Academic Goals, Initiatives and Impact

Here is a summary of our academic goals and proposals for new developments. The details and potential impact of these initiatives will be described in the following sections. Recommendations for implementation of our Strategic Plan appear on the last page, followed by two Appendices.

1. Undergraduate Education
    a) Research Based Learning
    b) Restructuring the Undergraduate Program 
    c) Undergraduate Co-op Program
    d) Outreach to High Schools and the Community 

2. Graduate and Postgraduate Training
    a) New M.Sc. Program in the Mathematics of Finance
    b) Interdisciplinary Programs 
    c) Postdoctoral Research Program

1. Undergraduate Education

(a) Research Based Learning

We want to provide a coherent teaching and learning environment for Mathematics on campus.  Research based learning means bringing students into regular contact with the research process by means of inquiry courses and an increased emphasis on the applications of mathematics.

Those who study mathematics even at the introductory level can gain a sense of future career options from contact with more advanced students. Until recently most mathematics graduates were employed as teachers at the high school or university level, or as actuaries in life insurance companies. But the advent of the computer and the accelerating pace of technology in our society has opened up new possibilities. Today, students with mathematical skills can find rewarding careers in business and industry sectors such as finance, statistics, computer graphics, communications, air and space.

Use of the lecture theatres for large Level I courses will introduce all mathematics students to the Centre, and provide them with the opportunity to form lasting connections. In Level 2 and beyond, students will benefit from the more specialized facilities of the Centre: classes in smaller seminar rooms, interactive learning through the use of technology, and expository guest lectures by mathematicians working in business and industry.

We plan to include a Mathematics Home Base for undergraduate students, an idea inspired by the facilities of the Arts & Science program. This will be a congenial drop-in place for students at various stages of their studies to meet informally with each other and with faculty. The Home Base will provide a resource area for supplementary course materials, such as mathematical software, multimedia tools and problem collections.

(b) Restructuring the Undergraduate Program

Our new Level I program in Mathematics & Statistics has just been approved by Undergraduate Council, and next year we will compete with the University of Waterloo for applications from the best high school graduates specifically interested in studying mathematics. The program is very flexible: students would take Math 1A3, Math 1AA3, and Math 1B3 in Level I and, depending on their interests, include a wide range of prerequisites for further study. In preparation for the new Level I program, we have just introduced a new first year inquiry course Math 1C3, which we expect to be required for all students in the program.

The Department is also considering a simplification of our upper year programs by identifying thematic streams to add emphasis in particular areas of Mathematics and Statistics around a common core. For example, many of our students want to combine mathematics and computing. Our proposed new stream in Computational Mathematics will meet this need by integrating computing skills, numerical methods and applications into some of our existing courses. In addition, we plan to introduce a new Level I course in Computing and Simulation to support this stream. Other possible streams include Mathematics & Physics, Biology & Mathematics, and Actuarial Science. The success of our recently introduced course Math 2U3 (Teaching Mathematics) indicates that we could also have an important role in training future teachers at the elementary or high school level to meet the large expected demand over the next five years. 

(c) Undergraduate Co-op Program

The Department will introduce a Co-op program in Mathematics and Statistics in order to promote mutually beneficial connections between our department and business in the greater Hamilton -- Toronto region. Our students should be able to get work placements in the nearby manufacturing, statistical consulting, environmental and financial service industries.

The Department will probably adopt the academic and work term sequence recommended by the Science Cooperative Education department. Students would be admitted into the program from Level II and would start the first of two 8 month work terms during January of Level III. The second work term starts in May of Level IV. If planning begins this summer and fall and approval for the program is granted, then students in second year could be admitted to the program in the spring of 2001 and the first 8 month work term would commence in January 2002.

(d) Outreach to High Schools and the Community

Successful universities recognize the need to be good citizens in their community. Our department has an excellent opportunity to contribute to this part of the university's mission by using the Centre for Mathematics as a focal point for outreach to the community. This Centre would encourage other members of the University to see our department as a resource, open to the questions and interests of non-mathematicians. The Centre will be seen as a meeting place for the many members of the University community who use or have an interest in mathematics and statistics.

Interdisciplinary programs and research collaborations within the university support these objectives, and many connections of this kind already exist. For example, several members of the Department currently collaborate with members of the Engineering Faculty, while others are developing links with the departments of Biology and Psychology. In addition, we have joint programs with Biology, Computer Science, Economics and Physics. Our Statistics group also has long-standing links with the departments of Chemical Engineering and Clinical Epidemiology and Biostatistics through its joint M. Sc. program.

Contemporary experimental work in science, engineering and medical research labs all require sophisticated mathematical and statistical techniques. Population studies in medical research or economic and social studies require statistical modeling and analysis of data. Valuable experience has already been gained in recent years through contract consultations or M.Sc. projects supervised by members of our Statistics group. The Centre for Mathematics will enable us to establish a more formal Statistical Consulting service to provide statistical advice on many of these problems. It will include an area for initial contacts with clients, project development, and presentation of results.

It is also extremely important to reach the wider community. The Department plans to establish a liaison program for building contacts with our local high school mathematics teachers, students and parents. Our department can help the public become aware of the increasing opportunities in mathematics and its importance in business and industry. We plan to organize and establish professional development programs for teachers and special programs, such as summer camps and math contests for students in our local school system. We are all aware of the success of similar activities in our Faculty of Engineering and also in the Faculty of Mathematics at the University of Waterloo.

2. Graduate and Postgraduate Training

For the first time in 30 years, we are entering a period of great upheaval in graduate Mathematics education. Because of the increasing use of Mathematics and Statistics in the outside world and the likely shortage of qualified academic personnel in the near future, we expect that the demand for graduate study at Ontario universities will greatly increase in the near future. Because of the strong research profile of our department, these circumstances provide the first real opportunity in many years to strengthen our graduate program.

a) M.Sc. Program in the Mathematics of Finance

The present M.Sc. program is being expanded to accommodate students in financial mathematics -- mathematical finance and risk management, energy finance, and stochastic methods in data analysis. We are also considering the introduction of an internship in business or industry, such as now exists in the Statistics M.Sc. program, for finance and other fields of applied mathematics. A new M.Sc. program with significantly different requirements may soon be necessary in order to accommodate these developments.

Financial mathematics is a rapidly developing area, whose growth has been generated by the needs of the finance industry for highly sophisticated models for risk management based on fundamental mathematical areas such as analysis, probability and mathematical physics. Our Financial Mathematics group will engage in contract research and seek joint projects with industry and the DeGroote School of Business with possibilities for commercial ventures. This new blend of applicable mathematics requires a new way of working: team-oriented rather than individual research, goal-oriented rather than curiosity-driven. The proposed Financial Mathematics program will provide the atmosphere and facilities for this exciting development.

A significant start has already been made in this direction with the establishment of PhiMac, a group of faculty, postdoctoral fellows and graduate students in our department, working together on problems in financial mathematics such as statistical properties of financial time series, derivative pricing models in markets, fixed interest rate models, and portfolio risk management. Funding by Mitacs now partly funds a postdoctoral student and a central location for their activities. These activities are described in more detail at the web site  www.math.mcmaster.ca/phimac.

(b) Interdisciplinary Programs

Faculty with applied interests are already involved in interdisciplinary work, and we would like to develop more such contacts. At the same time, engineers and physicists have asked our department for graduate level courses in areas such as asymptotic analysis or numerical methods, while biologists and psychologists would similarly appreciate courses in mathematical modeling appropriate for their students. The provision of joint courses is another way to improve our graduate program and raise the profile of mathematics at McMaster.

The Department will encourage graduate students with other backgrounds (e.g. engineering, physics or theoretical biology) to pursue a joint graduate degree in mathematics. In such cases joint supervision (e.g. by a mathematician together with an engineer or biologist) is appropriate. In addition, an M.Sc. program in "Mathematical Modeling & Industrial Applications" is being planned, which will provide training in data analysis, mathematical modeling, and numerical analysis; it would necessitate special academic requirements and am internship.

(c) Postdoctoral Research Program

Another major goal of the Department is the establishment of the Centre for Mathematics as one of the major research institutes in Canada. The Department of Mathematics and Statistics has strong researchers in main-stream areas of mathematics, including analysis, geometry and topology, algebra and number theory, mathematical biology, fluid dynamics, probability and statistics. We also have the most active postdoctoral fellowship program in Canada. Graduates from the best mathematics programs in the world (e.g. Harvard, Columbia, Princeton, Berkeley, Cambridge) come to McMaster for two years of research and some teaching (6 units/yr). The size of our program (about 15 each year) makes it comparable to the best research institutes, and the positive effect on our research activity can be felt by every group in the department. After leaving McMaster, our postdoctoral fellows have almost all gone on to tenure-track jobs at good universities or high-level positions in business. This is a real success story at McMaster and we should do everything possible to support this excellent program.

The creative potential of our faculty and postdoctoral fellows will be enhanced in the Centre for Mathematics by the coherent space, seminar rooms, consulting centre, library and web resources, and the new student-based learning initiatives.

The mathematical scene in Canada has recently been radically transformed by the creation of new nation-wide research networks linking the Pacific Institute in British Columbia and Alberta, the Fields Institute in Toronto and the Centre de Recherches Mathematiques in Montreal. McMaster was one of the founding universities of the Fields Institute and this relationship has had a significant impact on the research and training opportunities in our department. More than half of the thematic programs at the Fields Institute since its inception have been organized by faculty from McMaster, from which graduate students and postdoctoral fellows in these areas have benefited greatly.

Until the present time, almost all of this activity has occurred at the Field Institute itself. This will change during the next phase of the relationship between Fields and McMaster. The new National Centre of Excellence for Mathematics and "Mitacs" (the Mathematics of Information Technology and Complex Systems) will give every research-intensive university in Canada the opportunity to participate in university-industry collaborations. These activities will be integrated into our new graduate programs.

3. Faculty Appointments

Because of the diverse range of applications of the mathematical sciences, there is already a heavy demand for our courses from every Faculty in the University other than Humanities. In addition, as a result of the outreach activities of the Centre for Mathematics, we expect an increasing demand on our higher level courses and research seminars as the relevance and applicability of the more sophisticated tools from practically all branches of the mathematical sciences become better known.

We observe that applications in the mathematical sciences are often made possible by advances in curiosity-driven research, and that practical problems often stimulate the necessity of more basic research. The interactive relationship between the pure and applied parts of the mathematical sciences can only flourish when both parts are valued and nurtured.

The second important factor which influences the assessment of our human resource needs for the next five years is the impact of the enrollment increase¹ expected in our universities due to the double cohort and baby-boom echo. These increases will affect our department more severely than other departments in the University. Even now before these effects materialize, we have been forced to deal with the increase in teaching units resulting from the enrollment increase by the Faculty of Engineering and new undergraduate programs in Health Sciences, Kinesiology, and Radiation Technology.

The third important factor is our intention to support enthusiastically the recommendations of the Boyer Commission Report, which are endorsed in the report on Academic Planning by the University Planning Committee. Specifically, the recommendations to make research-based learning the standard and to remove barriers to interdisciplinary education will call for more research faculty. Students engaged in research-based learning require more individual attention as well as the feedback which only active researchers can supply.

The Department of Mathematics and Statistics teaches students from all Faculties and all programs of the University. Any increases in enrolment and any new programs introduced elsewhere in the University have an immediate effect on the resources of our department. We hope that the Administration will give due consideration to the impact of the academic plans of other units in the University on our resources in this round of academic planning.

(a) New Tenure-Track Appointments

Faculty members at McMaster are expected to be both excellent researchers and excellent teachers. They are the people who enhance the reputation of McMaster as a research-intensive university through their own research and through the training of post-doctoral fellows and graduate students. Equally importantly, they are the people who have the experience, breadth of knowledge, and personal contacts to implement research-based learning and interdisciplinary education at the undergraduate level.

In order to establish the Centre for Mathematics on the scale outlined in our strategic plan, we estimate that a 25% increase in our present faculty complement² of 32, or 8 new faculty positions, will be needed over the next five years.

In trying to create these positions, we will pursue vigorously all available opportunities for external funding. For instance, the recently announced Federal program to establish the Canada Research Chairs can be used to bring distinguished senior faculty into the department who will play a leadership role in some of the new initiatives described above. The McMaster campaign has also identified an endowed Chair in Mathematical Biology as a target for fund-raising. 

The Department has identified a number of areas (see the Appendix 1) where the hiring of new faculty members could make a strong contribution to our strategic plan. We emphasize that it remains the responsibility of the Chair and the Appointments Committee of the Department to set priorities and recommend appropriate action to the Dean when opportunities arise. Attention should be paid primarily to excellence in research and potential for excellence in teaching, maintaining or enhancing research strength across a broad range of active mathematical areas, and hiring young people to continuously renew our scientific energy.

(b) Teaching Appointments

By "Teaching Appointments'' we mean faculty members who are primarily teachers at the undergraduate level. They are people who enhance the quality of undergraduate education at McMaster by virtue of their superior teaching skills, their involvement in course development, and their exploration of new means of teaching. In the context of the Department of Mathematics and Statistics, their normal teaching load would be 18 units per academic year spread over 3 semesters (fall, winter and summer). We currently have two faculty members (one on a CLA) who were hired for this role, and several others whose primary contributions are now in teaching because they have given up research.

Since McMaster currently does not have a well-defined career path for faculty members in a tenure-stream appointment who contribute primarily through teaching³, we strongly urge the University toset up a structure within which "Teaching Appointments" might be made, one which allows for appropriate rewards and job security. We see this first as"good management" to maximize the job satisfaction and productivity of people already at McMaster, and secondly as a preparation for a possible need to hire extra teaching staff in a time of enrolment increases and shortages of research mathematicians in the job market.

4. Staff Appointments

(a) An Undergraduate Coordinator

The increase in student numbers during the next few years makes it mandatory that much greater help be provided in the teaching of undergraduate courses, in particular to those teaching the increasing number of large first year classes. Because of the huge number of students taking different sections of the same course, especially in Year 1, practically all faculty members in the Department of Mathematics and Statistics must coordinate a multi-section large-enrollment course each year.

While all the tasks involved in the coordination are important, many do not require expertise in research or scholarship, although they do require some background in mathematics ands ome knowledge of how courses are taught, as well as computer skills (TeX, HTML, web page design). Many of the tasks are the same across different courses and remain the same from year-to-year. We recommend the appointment of an Undergraduate Coordinator to support large course instructors in the scheduling, invigilation and marking of midterm tests, supervision of teaching assistants and graders, construction of course web-sites, and the editing and compiling of custom courseware.

(b) Internet and Computing Support Staff

We recommend that one additional support staff person be appointed to assist the Undergraduate Coordinator in the tasks described above. This staff member would also work with the rest of the secretarial staff to establish standards for web page design, grade-keeping spreadsheets, and handling of data in support of our undergraduate courses.

5. Infrastructure

(a) Space and Facilities

The move to Hamilton Hall and the establishment of the Centre for Mathematics will require an extensive planning process. We recommend that the Dean set up a Transition Committee as soon as possible to produce detailed plans in consultation with architectural and building consultants. The Transition Committee should report to the Dean and consist of the Chair (ex officio), Dr. James Stewart, and 3-4 faculty members of Mathematics & Statistics appointed by the Chair.

The broad outlines of our space and facilities requirements in Hamilton Hall are clear:  the large lecture theatres should be retained and enhanced with technological aids, and smaller classrooms and seminar rooms will be needed to provide teaching space for a full range of our courses. We will need space for the Mathematics Home Base, office and research space for faculty and staff, and space for graduate students. One of the roles of the Transition Committee will be to co-ordinate input from the department on the optimal use of space in a renovated Hamilton Hall.

(b) Computing Resources

We recommend that the Centre for Mathematics include a medium-size computing cluster (30-50 screens) to provide facilities for interactive computing for a variety of mathematics and statistics courses. A mathematics sub-net will be needed to link the computing resources in individual offices and provide access to common software packages.

(c) Access to Library and Audiovisual Resources

The Centre for Mathematics should contain a library resource area, consisting of a small collection of books, preprint series and reference materials, including access to MathSciNet and other internet resources. We would like to investigate the possibility of displaying a selection of the incoming mathematics and statistics journals for 2 weeks before they are sent to Thode Library. In this way, researchers can have immediate access to the latest published work in their areas.

Audiovisual resources will be increasingly important in the future as we develop new ways to teach. The classrooms in Hamilton Hall should be equipped with display projection facilities from computers or the internet, by some combination of fixed and portable devices.

Recommendations Summary

1. That the Faculty of Science endorse this Strategic Plan for the establishment of the James Stewart Centre for Mathematics in Hamilton Hall and support the academic initiatives proposed by the department.
   
2. That the Faculty of Science give a high priority to fulfilling the faculty, staff and resource requirements of the Centre for Mathematics so that its academic goals can be achieved.
   
3. That we maintain our commitment to renewal by hiring young faculty and our commitment to excellence by building research strength in a broad range of major mathematical areas.

Appendix 1

Some Hiring Opportunities in Mathematics & Statistics

1.Discrete and Computational Mathematics

Technology today poses a great number of problems that require the construction of complex systems through specific arrangements of their components. These include problems in the scheduling of industrial processes, communication systems, electrical networks, organic-chemical identification, economics and numerous other applied areas. Discrete Mathematics, and in particular Graph Theory and Combinatorics provide many of the mathematical tools used for solving these types of problems and as such are important and vital areas of mathematics. Appointments in these areas could assist in the development of a stream of study in Computational Mathematics. By putting in place such a stream of study, the Department will be able to attract a larger group of students into the mathematical sciences.

2. Geometric Analysis

The theoretical background behind many of the applied priority areas in our academic plan is Analysis, in particular the sub-areas of nonlinear partial differential equations, geometric analysis, Fourier analysis and wavelets. Geometric analysis in particular is one of the most important areas of modern mathematics, but Canadian mathematics departments have been slow to acquire research strength in this subject. An appointment in geometric analysis would be an excellent strategic initiative for our department, would perfectly complement our renewal efforts over the last 3 years in Applied Mathematics, and would support our proposal to develop engineering and industrial mathematics connections.

3. Algebraic Geometry

Most problems in Mathematics, applied or theoretical, involve, in the last analysis, the solution of a system of equations. Algebraic Geometry is the study of the solution of systems of algebraic equations. The recent spectacular proof of Fermat's Last Theorem involves establishing certain properties of an algebraic-geometric object (an elliptic curve) associated to a non-trivial solution of Fermat's equation. On the applied side, Algebraic Geometry has become important in Coding Theory and Cryptography, again through algebraic curves. Unfortunately, Algebraic Geometry represents a serious gap in the expertise of our department. An expert in this area would enhance simultaneously the research and instructional activities in our department in Number Theory, Logic, Geometry and Topology, and Mathematical Physics. Furthermore, the computational aspects of Algebraic Geometry -- such as algorithms for finding solutions and enumeration of invariants -- have a natural link to research areas in computer science such as computation complexity.

4. Statistics

An appointment in a major area of active research in Statistics would support our undergraduate program, our interdisciplinary M.Sc. program in Statistics and provide critical research strength to establish a Ph.d. program in Statistics. Some active areas of Statistics include: information and robustness (extraction of information from large data methods), biostatistics, inference for stochastic processes, time series (applications to environmental studies, economic data), multivariate analysis (data mining, data sets with highly correlated variables, latent variable methods ,applications to chemical engineering), and survey sampling.

5. Numerical Analysis

The availability of increasingly powerful hardware and software has made it possible to perform large-scale computations in many areas of science and engineering, statistics, and finance that were earlier inconceivable. At the moment we are unable to fill the needs of many departments in science and engineering for graduate courses in numerical and asymptotic analysis. Appointing a numerical analyst would not only make such expertise available to the University at large, but also it would strengthen our own research groups in applied mathematics and statistics, and widen our course offerings at the undergraduate level.

6. Probability Theory

This area provides theoretical support to many areas in Statistics, and applied areas such as Financial Mathematics. Probability theory is a critical ingredient in the construction of error correcting codes, which are fundamental to the storage and retrieval of digital information. The study of random phenomena is important as well in statistical mechanics and material science. In the past, many of our own undergraduates have gone on to a career in Actuarial Science. An appointment in this area not only strengthens research and teaching in our department, but also will be of interest to Electrical Engineering, Material Science, Business, and Economics.

7. Financial Mathematics

Our department has put in place a group in Financial Mathematics out of existing faculty. This group will be seeking research contracts with financial institutions and industry, and will develop a program in Financial Mathematics at the graduate level. An additional faculty member whose expertise is in stochastic analysis \& risk management, preferably with relevant work experience and contacts in financial institutions or industry, would strengthen our new program and support interactions with the Faculty of Business as well as the Department of Economics.

8. Mathematical Biology

An endowed Chair in this subject is featured in the current McMaster Development Campaign. We view this as an extraordinary opportunity to further interdisciplinary studies and research with departments in the Life and Health Sciences.

9. Mathematics of String Theory

String theory is one of the most promising grand unification theories in Theoretical Physics. Its mathematical formulation requires deep ideas and sophisticated tools from areas such as differential geometry, algebraic geometry, topology, and infinite-dimensional Lie theory. Its study has led to striking answers to some long-standing mathematical problems. In view of our department's research strength in geometry & topology, a mathematician working on problems in string theory would complement our strengths and provide yet another link to Physics and Astronomy at McMaster and Toronto.

Last updated May 2, 2000.

¹ The report "Ontario's Students, Ontario's Future'' authored by the Council of Canadian Universities projects 18% growth in full time university enrollment in Ontario over the period 1998-2010.

² This count assumes the successful replacement of all recent resignations by active researchers, and is down from an all-time high of 42 through losses and retirements.

³ See discussion under Objective 3 in the McMaster Academic Plan, Jan. 2000.