HANDBOOK
FOR
MATHEMATICS
MAJORS

2014 – 2015

 


 


 
 

CONTENTS

INTRODUCTION

ACADEMIC PROGRAMS

MISCELLANEOUS INFORMATION


Preface

This handbook has been prepared for the use of mathematics majors at Oberlin College. It contains suggestions for planning a viable major and for starting a career, as well as information about the mathematics program and staff that will be of interest to students. However, no booklet is a substitute for individual advising. The Department strongly encourages students to seek out faculty members on an individual basis to discuss careers and plans. These discussions need not and should not be limited to your advisor; all members of the Department will be happy to discuss the program, as well as career options in mathematics, with anyone who cares to inquire. One of the reasons for coming to Oberlin is the personal attention available from faculty members; students are encouraged to take advantage of this opportunity.

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INTRODUCTION

What Is Mathematics?

Mathematics is the study of structure and the way it can be applied to solve specific problems. The mathematics one sees in high school and the first year or so of college––techniques for solving equations, trigonometry, analytic geometry and calculus––represents only a small corner of the discipline. Some of the structures discussed in more advanced courses include algebraic systems such as groups and vector spaces, geometric notions such as surfaces, manifolds and topological spaces, and spaces of differentiable or integrable functions. Such structures are used to construct mathematical models that may explain and predict events in a wide variety of disciplines. Mathematics has been with us since antiquity and is a pervasive force in our society; it is a diverse field encompassing many subjects that are, unfortunately, largely unknown outside mathematics. The major branches of mathematics and a few of their applications are briefly described below.

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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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Career Opportunities

Many people believe that the only career open to a mathematics major is teaching. This is not the case, as our graduates with jobs in industry and government can certainly attest. Modern society has become intensely technological and, as a result, there are jobs awaiting anyone with solid training in mathematics. Add to a strong mathematics program a good background in the arts and sciences, which any diploma from Oberlin should imply, and you will have a very marketable undergraduate degree. Studies have shown that the starting salaries of undergraduate mathematics majors are only slightly lower than those of engineers and are considerably higher than those of business, economics and accounting majors (see, for example, the report on the College Placement Council's salary survey in the Notices of the American Mathematical Society, November, 1984). Oberlin mathematics graduates in recent years have gone on to careers in a variety of business and industrial settings; others have entered graduate programs in medicine, law, economics and engineering, as well as in fields more closely related to mathematics. Many career paths are open to a mathematics major, only a few of which are mentioned below. For further information you should speak with one of the department faculty or consult the pamphlet "Professional Opportunities in the Mathematical Sciences", which is published by the Mathematical Association of America.

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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 Office of Career Services.

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 Office of Career Services to determine exactly which examinations you need to take and when it would be best for you to take them.

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What Do Our Math Majors Really Do?

Lots of different things. To get an idea of exactly what Oberlin mathematics majors do, the Department conducted an employment census, in the Fall of 1993, of Oberlin mathematics majors who had graduated between 5 and 15 years earlier. Here is what these graduates were doing.

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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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ACADEMIC PROGRAMS

Intermediate and Advanced Courses

The following is a list of all intermediate and advanced course offerings in mathematics. Some courses are listed under more than one heading. The list is broken down by area so that you can more easily plan your own program of study.

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Pure Mathematics
Algebra
MATH 232
Linear Algebra 

MATH 317
Number Theory 

MATH 327
Group Theory

MATH 328
Computational Algebra and Algebraic Geometry

MATH 329
Rings and Fields 
Analysis
MATH 231
Multivariable Calculus 

MATH 301
Foundations of Analysis 

MATH 302
Dynamical Systems 

MATH 356
Complex Analysis

MATH 358
Real Analysis
Geometry & Topology
MATH 350
Geometry

MATH 353
Topology
Logic & Set Theory
MATH 220

Discrete Mathematics 


MATH 340

Mathematical Logic 

Applied Mathematics
Applied Analysis
MATH 234
Differential Equations 

MATH 305
The Mathematics of Climate Modeling

MATH 345
Information Theory 
Combinatorics
MATH 220
Discrete Mathematics 

MATH 343
Combinatorics
Modeling
MATH 342
The Mathematics of Social Choice

MATH 348 Graphical Models

MATH 397
Seminar in Mathematical Modeling 
Operations Research
MATH 331
Optimization 

MATH 338
Probability Models and Random Processes 
Probability & Statistics
STAT 213
Statistical Modeling 

STAT 215 Statistics and Modeling

STAT 237
Bayesian Computation

MATH 335
Probability

STAT 336
Mathematical Statistics

STAT 337
Data Analysis 
Other

MATH 399
Seminar (topic varies)

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Requirements

Students sometimes confuse requirements with recommended programs. The requirements for the mathematics major were set up with the needs of double-majors and double-degree students in mind. These requirements represent a minimal level of knowledge that the Department expects of all majors, regardless of their talents or interests. Most mathematics students are strongly urged to take courses in excess of these basic requirements. The suggested programs below could serve as a guideline for designing a major in mathematics.

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.  Four 300-level mathematics (MATH) and statistics (STAT) courses, which must include

      1. MATH 301
      2. MATH 327
      3. A modeling course from the following list:  MATH 331, 335, 338, 342, 343, 345 or 348, or STAT 336, 337.

C.  Two additional mathematics (MATH) or statistics (STAT) courses numbered 200 or above.

Note: One or both of the courses in item C above may replaced by a course or courses from the following list:

Computer Science: 
(a)  CSCI 150, 151, 210, 241, 275.  Only one of these courses may count toward the mathematics major.
(b)  CSCI 280, 357, 365, 383.
Physics & Astronomy: 
ASTR 301, 302 and PHYS 212, 290, 310, 311, 312, 316, 340, 410, 411–12*
Chemistry: 
CHEM 339, 349
Economics: 
ECON 342, 351, 353, 355.

*The module courses PHYS 411 and 412 together count as one course.

The department frequently 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 B.3 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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Recommended Programs

The following programs represent suggestions for coherent sequences in mathematics. These may be used as models, but they are by no means the only ones possible. Students are strongly encouraged to work out, with the help of their advisors, the program that best fits their needs and interests.

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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:


Fall
Spring
First year
MATH 133
MATH 134
Second year
MATH 220
MATH 231, 232

or


Fall
Spring
First year
MATH 133
MATH 134
Second year
MATH 220, 231
MATH 232

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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Honors Work

Each spring the Department invites a few of the most talented juniors to participate in our Honors program during their senior year. Honors consists of a year of work or research in an advanced area of mathematics, under the close supervision of a member of the faculty. For some students this takes the form of two one-semester projects; for most student, however, a single year-long project is undertaken. At the end of Winter Term, Honors candidates take a written examination, administered by a mathematician from outside the College; during the spring semester they complete a paper on some aspect of their Honors project and the external examiner visits campus to conduct an oral examination on the project. The results of the exams, as well as the quality of the individual project and overall achievement in mathematics, are used to determine the level of Honors awarded.

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

The other important feature of the Honors program is that it presents an opportunity for close work between a faculty member and a student on a topic of interest to them both. This is valuable preparation for students interested in graduate work in either mathematics or statistics.

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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MISCELLANEOUS INFORMATION

Student Research

Although some areas of research in mathematics may be difficult for undergraduates to undertake, many opportunities do exist for "hands-on" experience. In this section we catalog a few possibilities for research activities that are available to qualified students. You should also feel free to talk with faculty members about opportunities beyond those mentioned below.

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. 

Other Research. The National Science Foundation regularly sponsors undergraduate research in mathematics by funding summer work at a number of institutions through its Research Experiences for Undergraduates (REU) program. For addition information, check the NSF's website http://www.nsf.gov/home/crssprgm/reu/.

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Contests and Competitions

The Mathematics Department sponsors teams for two national competitions:

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 Orr Prize

Each year the Department awards the Rebecca Cary Orr Memorial Prize in Mathematics to an outstanding graduating senior. This prize was established by the family and friends of Rebecca Orr, an Oberlin College freshman who was killed in a tragic accident in the spring of 1982 and was a promising mathematics student. The prize in her memory is awarded on the basis of outstanding achievement in undergraduate mathematics and promise of future professional accomplishment.



 
 

The Baum Prize

The John D. Baum Memorial Prize in Mathematics is awarded annually to the Oberlin student with the highest score on the Putnam examination. This prize was established by the Department faculty in 1988 in fond memory of a long-time colleague, mathematician, and friend.

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Invited Lectures

From time to time throughout the academic year, the Mathematics Department sponsors lectures by outside speakers. These speakers are often well-known and very eminent mathematicians with national reputations. Speakers have included Thomas Banchoff (Brown University), Steven J. Brams (New York University), Robert Devaney (Boston University), Frank Morgan (Williams College), Ragni Piene (University of Oslo), and Carl Pomerance (Dartmouth College).

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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Majors Committee

Composed of undergraduate mathematics majors, the Majors Committee acts as an interface between students and the Department faculty. Its members administer course evaluations at the end of each semester, interview candidates for faculty positions in the Department, make recommendations concerning tenure decisions, and consider various student concerns as they arise. The Majors Committee also helps organize a student-staff picnic every spring. A list of the committee members may be found on the bulletin board outside the Department Office. Students with questions about the mathematics program are encouraged to contact any member of the Majors Committee, as should anyone interested in serving on the committee.

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Faculty Interests

Below is a list of Department faculty and their mathematical interests. You might wish to use this list when planning reading courses or Winter Term projects, or if you would like more information about a particular area of mathematics.
 
Robert Bosch  Operations research, discrete mathematics. 
Jack Calcut Low-dimensional topology, algebraic topology.
Susan Colley Algebraic geometry, algebra, topology. 
Christoph Marx
Mathematical physics, analysis.
Kay Knight Precalculus, calculus, tutor training.
Michael Raney Analysis, operator theory. 
Lola Thompson
Number theory, algebra
James Walsh Dynamical systems. 
Elizabeth Wilmer Combinatorics, probability.
Jeffrey Witmer Statistics.
Kevin Woods Combinatorics, discrete and computational geometry.
Robert Young Complex analysis, functional analysis.

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