysis of non-deterministic phenomena. Statistics deals with the collection and analysis of data and with the making of decisions in the face of uncertainty. It is frequently used in the social sciences as well as in all areas of experimental science.

Operations research involves applications of mathematical models and the scientific method to help organizations or individuals to make complex decisions. Traditionally, it has focused on mathematical optimization theory. Typical problems in operations research include the development of an optimal flight schedule for an airline and finding the best inventory policy for a bookstore.

Actuarial mathematics is one of the early examples of mathematical modeling. This field uses the methods of probability and statistics, along with a study of economic factors, to estimate the financial risk of future events.

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Academic Research

Research in pure mathematics is conducted primarily by college and university faculty. If a career in research mathematics and teaching at the college level appeals to you, you will need to attend graduate school and ultimately earn a Ph.D. in mathematics. An Oberlin degree is a good first step in this process. In fact, during the period 1920–1990, Oberlin produced more graduates who have gone on to earn Ph.D.'s in mathematics than any other primarily undergraduate institution in the United States. The nature of research is such that it is vital for one to have a thorough background in theoretical and foundational areas. Thus, even if you want to pursue research in an applied area such as statistics or operations research, you will best prepare for graduate school by taking as much pure mathematics as you can during your undergraduate years.

Industry and Government

Career possibilities in industry include employment in scientific and engineering firms, as well as work in financial and management companies. In addition to a career in the private sector, there is also the possibility of government work. For a mathematics graduate, there are opportunities at government laboratories (such as that in Los Alamos, NM), federal research centers, (e.g., the Center for Naval Analyses), and the Federal Reserve Bank, to name a few. Mathematics students are highly regarded by industry and government for their general problem-solving abilities, their analytical training, and their ability to think abstractly. Most companies are willing to provide some specific training on the job. In fact, some companies even prefer to do this, for then you may be more thoroughly indoctrinated into the company's standard methodologies. Computer experience is very desirable for such a career, and some training in an area to which mathematics can be applied is useful. The best preparation, however, is a solid program in both pure and applied mathematics.

Actuarial Work

Actuaries evaluate the current financial implications of future contingent events. Sometimes called "social mathematicians", they project the financial effects that various human events--birth, sickness, accident, retirement, and death--have on insurance and other programs. Salaries are high in this profession and successful actuaries are in constant demand. Professional advancement in this field comes through passing a series of examinations administered by the Society of Actuaries and the Casualty Actuarial Society. The first of these examinations covers the calculus of one and several variables, differential equations, and probability. Questions on the examination are presented primarily in the context of risk management and insurance. With some study, any mathematics major should be able to pass this examination and thus become eligible for both summer jobs and an entry-level position upon graduation.

Pre-Collegiate Education

There is a very real and disturbing shortage of qualified mathematics teachers and mathematics supervisors in the nation's elementary and secondary schools. Oberlin graduates typically have both the academic and the personal strengths to make a tremendous contribution in this field. To teach in the public schools, one must be certified; this means approximately a year of courses in mathematics education and practice teaching. Oberlin entrants into the teaching field typically take this program at the post-graduate level, obtaining a master's degree in mathematics and/or education along with certification. A post-graduate year, or more, is strongly recommended as suitable grounding for the kind of leadership that Oberlin graduates can expect eventually to exert in the field. Due to the above-mentioned personnel shortage, generous fellowships are often available. Work with school students within the mathematics program in the Oberlin City Schools is also available, usually for credit.

Another option is that of beginning teaching immediately upon graduation in a private school, for which certification is not required. One might plan on teaching for only a few years in a private school before going on with the next stage of one's career, or on staying with it indefinitely (in which case some post-graduate work will soon become desirable). There are several placement services for teaching positions in private schools. With the help of these services, Oberlin students have been very successful in finding placements. For additional information, see a member of the Department, or consult the Center for Engaged Liberal Arts (CELA), Career  Exploration and Development.

A Note for Double Majors

We shouldn't leave this section on career opportunities without mentioning what an attractive package a double major in mathematics can be. At Oberlin, many students have taken double majors in mathematics and either physics or economics, but these are not the only possibilities. For example, biomathematics and biostatistics are fields which apply mathematical techniques to the study of biological systems. Good undergraduate preparation for work in such areas would naturally include study in both biology and mathematics. Rigorous mathematical techniques have only recently begun to be employed in some of the social sciences, so such a double major might be very attractive to a prospective employer or graduate school. Many other combinations are possible. We have had double majors with branches in almost every department in the college. In addition, many students have completed double degrees in mathematics (in the College of Arts and Sciences) and music (in the Conservatory). The mathematics major at Oberlin gives you the opportunity to design a strong and personal program of study. To take full advantage of this opportunity, you should start planning your own program early in your college career.

Examinations for Graduate Schools and Careers

You should be aware of some of the standardized tests that you may have to take, so that you can plan your academic program in order to take these examinations at the appropriate times. For example, if you are considering graduate study, you should take the Graduate Record Examination (GRE) by the fall of your senior year. There are two types of GRE's: (a) the general aptitude exam, which is much like the SAT––there are verbal, analytical, and quantitative portions; and (b) the advanced exam, which is given in a wide range of academic areas. If you are planning to do graduate work in mathematics or computer science, you will normally need to take both the aptitude examination and an advanced examination.

If you are interested in an actuarial career, you will want to begin taking the series of professional actuarial examinations. The first examination concerns freshman and sophomore mathematics and probability.

Other standardized tests you might need to take include the MCAT (for medical school), the LSAT (for law school), and the GMAT (for business school). You should consult with your advisor or talk with a consultant in the CELA/Career Exploration and Development to determine exactly which examinations you need to take and when it would be best for you to take them.

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There were a total of 276 mathematics graduates in the classes from 1978 to 1988. An unexpectedly large number (82) of these graduates were in the financial world as consultants, analysts, actuaries and the like. A substantial subgroup (34) of these were in managerial and/or directorial positions. Their employers ranged from banks and insurance companies through industrial and computing firms including the likes of Shearson Lehman Hutton, AT & T, Bankers Trust Co., Chemical Bank, and Allstate.

Beside this group, the next largest category were graduates who remained in academia either as teachers (39) or as graduate students (35). Fields taught and/or studied by these grads ranged over all the physical sciences. The largest group (naturally) consisted of mathematics teachers and students, but other subjects covered included statistics, physics, chemistry, economics, electrical engineering, biostatistics, operations research, and psychology. Another large group (37) was employed in the computer industry. Employers of this group included Microsoft, Sun Microsystems, Bell Labs, and Hewlett-Packard.

Beyond these groups, graduates spread out into a remarkable variety of professions and specialties. There were doctors (5) and lawyers (11). Nine others were involved in the legal or medical professions as residents, law clerks, and medical/legal researchers.

Not surprisingly, there were artists among our graduates. Most of them were musicians (16): performers and teachers, but there were also graphic artists including an architect and a freelance cartoonist.

Fewer graduates than we expected (26) worked in government. These alumni included three employed by NASA, several employees of the Justice Department, and four graduates in the military.

Some unusual jobs our mathematics graduates have had are: helicopter pilot, aircraft commander, farmer, minister, rabbi, clinical psychologist at a federal correctional institution, and wardrobe staff person for "Miss Saigon".

Ten percent (24) were "missing data" and, in addition, one graduate reported being unemployed. All in all, we think, an impressive variety of careers.

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Major in Mathematics

A major in mathematics consists of at least 9 full academic courses, which must include:

A.  MATH 220, 231, and 232.
B.  CSCI 150 or 151.
C. MATH 301 and 327.
D. A modeling course from the following list:  MATH 305, 318, 331, 332, 335, 342, 343, 397, or STAT 336, 339
E.  One 300-level mathematics (MATH) or statistics (STAT) course.
F.  One additional 200- or 300-level approved elective.

Note: One of the courses in item F above may include a course from the following list:

The department occasionally offers a 300-level seminar in addition to its regular offerings. Students should check with the instructor to find out whether the seminar can be used to fulfill requirement D above.

Minor in Mathematics

A minor in mathematics consists of at least five full academic courses in mathematics (MATH) or statistics (STAT) numbered 200 and above, including at least two courses numbered 300 and above.

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For All Students

After completing the introductory calculus sequence (MATH 133–134), students who would like to major in mathematics are strongly encouraged to enroll first in Discrete Mathematics (MATH 220), Multivariable Calculus (MATH 231) and Linear Algebra (MATH 232) during their sophomore year. We also strongly urge students enroll in Discrete Mathematics before enrolling in Linear Algebra. For students with no previous background in calculus, the following 2-year sequences should be typical:

or

Naturally, students who enter Oberlin with some credit for calculus should place themselves at the appropriate position in this sequence, and also consider enrolling in other mathematics courses as well.

Graduate School in the Mathematical Sciences

Students who plan to attend graduate school in any mathematical discipline should concentrate as undergraduates on core mathematics. We recommend at least one advanced course in each of these areas: algebra, analysis, and applied mathematics, and at least one additional advanced course in one of these areas. Graduate faculty in pure mathematics normally expect students to have taken a full year of both analysis and algebra, as well as courses in geometry or topology. Graduate faculty in applied fields will expect a somewhat different background, but a year of analysis, a course in algebra, as well as courses in applied mathematics (probability/statistics, operations research, etc.) are essential. Students considering graduate school in statistics should take some undergraduate statistics, at least one computer science course, and, of course, lots of mathematics.  All students planning graduate work should take considerably more than the required 9 mathematics courses. It is also advisable for such students to take courses in other disciplines that make use of advanced mathematics, such as physics or economics.

Non-academic Careers in Mathematics

Experience in computer science at least at the level of CSCI 150 is recommended for anyone planning a career in industry or government. As mathematical preparation for such a career, we recommend at least two advanced courses in pure mathematics as well as several advanced courses in applied mathematics. Among the latter the Probability course is particularly recommended, since non-deterministic models are important in almost every branch of applied mathematics.

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The Honors program serves two important roles. Although the Department offers a rich set of courses in pure and applied mathematics, talented students occasionally exhaust our course offerings. The Honors program allows such students to explore particular areas of mathematics in greater depth. A few recent projects have been in

• Chaotic Dynamical Systems
• Algebraic Geometry
• Number Theory
• Mathematical Logic
• Optimization
• Low-dimensional topology
  
• Computational algebra.

Students are admitted into the Honors program at the invitation of the Department. The factors considered most heavily by the Department in selecting candidates are: success in course work, enthusiasm for mathematics and interest in graduate work, and a broad base of completed courses. Students interested in Honors work should normally complete a substantial number of 300-level courses, including some of the theoretical courses, by the end of their junior years.

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MATH 550 and 551. There are two research courses in the Mathematics Department, MATH 550 (fall) and 551 (spring) that enable students to pursue research projects under the supervision of faculty for academic credit (either as a full or half course). 

Senior Scholars. The Senior Scholars program permits a few exceptional students to spend their senior years working on research rather than on course work. The Mathematics Department has only rarely had senior scholars––just two in the past forty years.

Summer Research.  From time to time, funds are available to support students to work with faculty at Oberlin during the summer months. 

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The William Lowell Putnam Examination is administered each year by the Mathematical Association of America. There are morning and afternoon sessions, with six problems assigned per session. Each college and university in the nation is allowed to enter a team of three. (In addition, any undergraduate may participate as an individual.) The team members work separately; the score for a team is the sum of the individual members' ranks. The examination is difficult, but Oberlin has at times done quite well, placing second in the nation in 1972 and tenth in 1991. A cash prize is given to the highest scoring Oberlin student. (See The Baum Prize section below.)

The Consortium for Mathematics and its Applications has recently started a national contest in mathematical modeling. The College has sponsored a team of three undergraduates. These students are given a choice of one of two unstructured problems which they must carefully formulate and attempt to solve over a weekend. We have sponsored a team for this contest each year. For more information, consult the Chair of the Department or Robert Bosch.

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The Department is also a member of the Ohio Colleges Speaker's Circuit, a consortium of area colleges whose purpose is to promote contact by exchanging speakers from among their respective mathematics faculties.

Distiguished Visiting Scholar. Beginning in 1995–96 and thanks to the generosity of alumni, the Department is able to sponsor an annual visit by an eminent mathematical scientist who will conduct classes as well as deliver the Fuzzy Vance Lecture (a public lecture).

Lenora Lecture. Beginning in 2013–14 and thanks to the generosity of Robert Young and in memory of his Aunt Lenora, the Department is able to bring a mathematical scientist to campus annually to collaborate with a faculty member and to deliver a public lecture.

Tamura/Lilly Distinguished Lecture. Thanks to the generosity of Roy Tamura '78 and the Eli Lilly Company, the Department sponsors an annual visit by a mathematical scientist to deliver a public lecture.

Attending lectures is an important and fun way to learn about mathematics outside of the classroom. Students at all levels are strongly encouraged to go to talks and to meet the speakers.

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Updated August 20, 2024