Mission Statement As part of a large public research university, the Department performs many different roles in fulfilling its mission to discover and communicate mathematical knowledge. In the broadest terms, the Department is strongly committed to the following goals. To provide a wide range of opportunities for first- and second-year students to acquire the mathematical skills they will need in their chosen field of study. Diverse levels of student preparation and career goals require a variety of content and pedagogy. To offer a range of undergraduate concentration programs designed to prepare students for leading roles in education, science, business, industry, the professions, and other careers. Of equal importance with mathematical content in these programs are the analytical and problem-solving skills which find application in nearly every field of endeavor. To offer graduate programs leading to the M.S. degree which include a comprehensive level of specialized training in mathematics applicable to science, engineering, and business. To offer graduate programs leading to the Ph.D. degree which enable students to reach and expand the frontiers of mathematical knowledge through original research while acquiring a broad grasp of the current state of the field. An important component of these programs is training in the skills of teaching and communicating mathematics. To provide in all of these programs a supportive learning environment in which students can develop to their maximum potential. To conduct research in mathematics, individually and in collaborative groups with colleagues around the nation and the world, with the goal of increasing human understanding of all mathematical notions and their applications. To cooperate with agents of the University, the State of Michigan, other levels of government, representatives of industry and business, and local, national and international societies to further the field of mathematics and bring its power to bear in the solution of problems for the good of mankind. In particular, the Department is always cognizant of its role as the partner of the public schools of Michigan in the mathematical education of its youth. The rest of this document describes current and planned activities directed towards the realization of these goals. Courses intended for first- and second-year students serve a very broad clientele. The vast majority of students at this level are learning mathematics as an adjunct to another discipline in which they will major. They arrive at the University with preparation ranging from the barest minimum level of algebra and arithmetic to Advanced Placement credit for several courses. The goal of the Department is to offer a selection of courses for the entering student which will enable him/her to progress as rapidly as possible to the desired level of mastery. To this end, in addition to the main calculus sequence we offer a precalculus course which stresses treatment of data, a course designed expressly for students with AP credit for one semester, and four honors sequences of varying focus and challenge. Two of these pursues more deeply the theory behind the problem-solving techniques of the calculus, one stresses harder and more realistic problems from science and engineering, and one introduces students to combinatorial mathematics with a flavor quite distinct from calculus. All of these courses are open to students from all segments of the University and counseling is available to help each student make the appropriate choice. The Department is constantly working to improve both the content and pedagogy of courses at all levels. Many of the recent efforts in this direction have been focussed on the first-year courses. In addition to the development of the alternative courses mentioned above, there have been some major changes in instructional methodology. All of these classes are relatively small, with at most 32 students, and most emphasize active participation of the students in cooperative learning projects. Appropriate technology (graphing calculators, personal computers, and workstations) is used and there is a focus on fundamental concepts and communicating mathematical ideas. In these efforts we are playing a strong leading role in the national curricular reform movement and our methods have become a model for other universities. We place a very high priority on undergraduate instruction and expect to continue to change and improve our instructional program. An undergraduate student who intends to major or concentrate in mathematics should make this decision, at least tentatively, by the middle of the sophomore year. This is the time for most mathematics students to begin the transition from the problem-solving orientation of the calculus courses to the abstraction and rigorous thinking that are the hallmarks of more advanced mathematics. The Department offers several quite distinct programs leading to a BA or BS degree. Applications of mathematics to many different fields are featured in the options of the Mathematical Sciences program. Students may choose courses organized around one of eight different themes including Operations Research and Modeling, Mathematical Economics, or Discrete and Algorithmic Methods. For the more theoretically minded, there is the Pure Mathematics program. The Actuarial program offers preparation for a career in private and social insurance and employee benefit plans. The Teaching Certificate program includes courses in the School of Education and leads to a career in teaching mathematics in grades 7--12. These programs are regularly revised and modified in response to changing needs of our undergraduates. For example, we have recently introduced a new option in the Mathematical Sciences program in the Mathematics of Finance and Risk Management. This program includes courses in Economics and Statistics as well as a solid mathematical background including special new courses in financial mathematics and risk theory. Plans for the near future include an option in Mathematical Biology. Undergraduate students are also offered many opportunities beyond the classroom. Faculty counselors help them to select appropriate courses and develop a coherent program in accord with their intellectual and career goals. The University's Research Opportunity and Research Experience for Undergraduates programs offer experience in conducting individual research under the guidance of a faculty mentor. The Undergraduate Math Club and the Actuarial Club sponsor frequent meetings, which often feature mathematical lectures accessible to undergraduates or visits by representatives from business and industry. Local and national problem-solving competitions allow undergraduates to test their skills against their contemporaries. The BA or BS degree in mathematics is an excellent preparation for a wide range of careers. Although few companies have positions labeled "mathematician", they are increasingly eager to hire students trained in mathematics for many technically oriented jobs. It is almost universally recognized that the analytical and problem-solving skills that come with a degree in mathematics are easily transferable to other areas. Indeed, many of our graduates go on to careers in non-mathematical fields such as business, law and medicine. One of the important roles of a research mathematics department is the training of the next generation of doctoral-level mathematicians. Although the majority of the Ph.D. recipients from the Department go on to careers as college and university faculty, an increasing number take jobs in industry. Admission to graduate studies in our Department is highly selective: from around 450 applicants we usually enroll around 25-30 new graduate students each year. Entering students may choose from among over 20 graduate courses each semester. A typical program includes two courses in preparation for the Qualifying Review, a comprehensive exam, and a third elective course. During the second year, a student chooses an area of mathematics in which to concentrate, and after attending courses and seminars designed to bring him/her to the frontiers of current knowledge, begins doing research under the direct supervision of a member of the faculty. The process culminates with the writing of a dissertation containing original mathematical results of publishable quality. Most students complete the Ph.D. program in 5-6 years. An important element of the teaching function of the Department is the program of seminars, colloquia, and special lectures. Most research areas (see below) offer a weekly seminar at which local faculty and graduate students and visitors expound recent results. These are an important source of current information to faculty and students as well as colleagues from neighboring colleges and universities. A weekly Colloquium and frequent special lectures provide opportunities to profit from the insights of prestigious visitors from around the world. All members of the faculty share in all of the various teaching responsibilities: undergraduate and graduate courses, seminars, counseling, and individual guidance of research students. The normal responsibility of a faculty member is the teaching of two regularly-scheduled courses each semester. Some reductions from this schedule are given for administrative duties and to provide especially productive faculty additional time for research. A special mission of a mathematics department of a research university is the discovery and creation of new mathematical knowledge. Beginning with the research for their Ph.D. dissertations, members of our faculty find enormous satisfaction and excitement in ``doing mathematics" --- proving new theorems, formulating new concepts, discovering new connections, and solving problems from other areas of science and engineering. Although some mathematical research is directly applicable to other fields, much of it gets its significance from its place in the magnificent intellectual edifice that is modern mathematics. Moreover, history has shown repeatedly that parts of mathematics once considered of little practical value frequently turn out to be crucial for the understanding and description of scientific phenomena. Mathematics is as much a science of form as it is of number, and many areas of mathematical research may be distinguished by the types of forms they study. Most of the main research areas are represented in the Michigan Department. The core areas, those represented on the graduate Qualifying Review, are Analysis, Algebra, Geometry/Topology, and Applied Mathematics. Some other areas are Mathematical Logic, Number Theory, Combinatorics, and Algebraic Geometry. Many of these areas are in turn subdivided into fields such as Complex analysis, Differential Geometry, Representation theory, or Group Theory. Faculty associated with each of these areas collaborate with each other and with colleagues around the world to solve old problems and formulate new ones. Mathematics is a culturally independent subject; the same results are obtained and appreciated on every continent. Our faculty participate actively in hundreds of local, national, and international conferences each year at which they communicate the results of their research and learn from others. They also publish articles in the literally thousands of mathematics journals or, increasingly, in electronic form. The Department is committed to the position that excellent research and excellent teaching are complementary skills. At every level of instruction, the teacher who is also a researcher and scholar brings added understanding, depth, and intensity to the classroom and is able to infuse a course with perspectives derived from the most recent advances. The Department has a very distinguished faculty including two members of the National Academy of Sciences and many recipients of the most prestigious national and international fellowships and prizes. Among all departments of mathematics nationally, the Michigan Department has consistently ranked in the top ten. We will continue to recruit junior and senior faculty with both extraordinary accomplishments and promise in both research and teaching. Faculty in the Mathematics Department are involved in many sorts of service not directly related to teaching or research. Operating a department of over 90 faculty, 130 graduate students, 200 undergraduate majors, and many thousands of first- and second-year students requires a good deal of administration, which is shared between faculty and administrative staff. Similarly the larger College of Literature, Science, and the Arts and the University as a whole have many administrative tasks which must be shared by members of the various departments. Most of the mathematics faculty belong to one or more of the national mathematics organizations, such as the American Mathematical Society (AMS) or the Society for Industrial and Applied Mathematics (SIAM), and many serve on committees for the administration of these organizations or as editors of the professional journals they publish. Other faculty serve on review panels for funding agencies, such as the National Science Foundation, for the evaluation of other mathematics departments, and for other government agencies. Faculty also serve as technical consultants to many branches of industry and government, evaluate and review research articles for journals, and even advise political leaders.