Department of Mathematics and Statistics - University at Albany-SUNY

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# Department of Mathematics and Statistics

#### Faculty

Distinguished Professor Emeritus
Charles A. Micchelli, Ph.D.
Stanford University

Distinguished Service Professor Emeritus
Timothy L. Lance, Ph.D.
Princeton University

Professors Emeriti
Louis Brickman, Ph.D.
University of Pennsylvania
Lindsay N. Childs, Ph.D. (Collins Fellow)
Cornell University
Nathaniel A. Friedman, Ph.D.
Brown University
Richard Z. Goldstein, Ph.D. (O’Leary Fellow)
University of Pennsylvania
Benton N. Jamison, Ph.D.
University of California, Berkeley
Melvin L. Katz, Ph.D.
University of California, Berkeley
Violet H. Larney, Ph.D.
University of Wisconsin
Thomas H. MacGregor, Ph.D.
University of Pennsylvania
George E. Martin, Ph.D.
University of Michigan
Hajimu Ogawa, Ph.D.
University of California, Berkeley
Richard O’Neil, Ph.D.
University of Chicago
R. Michael Range, Ph.D.
University of California, Los Angeles
Edward C. Turner, Ph.D.
University of California, Los Angeles
Donald R. Wilken, Ph.D.
Tulane University

Professors
Martin V. Hildebrand, Ph.D.
Harvard University
Cristian Lenart, Ph.D.
University of Cambridge
Michael I. Stessin, Ph.D. (Department Chair)
Moscow State University
Rongwei Yang, Ph.D.
State University of New York at Stony Brook
Kehe Zhu, Ph.D.
State University of New York at Buffalo

Associate Professors Emeriti
Guy D. Allaud, Ph.D.
University of Wisconsin
Herbert I. Brown, Ph.D.
Rutgers University
William F. Hammond, Ph.D.
Johns Hopkins University
Lloyd L. Lininger, Ph.D.
University of Iowa
Ricardo Nirenberg, Ph.D.
New York University
Associate Professors
Ivana Alexandrova, Ph.D.
University of California, Berkeley
Boris Goldfarb, Ph.D.
Cornell University
Antun Milas, Ph.D.
Rutgers University
Steven Plotnick, Ph.D.
University of Michigan
Karin B. Reinhold-Larsson, Ph.D.
Ohio State University
Carlos C. Rodriguez, Ph.D.
Columbia University
Malcolm J. Sherman, Ph.D.
University of California, Berkeley
Anupam Srivastav, Ph.D.
University of Illinois, Urbana-Champaign
Mark Steinberger, Ph.D.
University of Chicago
Alexandre Tchernev, Ph.D.
Purdue University
Marco Varisco, Ph.D.
University of Munster
Yiming Ying, Ph.D.
Zhejiang University

Assistant Professors
Marius Beceanu, Ph.D.
University of Chicago
Justin M. Curry, Ph.D.
University of Pennsylvania
Yunlong Feng, Ph.D.
University of Science and Technology of China
Joshua Isralowitz, Ph.D.
State University of New York at Buffalo
Hyun-Kyoung Kwon, Ph.D.
Brown University
Michael Lesnick, Ph.D.
Stanford University
Matthew C. B. Zaremsky, Ph.D.
University of Virginia
Changlong Zhong, Ph.D.
University of Southern California

Adjuncts (estimated): 7
Teaching Assistants (estimated): 30

The Mathematics Department provides a broad offering of courses from which a student can make a selection designed to satisfy any of a large variety of objectives. In addition to including the standard courses in pure and applied mathematics, our course offerings are unusually strong in statistics and actuarial mathematics. The department offers two majors: the major in mathematics and the major in actuarial and mathematical sciences. A third major, the major in computer science and applied mathematics, is offered jointly with the computer science department.

#### Careers

Careers in mathematics typically require study beyond the undergraduate level. Examples include careers in the insurance industry (as an actuary), in the pharmaceutical industry (as a statistician), in the financial industry (as an analyst) or in education (as a mathematics teacher or professor). However, a student with an undergraduate degree in mathematics has an excellent fundamental preparation for careers in many fields. Many areas of business require high-level quantitative and analytical problem-solving skills, and a major in mathematics exercises those skills at a high level. A degree with a major in mathematics signifies that the graduate is well prepared to handle the mathematical demands of positions in a wide range of fields. The Mathematical Association of America (www.maa.org/careers) has a number of resources on careers in or related to mathematics.

The Department has several options or concentrations that focus on particular career tracks. The Department offers a major in Actuarial and Mathematical Sciences for students interested in pursuing a career as an actuary. The undergraduate major includes coursework that constitutes substantial preparation for the four preliminary examinations set by the Society of Actuaries and the Casualty Actuary Society that are required for professional advancement in the field.

For a career as a statistician, a student can choose within the mathematics major a concentration in statistics and probability that, along with appropriate courses in analysis and algebra, is excellent preparation for a graduate degree in statistics or biostatistics.

A major in mathematics followed by a master's degree in education or in mathematics with appropriate courses in education provides the prerequisites for a career as a secondary school mathematics teacher. A mathematics major followed by graduate work at the doctoral level provides suitable preparation for a career as a college or university professor.

In addition to these career options, a major in mathematics and a major or minor in a social or natural science or business are good preparation for graduate work in quantitative or mathematical areas of the other (or a related) discipline. Some examples include economics, physics, atmospheric science, biology, marketing, geography and planning, engineering, cryptography, sociology, demography, management, nanotechnology, epidemiology, biophysics, education, and finance. Mathematics majors have also gone on to law school and to medical school.

#### Placement and Proficiency Credit

The University awards up to 8 credits and advanced placement in its sequences of calculus courses based on performance on the advanced placement calculus examinations administered by the College Board. Details concerning the decisions on credit and placement are available from the Admissions Office.

#### Admission

Students may not declare a major in either mathematics or actuarial and mathematical sciences until they have completed at least one of A MAT 113, 119, 214, or 218 with a grade of C or above, or S. Transfer credits and grades may be used to satisfy the requirement.

#### Mathematics Major

Students majoring in mathematics may choose to complete the requirements for either the B.A. or B.S. degree. With either degree option, a student may apply for admission to the honors program.

Since advanced courses in mathematics typically have long chains of prerequisites, students considering a major in mathematics or actuarial sciences are encouraged to visit the department office (ES 110) as early as possible for informal advice. Information is also available on the Mathematics Department’s website http://math.albany.edu/.

To complete a mathematics major comfortably in four years, the calculus sequence, A MAT 299, and A MAT 220 or a suitable 300 level course (A MAT 326, 308 or 362) should be completed by the end of the sophomore year.

### Degree Requirements for the Major in Mathematics

General Program B.A.: The purpose of the B.A. program is to provide the student a broad view of mathematics and statistics. The B.A. degree requires a minimum of 37 credits including:

Required 6 courses: A MAT 111 or 112 or 118; 113 or 119; 214 or 218; 220 or 222; 299 and corequisite U UNL 299

3 credit course numbered above 300 in each of four areas:

- Algebra: A MAT 326, 327, 328, 424
- Analysis: A MAT 311, 312, 314, 409, 412, 413, 414
- Geometry/topology: A MAT 331, 342, 432, 441, 442
- Probability/statistics: A MAT 308, 362 or 367*, 363, 370, 464, 467

Electives: 6 credits in Mathematics at the 200 level or above

* Only one of A MAT 362 and 367 may be applied toward any major or minor in mathematics or a minor in statistics.

General Program B.S.: A minimum of 43 credits including:

Required 6 courses: A MAT 111 or 112 or 118; 113 or 119; 214 or 218; 220 or 222; 299 and corequisite U UNL 299

Two of the following five sequence options:

- Any two of A MAT 326, 327, and 328
- Either (a) both A MAT 314 and 315 or (b) any two of 312, 412, 413, 414
- Any two of A MAT 331, 342, 432, 441, 442
- Either (a) any two of A MAT 308, 362 or 367*, 363, 369, 464, 465, 467, 468 or (b) A MAT 370 and one of A MAT 367, 369, 464, 465, 467, 468.
- A MAT 312 and A MAT 424 (A MAT 312 cannot be part of two sequence options.)

* Only one of A MAT 362 and 367 may be applied toward any major or minor in Mathematics or a minor in Statistics.

With departmental approval, other 400 level or 500 level courses may be substituted for the courses listed above.

Electives: 6 credits in Mathematics at the 200 level or above

Computer Science: 6 credits from I CEN/I CSI 201, I CEN/I CSI 213, I CSI 101, 203, 204, 205

A minor in: Atmospheric Science, Biology, Business, Chemistry, Computer Science, Economics, Electronics, Informatics, or Physics.

NOTE: The Statistics minor is not open to students with a major in Mathematics.

#### Career Tracks

With suitable advisement, students can design programs that will best meet their particular interests and career goals. To guide students in their planning, a number of options are presented here.
1. Secondary School Teaching
Students planning to become mathematics teachers at the secondary level in New York State normally need to obtain a master's degree in mathematics or education to attain a professional level of certification. For undergraduate preparation, students are urged to include the following courses in their programs:

- A MAT 312 (analysis), 326, 327, or 328 (algebra), 331 or 432 (geometry), 362 (probability), 308 or 363 (statistics), and 452 (history)
- Six credits of computer science or of one science
- One year of a language

Also recommended is E TAP 201 (Exploring Teaching as a Profession)

A grade point average of 3.0 or higher overall and in mathematics is normally required for admittance to master's programs. Seniors interested in pursuing a master's degree should contact advisers in the Pathways into Education (PIE) Center.

2. Graduate School Preparation
The department offers excellent opportunities for students who plan to go on to graduate work in mathematics and statistics as well as other areas such as computer science, the natural sciences, and the social and behavioral sciences. Students whose goal is to obtain a graduate degree in mathematics or statistics should try to include in their programs a core of advanced undergraduate courses chosen from at least three of the following four areas:

- Algebra: A MAT 327, 328, 424
- Analysis: A MAT 413, 414
- Geometry/Topology: A MAT 342, 441, 442
- Probability/Statistics: A MAT 467, 468

For additional advanced study, students have two choices. One option is to pursue an independent study project leading to an undergraduate thesis, either (preferably) within or (optionally) independent of the honors program. This option can provide an opportunity for the student to pursue research in mathematics. Students considering an undergraduate research/thesis should meet with the Director of Advisement or the Director of Undergraduate Studies during the junior year to begin planning this option.

Another option is to take graduate courses. Qualified undergraduate mathematics students may be admitted to the Department’s 500 level graduate courses in algebra, real analysis, complex analysis, topology, and other areas. (Students admitted to the combined bachelor's/master's program may take graduate courses for graduate credit — see below.)

3. Statistics
Statistics is a widely applied area of mathematics and the demand for statisticians is high. A concentration in statistics may begin with either of two sequences: A MAT 308, 362, or A MAT 362, 363. (The former sequence can be started after completion of Calculus II.) Either sequence can be followed by courses in applied statistics (A MAT 369, 465), applied probability (A MAT 464) and mathematical statistics (A MAT 467, 468). Students who plan to pursue the further study of statistics at the graduate level are advised to take A MAT 312 (Basic Analysis) and A MAT 424 (Advanced Linear Algebra). Statistics in practice usually involves heavy use of the computer, so students should include I CSI/I CEN 201 and additional computer science in their programs.

4. Applied Mathematics
Although it is common to classify mathematics as either “pure” or “applied,” the division is often arbitrary. In recent years some extremely abstract mathematics has turned out to be useful in areas outside mathematics. Students preparing for a career in applied mathematics area advised to acquire as strong a background as possible in the pure mathematical areas of analysis, algebra, and geometry/topology. On the other hand, students concentrating in pure mathematics should have some understanding of how to apply mathematical methods to other disciplines.

Here are some mathematical areas that are more commonly applied to problems in other fields, along with the corresponding courses in which methodology or applications are treated.

- Applied algebra: A MAT 326, 328, 372, 374
- Applied analysis: A MAT 311, 314, 315, 409, 412, 416
- Numerical Methods: A MAT 301, 313, 401
- Probability/Statistics: A MAT 308, 362, 363, 369, 464, 465

#### Honors Program

The honors program is designed for the talented and committed student of mathematics in either the B.A. or the B.S. major. Successful completion of the program is excellent preparation for graduate work in mathematics or a mathematics-related discipline.

At the lower division level, the Department offers Honors Calculus II (A MAT 119), III (A MAT 218), and Honors Linear Algebra, A MAT 222. A strong student may begin in whichever of these courses is appropriate based on the student’s previous coursework in calculus.

A student may be admitted formally to the Department’s honors program at any time after the sophomore year, and then will be formally advised by the Director of the Honors Program. However, any student who is interested in the program should see the Director of the Honors Program as early as possible for informal advisement. To be admitted, the applicant must have an academic average in all University courses of at least 3.30, and an academic average in all mathematics courses of at least 3.40.

In addition to the lower division core, the requirements for graduation with honors in mathematics include:

- Required courses: A MAT 413, 414, 424, and 9 additional credits from among A MAT 327, 416, 420, 425, 432, 441, 442, 464, 467, 468, 510A, 513A, 520A, 520B, 540A, 557A, 557B.
- A MAT 482 (Senior Seminar), A MAT 497 (Independent Study) or A MAT 499 (Undergraduate Thesis).
- An acceptable honors thesis presented in a public venue.
- An academic average of at least 3.30 in all University courses and at least 3.40 in all mathematics courses numbered 400 or above.
- For the B.S., 6 credits in Computer Science from I CSI 101, I CSI/I CEN 201, I CSI 203, I CSI 205, I CSI/I CEN 213 and a minor in atmospheric science, biology, business, chemistry, computer science, economics, electronics, informatics, or physics.

#### Actuarial and Mathematical Sciences Major

Actuaries are professionals who manage risk for insurance companies, pension plans and other financial institutions, and government and business. To attain professional standing as an actuary requires successful completion of a sequence of examinations offered by the Society of Actuaries and the Casualty Actuarial Society. The actuarial major is designed to prepare students for employment in the actuarial field by offering coursework related to the preliminary actuarial examinations. Past experience suggests that students who pass two actuarial exams while in college are likely to secure employment in the field, and some students have secured actuarial employment with fewer exams.

The actuarial exams described immediately below are interdisciplinary, testing material from several courses. Some of the courses listed as preparation for an exam are relevant to only a few questions on that exam, so students may reasonably take an exam before taking all the courses listed as relevant.

Exam P (probability). Students need A MAT 112, 113, 214, 362. A MAT 467 may also be useful, but for at most a few questions.

Exam FM (financial mathematics). Students need A MAT 301 (which in turn requires calculus). A ECO 110 and 111 may also prove useful.

Exam M (actuarial models). Students need A MAT 403 and topics from A MAT 464 for the life contingencies segment of the two-part Exam M.

Exam C (construction and evaluation of actuarial models). Several required courses are relevant, including A MAT 362, 363, 464, 465, 467, A ECO 300 and 301.

The Society of Actuaries grants VEE (validation by educational experience) credit in the indicated area to students completing the following listed courses with a grade of B- or better.

Applied Statistical Methods: A MAT 465 (or 565). Or students can take (i) both A ECO 621 and 720, or (ii) both A MAT 558 and 664 (which are equivalent to H STA 558 and 664).

Economics: A ECO 110 and 111. Or students can take both A ECO 300 and 301.

Corporate Finance: B FIN 300.

Students completing the B.S. program in actuarial science will have studied virtually all the material tested on Exam P, Exam FM and the life contingencies segment of Exam M. Students will also have studied about half the material tested on Exam C. Students who complete a B.S. in actuarial science will have taken courses carrying VEE credit in the three required areas: applied statistical methods, corporate finance, and economics.

Students who enter as freshman with AP credit for Calculus I and II (calculus BC) should be able to prepare for the financial economics segment of Exam M and for the topics of Exam C not covered by required courses. Doing so requires additional advanced courses in statistics, business and economics. Completing such courses and graduating within four years requires advanced planning.

Actuarial majors with AP credit in calculus are encouraged (but not required) to take A MAT 119 and 218, the honors versions of A MAT 113 and 214.

#### Degree Requirements for the Major in Actuarial and Mathematical Sciences

General Program B.S.: A combined major and minor sequence consisting of 64 credits as follows:

- 37 credits in Mathematics: A MAT 112 or 111 or 118; 113 or 119; 214 or 218; 220 or 222; 299 and corequisite U UNL 299; 301 or A ECO 351; 362; 363; 403; 464; 465 or 565
- 6 credits: from I CSI/I CEN 201, I CSI 203, 204, 205, 310
- 6 credits: B ACC 211, B FIN 300
- 15 credits in Economics: A ECO 110, 111, 300, 301, and 466

Note: Actuarial majors automatically fulfill the requirement for a minor in economics (since A MAT 301 is equivalent to A ECO 351).

The following is the recommended schedule of required mathematics courses for students entering without AP credit in calculus:

Year | Fall | Spring
Fresh. | 112 | 113
Soph. | 214, 299 | 220, 301
Junior | 362 | 363, 469
Senior | 403, 465, 469 | 464, 469 (rec 404)

Note: A MAT 469, which may be repeated for credit, is an optional one-credit course that drills students on problems from one of the preliminary actuarial exams (either P, FM or MLC).

Students are advised to take A ECO 110 and 111 as freshmen, and in any event, not later than their sophomore year. By doing so, students will not need to take more than one upper division economics course during any single semester.

#### Honors Program in Actuarial and Mathematical Sciences

A student may be admitted formally to the Department’s honors program in actuarial science at any time after the sophomore year. However, any student who is interested in the program should see the Director of the Actuarial Science Program as early as possible for informal advisement concerning the honors thesis. To be admitted, the applicant must have an academic average in all University courses of at least 3.30, and an academic average in all mathematics courses of at least 3.40

The requirements for graduation with honors include:

- Required courses: A MAT 468 together with the other required courses of the Actuarial Science major.
- An acceptable honors thesis presented in a public venue.
- An academic average of at least 3.30 in all University courses and at least 3.40 in all mathematics courses numbered 400 or above.

#### Combined B.A./M.A. and B.S./M.A. Programs

The combined B.A./M.A. and B.S./M.A. programs in mathematics provide an opportunity for students of recognized academic ability and educational maturity to fulfill integrated requirements of undergraduate and master’s degree programs from the beginning of their junior year. A carefully designed program can permit a student to earn the B.A. or B.S. and the M.A. degrees within nine semesters.

The combined programs require a minimum of 138 credits, of which at least 30 must be graduate credits. In qualifying for the B.A. or B.S., students must meet all University and college requirements, including the requirements of the undergraduate major described previously, the minimum 90 or 60 credit liberal arts and sciences requirement, general education requirements, and residence requirements. In qualifying for the M.A., students must meet all University and college requirements as outlined in the Graduate Bulletin, including completion of a minimum of 30 graduate credits and any other conditions such as a research seminar, thesis, comprehensive examination, and residence requirements. The main benefit of the combined program is that up to 12 graduate credits in mathematics may be applied simultaneously to both the undergraduate degree program and the M.A. program.

Students may apply to the graduate committee of the department for admission to either combined program in mathematics at the beginning of their junior year or after the successful completion of 56 credits. A cumulative grade point average of 3.20 or higher and three supportive letters of recommendation from faculty are required for consideration.

#### Related Program: Interdisciplinary Major in Computer Science and Applied Mathematics

This major prepares a student to handle mathematically oriented computer applications in engineering and business. Students have two options: a) the general sequence and b) the Data Analytics sequence. Both require at least 66 credits.

• The general sequence combines a strong sequence in computer science with relevant courses in mathematics.
• The Data Analytics sequence combines a strong sequence in probability and statistics with a strong sequence in computer science.

Details of the program are listed under The Department of Computer Science in the College of Computer Engineering and Applied Sciences.

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