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Mathematics

(For 2013–2014 academic year)

Professors Hart, Lantz, Robertson, Saracino, Schult (Chair), Strand, Valente
Assistant Professors Ay, Christensen, Howard, M. Ionescu, Seo
Visiting Assistant Professor Jiménez Bolaños

There are many good reasons to study mathematics: preparation for a career, use in another field, or the beauty of the subject itself. Students at Colgate who major in mathematics go on to careers in medicine, law, or business administration as well as areas of industry and education having an orientation in science. Non-majors often require mathematical skills to carry on work in other disciplines, and all students can use the study of mathematics to assist them in forming habits of precise expression, in developing their ability to reason logically, and in learning how to deal with abstract concepts. There are also many people who view mathematics as an art form, to be studied for its own intrinsic beauty.

All mathematics courses are open to qualified students. Entering first-year students who have successfully completed at least three years of secondary school mathematics, including trigonometry, should be adequately prepared for MATH 111. Students who have studied calculus in secondary school are often ready to enter MATH 112 or 113.

Major Program

MATH 111, 112
(or equivalent calculus experience approved by the department) is required for admission to a major or minor in mathematics.

The requirements for a major in mathematics are as follows:
1. MATH 113 and 214 completed by the end of the sophomore year.
2. MATH 250 completed by the end of the sophomore year with a grade of C or better.
3. MATH 320 and 323
4. Five additional mathematics courses numbered 300 or above. One of these must be a senior research experience which will normally be MATH 399 or a 400-level course. With department chair pre-approval, on occasion the research experience may be satisfied by an independent study. See “Advanced Placement and Transfer Credit” below for additional information.
Majors who are planning to undertake graduate study in mathematics are advised to take MATH 421 and 424.

Minor Program in Mathematics

The requirements for a minor are as follows:
1. MATH 113 and 214.
2. MATH 250 completed with a grade of C or better.
3. Either MATH 320 or 323.
4. Two other mathematics courses numbered 300 or above. See “Advanced Placement and Transfer Credit” below for additional information.

Minor Program in Applied Mathematics

The requirements for a minor are as follows:
1. MATH 113 and 214.
2. MATH 250 completed with a grade of C– or better.
3. MATH 308.
4. Two of the following courses: MATH 307, 310, 311, 312, 313, 315, 316, 317, 329, MATH/BIOL 334, MATH 458 (formerly MATH 407), MATH/BIOL 481. See “Advanced Placement and Transfer Credit” below for additional information.

In order to graduate with a major in mathematics, a minor in mathematics, or a minor in applied mathematics, the student must have a GPA of at least 2.00 in mathematics courses counted for the major or minor.

The department also strongly recommends that students pursuing a major or a minor in mathematics complete COSC 101 or its equivalent.

Honors and High Honors

To qualify for honors in mathematics, majors must take, as one of the courses required for the major, a course at the 400 level. Majors must have a GPA of at least 3.30 in the following courses: MATH 113, 214, 250, 320, 323, the 400-level course just described, and three other math courses numbered 300 or above. For high honors, the corresponding GPA must be at least 3.70.

Candidates for honors must also perform satisfactorily on the honors examination, which is given once each semester and covers MATH 320 and 323.

Based upon the result of the honors examination, a student may be invited to stand for high honors. A candidate for high honors must, under the guidance of a faculty member of the department, write a high honors paper during the senior year and make an oral presentation of the results. In order for high honors to be awarded, the department must accept this paper and presentation as being of high honors quality. The high honors candidate may register for an independent study course so that the paper satisfies the senior experience requirement.

Awards

See Honors and Awards: Mathematics in Chapter VI.

Advanced Placement and Transfer Credit

Students who have taken the Calculus-BC, Calculus-AB, or Statistics Advanced Placement exam of the College Entrance Examination Board will be granted credit according to the following policy:
1. Students earning 4 or 5 on the Calculus-BC Advanced Placement exam will receive credit for MATH 111 and 112. Students earning 3 on the Calculus BC exam will receive credit only for MATH 111.
2. Students earning 4 or 5 on the Calculus-AB Advanced Placement exam will receive credit for MATH 111.
3. Students earning 4 or 5 on the Statistics Advanced Placement exam will receive credit for MATH 102.
4. There are no other circumstances under which a student will receive credit at Colgate for a mathematics course taken in high school.
Transfer credit for a mathematics course taken at another college will be granted upon the pre-approval of the department chair. Mathematics majors or minors may not receive transfer credit for MATH 250, 320, or 323, but must pass these courses at Colgate and must take them as regularly scheduled courses, not as independent studies. At most, two transfer or independent studies courses may be counted toward a major or minor.

Teacher Certification

The Department of Educational Studies offers a teacher education program for majors in mathematics who are interested in pursuing a career in elementary or secondary school teaching. Please refer to “Educational Studies.”

Computer Science/Mathematics Major

See “Computer Science.”

Mathematical Economics Major

See “Economics.”

Mathematical Systems Biology Minor

K. Belanger (Chair of the Department of Biology)
D. Schult (Chair of the Department of Mathematics)

Mathematical systems biology describes a field of inquiry in which mathematical and computational methods are used to examine complex, large scale interactions between components of biological systems and to predict how these interactions influence the properties of those systems. The systems examined may range in scale from molecular through cellular and tissue levels to the scale of organisms and entire ecosystems. The unifying feature of this field is quantitative description of interactions between components of biological systems.

The interface between mathematics and biology is one of the most rapidly expanding areas of research in the sciences. The technological development of methods for generating large amounts of biological data — including genome sequence information, total protein analysis, metabolic information, etc. — demands the development of mathematical and computational methods for analyzing these data and for developing predictive models that use such large data sets. The multidisciplinary field of systems biology requires an understanding of both mathematical and biological concepts, insights into interesting questions in biology, and comprehension of the mathematical methods that can be used to address many of those questions. The mathematical systems biology minor provides students with the coursework in mathematics and biology required to begin to gain insights and experience in this important new field.

Minor Program course requirements are described below (six courses).
1. MATH 113, Calculus III
2. MATH 214, Linear Algebra
3. BIOL 211, Ecology, Evolution, and Diversity or BIOL 212, Molecules, Cells, and Genes
4. MATH 315, Mathematical Biology or MATH/BIOL 334, Systems Biology
5. One additional biology course from the following:
BIOL 211, Ecology, Evolution, and Diversity
BIOL 212, Molecules, Cells, and Genes
BIOL 220, Biostatistics
BIOL 225, Bioinformatics
Any 300- or 400-level BIOL elective course

6. One additional mathematics course from the following:
MATH 307, Dynamical Systems and Chaos
MATH 308, Differential Equations
MATH 310, Combinatorial Problem Solving
MATH 311, Partial Differential Equations
MATH 312, Applied Mathematics: Social Sciences
MATH 315, Mathematical Biology
MATH 316, Probability
MATH 317, Mathematical Statistics
MATH 329, Numerical Analysis
MATH 334, Systems Biology
MATH 458, Real-time Nonlinear Dynamics and Chaos


Students declaring a minor in mathematical systems biology select an adviser from either the mathematics or biology department. Those students minoring in mathematical systems biology who have declared a major in either biology or mathematics are required to choose a minor adviser from the department in which they are not majoring. The chair of the minor adviser’s department approves and signs the mathematical systems biology minor declaration form. As with any minor at Colgate, no more than two of the courses applied to the minor may also be used for a major.

Study Groups

Colgate sponsors several study-abroad programs that can support continued work toward a major in mathematics. These include, but are not limited to, the Wales Study Group (U.K.), the Australia Study Group, the Australia II Study Group, and the Manchester Study Group (U.K.). For more information about these programs, see “Off-Campus Study Group Programs” in Chapter VI.

Course Offerings

MATH courses count toward the Natural Sciences and Mathematics area of inquiry/distribution requirement, unless otherwise noted.

101  Precalculus Mathematics
A. Strand
A study of the following types of functions, their properties, and their graphs: polynomials, rational functions, trigonometic, exponential, and logarithmic functions. This 0.25-credit course is intended for students whose background in mathematics may be deficient. Its objective is to lay a foundation for the study of calculus and concurrent enrollment is MATH 111 is required. Prerequisite: permission of the instructor. Offered in the fall only.

102  Introduction to Statistics
Staff
Introduces students to statistical thinking by examining data collected to solve real-world problems. A wide range of applications are considered. Topics include experimental design, descriptive statistics, the normal curve, correlation and regression, probability theory, sampling, the central limit theorem, estimation, hypothesis testing, paired observations, and the chi-square test. Particular emphasis is given to the models that underlie statistical inference. Prerequisite: three years of secondary school mathematics. Note: This course is not open to students who have either received credit for or are currently enrolled in MATH 317. This course is crosslisted as CORE 143S.

111  Calculus I
Staff
An introduction to the basic concepts of differential and integral calculus including limits and continuity; differentiation of algebraic, trigonometric, exponential, and logarithmic functions; applications of the derivative to curve sketching, related rates, and maximum-minimum problems; Riemann sums and the definite integral; and the fundamental theorem of calculus. Prerequisite: three years of secondary school mathematics including trigonometry.

112  Calculus II
Staff
A continuation of the study of calculus begun in MATH 111, including the calculus of inverse trigonometric functions, techniques of integration, improper integrals, l’Hôpital’s rule and indeterminate forms, applications of integration, and Taylor series. Prerequisite: MATH 111 with a grade of C– or higher or equivalent experience in a secondary school calculus course.

113  Multivariable Calculus
Staff
The calculus of functions of two or three variables. Among the topics considered are surfaces in three-dimensional space, partial derivatives, maxima and minima, and multiple integrals. Prerequisite: MATH 111 or MATH 112 with a grade of C– or higher or equivalent experience in a secondary school calculus course.

214  Linear Algebra
A. Ay, D. Lantz, D. Schult, A. Strand
A study of systems of linear equations, matrices, determinants, vector spaces, linear transformations, eigenvalues, eigenvectors, and diagonalization. Prerequisite: MATH 113 or co-registration in MATH 113.

250  Number Theory and Mathematical Reasoning
E. Hart, D. Howard, D. Saracino, K. Valente
Questions about the positive integers 1, 2, 3 . . . have fascinated people for thousands of years. The ancient Greeks noted the existence of right triangles with sides of integral length, corresponding to equations such as 32 + 42 = 52 and 52 + 122 = 132. Is there a way of describing all such “Pythagorean Triples”? As another example, we observe that 5 = 12 + 22, 13 = 22 + 32, 17 = 12 + 42, while none of the primes 7, 11, or 19 can be expressed as the sum of two squares. Is there a pattern? Does it continue forever? This course focuses on such equations as a means for introducing students to the spirit and methods of modern mathematics. The emphasis throughout is on developing the ability to construct logically sound mathematical arguments and communicate these arguments in writing. Prerequisite: MATH 112 or 113 with a grade of C or better, or permission of instructor.

307  Dynamical Systems and Chaos
M. Ionescu, D. Schult
A dynamical system is a set of mathematical rules which specify change over time. Examples include models of the solar system or of population levels. The goal of this course is to introduce some of the concepts and discoveries that have been made over the past fifty years about discrete dynamical systems. Repeated application of non-linear functions can lead to complex phenomena. Topics include fixed points, stability, chaos, symbolic dynamics, fractals, Lyapunov exponents, the Julia set, and the Mandelbrot set. Prerequisite: MATH 214. Offered in the fall only, in alternate years.

308  Differential Equations
A. Ay, D. Schult, A. Strand
A study of ordinary differential equations with associated initial condition. First order equations, linear second order equations with constant coefficients, systems of differential equations, other selected topics, and applications. Prerequisite: MATH 112 or MATH 113 or permission of instructor.

310  Combinatorial Problem Solving
D. Howard, A. Robertson
This course develops methods to solve combinatorial (finite) problems arising in mathematics, computer science, and other areas from the natural and social sciences. Enumeration and graph theory are the main subjects. Topics include recurrence relations, generating functions, inclusion-exclusion, modeling with graphs, trees and searching, graph coloring, and network algorithms. The emphasis is on problem solving rather than theory. Prerequisite: MATH 112. Offered in the spring only, in alternate years.

311  Partial Differential Equations
A. Ay, D. Schult
This course explores mathematics as it is applied to the physical sciences. Mathematical topics may include boundary value problems, partial differential equations, special functions, Fourier series and transforms, Green’s functions, and approximate solution methods. Prerequisite: MATH 113 and MATH 308 or permission of instructor. Offered in the spring only, in alternate years.

312  Applied Mathematics: Social Sciences
D. Schult
How do we translate problems from the world into solvable mathematical problems? Mathematical modeling is the art of creating mathematical problems whose solutions are useful for real world problems. Methods such as scaling, qualitative analysis, limits of predictability, and simple random models are discussed. Applications considered arise from economics, political science, and sociology. Prerequisite: MATH 113 or permission of instructor. Offered in the spring only.

313  Functions of a Complex Variable
D. Lantz, D. Schult
An introductory study of functions in the complex plane. Topics include complex numbers and functions, the theory of differentiation and integration of complex functions, sequences and series of complex functions, conformal mapping. Special attention is given to Cauchy’s integral theorem. Prerequisite: MATH 113. Offered in the spring only, in alternate years.

315  Mathematical Biology
A. Ay, D. Schult
This course provides an introduction to the use of continuous and discrete mathematical models in the biological sciences. Biological topics may include single and multispecies population dynamics, modeling of infectious diseases, regulation of cell function, molecular interactions, neural and biological oscillators, ecology, cancer biology, and virus dynamics. Mathematical techniques include modeling, phase plane analysis, bifurcation diagrams, perturbation theory, and computer simulations. Prerequisite: MATH 113. Note: This course is not open to students who have either received credit for or are currently enrolled in MATH 312. Offered in the fall only, in alternate years.

316  Probability
E. Hart, M. Ionescu, A. Robertson
An introduction to the basic concepts of discrete and continuous probability: axioms and properties of probability, standard counting techniques, conditional probability, important random variables and their discrete and continuous distributions, expectation, variance, and joint distribution functions. Additional topics may include: Poisson processes, Markov chains, and Monte Carlo methods. Prerequisite: MATH 112 or MATH 113. Offered in the fall.

317  Mathematical Statistics
E. Hart, A. Robertson
The standard methods in statistics are developed with mathematical rigor. Topics include parameter estimation, including Bayesian estimation, the Central Limit Theorem, hypothesis testing, regression, analysis of variance, and nonparametric statistics. Applications of these tools are studied, with the choice of topics determined by the instructor. Prerequisite: MATH 316. Offered in the spring only, in alternate years.

320  Abstract Algebra I
E. Hart, D. Lantz, D. Saracino, K. Valente
An introduction to the basic structures of abstract algebra including groups, rings, integral domains, and fields. Prerequisite: MATH 250 with a grade of C or better. Offered in the spring.

323  Real Analysis I
M. Ionescu, D. Lantz, A. Robertson
A rigorous treatment of the basic concepts of real analysis, including limits, continuity, the derivative, and the Riemann integral. Prerequisites: MATH 113 and 250 with a grade of C or better. Offered in the fall.

327  Geometry
D. Lantz
A study of several geometrical systems, with emphasis upon a development of Euclidean geometry that meets current standards of rigor. Prerequisite: MATH 250. Offered in the fall only, in alternate years.

329  Numerical Analysis
A. Ay, D. Schult
An introductory treatment of methods used for numerical approximation. Topics include roots of equations, simultaneous linear equations, quadrature, and other fundamental processes using high speed computing devices. Prerequisite: MATH 113. Offered in the fall only, in alternate years.

331  Number Theory II
D. Saracino
This course continues the study of number theory begun in MATH 250 and includes the Quadratic Reciprocity Law of Gauss, Diophantine equations, and topics from algebraic number theory. Prerequisite: MATH 320 or permission of instructor. Offered in the fall only, in alternate years.

334  Systems Biology
This course is crosslisted as BIOL 334. For course description, see “Biology: Course Offerings.”

342  Topology
E. Hart
An introduction to both point-set topology and basic algebraic topology. Topics include metric spaces, topological spaces, compactness, connectedness, the classification of surfaces, mod-2 homology, and the Jordan curve theorem. Additional topics that demonstrate connections with analysis, dynamics, and algebra are determined by the instructor based on student interest. Prerequisites: MATH 250 with a grade of C or better. Offered in the spring only, in alternate years.

399  Mathematical Problem Solving
M. Ionescu, D. Lantz, A. Robertson
This capstone seminar presents students with numerous and varied problems, drawn from many different mathematical areas, both pure and applied. There are weekly problem sets in addition to the presentation of a semester-long “project problem.” Fulfills the senior experience for the major, the course is open only to seniors.

421  Abstract Algebra II
D. Lantz, D. Saracino, K. Valente
A careful and intensive study of topics such as group theory, ring theory, field theory, and Galois theory. Prerequisite: MATH 320 with a grade of B or better or permission of instructor. Offered in the fall only, in alternate years.

424  Real Analysis II
A. Ay, M. Ionescu, A. Robertson
Topics for this course are selected from among the following: metric spaces, sequences and series of functions, the Lebesgue integral. Prerequisite: MATH 323 with a grade of B or better or permission of instructor. Offered in the spring only, in alternate years.

452  Mathematical Logic
D. Saracino
This course deals with one or more topics in mathematical logic, chosen from among the following: naive and axiomatic set theory, propositional and predicate calculus, completeness and compactness theorems, first-order model theory, recursive functions, and Gödel’s Incompleteness Theorem. Prerequisite: MATH 320 with a grade of B or better and permission of instructor. Offered in the fall only, in alternate years.

458  Real-time Nonlinear Dynamics and Chaos
D. Schult
An introduction to the techniques and concepts used to analyze real-time dynamic models that involve nonlinear terms. Applications are emphasized and demonstrate the universality of chaotic solution behavior. This course is team-taught by members of the physics and mathematics departments. Students should enroll through the department for which they intend to use the credit. (Formerly MATH 407.) Prerequisites: MATH 308 or PHYS 431 (formerly PHYS 302). Offered in the spring only, in alternate years. This course is crosslisted as PHYS 458 (formerly PHYS 407).

480  Special Topics in Mathematics
Staff
The topic for this course varies depending on the needs and backgrounds of students and interests of the instructor. Students should consult the instructor for the specific content of the course and prerequisites.

481  Modeling of Biological Systems
This course is crosslisted as BIOL 481. For course description, see “Biology: Course Offerings.”

291, 391, 491  Independent Study
Staff
Open to qualified students with permission of department chair.