Mathematics

Associate Professor Russo, Chair; Professors Georges, Mauro∙∙, and Stein; Associate Professors Sandoval and Wyshinski; Assistant Professor Skardal; Harold L. Dorwart Visiting Assistant Professors Adelstein, Goldwyn, and Wash; Visiting Assistant Professors Emerick, Pellico, and Ray; Visiting Lecturer Samuel; Aetna Quantitative Center Director and Lecturer Gingras; Ann Plato Fellow Nu’Man

The mathematics major is designed to expose students to the fundamentals of mathematics and to give students a solid mathematical foundation. The major is designed with enough flexibility to accommodate students who want to major in mathematics but whose postbaccalaureate plans may not be math-related and students who want to double major in physics, engineering, economics, computer science, or other fields, as well as students who intend to pursue graduate study in mathematics, statistics, or computer science, or students interested in careers requiring a strong mathematical background. Students intending to pursue graduate study in mathematics should supplement the basic major requirements with as many additional 300-level mathematics courses as possible and should consult with their adviser or with the department chair at the earliest possible date in order to plan their course of study.

Students are required to take nine courses, of at least one credit, at the 200-level and above, including MATH 228, 231, 307, 331, and 400. Courses counted towards the major require a grade of C- or better. Of the four electives, at least one must be a 300-level mathematics course, and at most one may be a cognate course chosen from the courses listed below, each of which is offered by another department.

Although a student may begin the mathematics major as late as the fall semester of the sophomore year, the department recommends that prospective majors adopt the following typical schedule:

Year

Fall

Spring




First

131

132

Sophomore

231, 205

228, elective

Junior

307 or 331

two electives

Senior

307 or 331

400

The Writing Intensive Part II requirement is fulfilled by taking either MATH 307 or MATH 331. In order to fulfill the requirement, one of these courses must be taken at Trinity.

Honors—Honors in mathematics, granted by departmental vote in the spring of the honor candidate’s senior year, is earned by:

The student must apply to the department chair for honors candidacy in the second semester of the junior year. Upon acceptance, the candidate, together with the department chair, will select an honors adviser who will supervise the honors thesis. The student will then submit a thesis proposal by the last day of classes for the spring semester of the junior year.

Honors theses need not be one of newfound mathematical results, but it is expected to be a balance of the historical, biographical, and mathematical aspects of the topic. The project will culminate with the submission of the final polished draft to the honors adviser no later than one week before the last day of classes of the spring semester. A formal presentation will be given by the candidate prior to the day on which senior grades are due. Complete guidelines for the completion of the honors thesis may be obtained from the department chair.

Study away—Students of mathematics have many opportunities to study abroad, but all of them require a certain amount of early planning. Students are encouraged to discuss their plans with their advisers or the department chair as soon as possible since many courses in the Mathematics Department are not offered every year. Well-prepared students should consider the Budapest semester in mathematics; more information on this program can be found on the study-away website.

Many study-abroad programs in English-speaking countries offer a wide range of mathematics courses that will count toward the major. For specific advice, please consult the department chair. Students who feel they are sufficiently proficient in a language to take mathematics courses in a foreign language should discuss this with their advisers. Students who take mathematics courses while abroad should be aware that universities that follow the European model cover the material in a somewhat different order than is done in the United States and that classes are primarily lectures with far less feedback from the instructor than is typical at Trinity.

Fall Term

Quantitative Literacy courses that were formerly designated with the MATH prefix (such as MATH 101) are now given the prefix QLIT and can be found in the schedule of classes.

Courses offered by the Aetna Quantitative Center

[101. Foundational Techniques for Quantitative Reasoning]— This course offers students new insights into important and widely used mathematical concepts, with a strong focus on numerical and algebraic relationships. (Enrollment limited)

Courses offered through the Mathematics Department

107. Elements of Statistics— A course designed primarily for students in the social and natural sciences. Topics include graphical methods, measures of central tendency and dispersion, basic probability, random variables, sampling, confidence intervals, and hypothesis testing. This course is not open to students with credit for Mathematics 131 or above, or who have placed into Mathematics 207 on the Mathematic Placement Examination. Prerequisite: A satisfactory score on the Mathematics Placement Examination or a C- or better in Quantitative Literacy 101. Students who qualify for Mathematics 131 or 207 will not be eligible to enroll in this course. (NUM) (Enrollment limited) –Ray, Samuel

114. Judgment and Decision Making— In this course, we consider the application of elementary mathematical analysis to various procedures by which societies and individuals make decisions. Topics may include weighted and unweighted voting, fair division of resources, apportionment of goods and representatives, and personal decision-making algorithms based upon utility, risk, probability, expectation, and various game-theoretic strategies in general. Examples may be drawn from medicine, law, foreign policy, economics, psychology, sports, and gambling. Prerequisite: A satisfactory score on the Mathematics Placement Exam (NUM) (Enrollment limited) –Pellico

[118. Mathematics of Games and Gambling]— We introduce at an elementary level the mathematics necessary to analyze and understand games of strategy and chance, including: lotteries, poker, craps, tournaments, the prisoner’s dilemma, and the Monte Hall problem. Prerequisite: A satisfactory score on the Mathematics Placement Exam (NUM) (Enrollment limited)

[123. Mathematical Gems]— An introduction to mathematical topics from number theory, geometry, game theory, infinity, chaos, and more. Not open to students who have received credit for Mathematics 131. Prerequisite: A satisfactory score on the Mathematics Placement Exam (NUM) (Enrollment limited)

[125. Functions and Limits]— The sequence Mathematics 125-126 provides an opportunity to study differential calculus while simultaneously covering the needed skills from precalculus. Students who finish both Mathematics 125 and 126 will be prepared to take Mathematics 132, Calculus II. Topics in Mathematics 125 will include: the real number system; linear, quadratic, polynomial, rational, exponential, and trigonometric functions; equations and inequalities; limits and continuity; applications. Not open to students who have received credit for Mathematics 131. Ordinarily, this course, to be followed by Mathematics 126, is elected by students who need to take a course in calculus, but whose backgrounds in algebra and trigonometry need strengthening. Prerequisite: A satisfactory score on the Mathematics Placement Examination or a C- or better in Quantitative Literacy 101. Students who qualify for Mathematics 131 or 207 will not be eligible to enroll in this course. (NUM) (Enrollment limited)

131. Calculus I— The real number system, functions and graphs, continuity, derivatives and their applications, antiderivatives, definite integrals, and the fundamental theorem of calculus. Mathematics, natural science, and computer science majors should begin the Mathematics 131, 132 sequence as soon as possible. Not open to students who have received credit for Mathematics 126 or who have received credit by successful performance on the Advanced Placement Examination of the CEEB (see Catalogue section “Advanced Placement for First-Year Students”). Prerequiste: A satisfactory score on the Mathematics Placement Examination. (1.25 course credits) (NUM) (Enrollment limited) –Adelstein, Emerick, Nu’Man, Pellico, Ray, Russo

142. Accelerated Calculus II— This course is an accelerated version of Mathematics 132, which will cover in greater depth topics from that course, along with selected other topics from single-variable calculus. It is intended for those with strong Calculus I backgrounds; in particular, first-year students who have received credit via the Calculus AB Advanced Placement Examination should register for this course. Open to other students with permission of the instructor. See the description of Mathematics 132. Prerequisite: C- or better in Mathematics 126 or 131, or an appropriate score on the AP Examination or Trinity’s Mathematics Qualifying Examination. (1.25 course credits) (NUM) (Enrollment limited) –Wash

205. Abstraction and Argument— This course deals with methods of proof and the nature of mathematical argument and abstraction. With a variety of results from modern and classical mathematics as a backdrop, we will study the roles of definition, example, and counterexample, as well as mathematical argument by induction, deduction, construction, and contradiction. This course is recommended for distribution credit only for non-majors with a strong mathematical background. (NUM) (Enrollment limited) –Georges

207. Statistical Data Analysis— An introductory course in statistics emphasizing modern techniques of data analysis: exploratory data analysis and graphical methods; random variables, statistical distributions, and linear models; classical, robust, and nonparametric methods for estimation and hypothesis testing; analysis of variance and introduction to modern multivariate methods. Students with a strong mathematical background are advised to take Math 207 in place of Math 107. Those who successfully complete Math 107 may take Math 207 for credit due to its increased depth of coverage and breadth of topics. Prerequisite: A suitable score on the Mathematics Placement Examination or a grade of C- or better in Mathematics 107.. (NUM) (Enrollment limited) –Goldwyn, Skardal

228. Linear Algebra— A proof-based course in linear algebra, covering systems of linear equations, matrices, determinants, finite dimensional vector spaces, linear transformations, eigenvalues, and eigenvectors. Prerequisite: C- or better in Mathematics 142 or 132, or a 200-level mathematics course, or permission of instructor. (NUM) (Enrollment limited) –Emerick, Wash

231. Calculus III: Multivariable Calculus— Vector-valued functions, partial derivatives, multiple integrals, conic sections, polar coordinates, Green’s Theorem, Stokes’ Theorem, and Divergence Theorem. Prerequisite: C- or better in Mathematics 132 or 142. (1.25 course credits) (NUM) (Enrollment limited) –Skardal, Wyshinski

305. Probability— Discrete and continuous probability, combinatorial analysis, random variables, random vectors, density and distribution functions, moment generating functions, and particular probability distributions including the binomial, hypergeometric, and normal. Prerequisite: C- or better in Mathematics 231. (NUM) (Enrollment limited) –Emerick, Russo

307. Abstract Algebra I— An introduction to group theory, including symmetric groups, homomorphism and isomorphisms, normal subgroups, quotient groups, the classification of finite abelian groups, the Sylow theorems. Prerequisite: C- or better in Mathematics 228 or permission of instructor. (WEB) (Enrollment limited) –Wash

314. Combinatorics and Computing— Introduction to combinatorics. Topics may include, but will not necessarily be limited to, computer representation of mathematical objects, enumeration techniques, sorting and searching methods, generation of elementary configurations such as sets, permutations and graphs, and matrix methods. Prerequisite: C- or better in Mathematics 228 or permission of instructor. (NUM) (Enrollment limited) –Georges

[318. Topics in Geometry]— Differential geometry, projective geometry, non-Euclidean geometry, combinatorial topology, or such topics as the department may specify. May be repeated for credit with different topics. Prerequisite: C- or better in Mathematics 228 and 231. (NUM) (Enrollment limited)

325. Special Topics in Mathematical Biology— This course provides an introduction to the development, application, and evaluation of biological models. Both deterministic and stochastic models will be developed through a case-study based approach at the molecular, cellular, and population levels. Topics include current application areas such as neurophysiology, cardiology, cellular dynamics and gene expression, spread of infectious diseases, conservation of endangered species, and cancer growth. Theory from differential equations, statistics, scientific computing, and linear algebra will be introduced as needed with topics to include basic modeling principles, discrete-time models, matrix models, dynamical systems techniques, Markov chains, pattern formation, and agent-based models. When necessary, students will implement models using a high-level programming language as well as engage with current biology research literature. Prerequisite: C- or better in a 200 level Mathematics course and permission of instructor. (NAT) (Enrollment limited) –Goldwyn

[326. Graph Theory with Applications]— Introduction to the theory of graphs, with applications to real world problems. Topics may include, but are not necessarily restricted to: connectivity, paths and cycles, trees as information structures, digraphs and depth-first search, stability and packing problems, matching theory and schedules, transportation networks, Max-Flow-Min-Cut Theorem, planar graphs, color ability, and the four color problem. Admission to this course is usually contingent upon a student’s having credit for Mathematics 228. Offered in alternate years. Prerequisite: C- or better in Mathematics 228 or permission of instructor. (NUM) (Enrollment limited)

[331. Analysis I]— Properties of the real number system, elementary topology, limits, continuity, uniform convergence, differentiation and integration of real-valued functions, sequences, and series of functions. Prerequisite: C- or better in Mathematics 228 or permission of instructor. (WEB) (Enrollment limited)

[341. Complex Analysis]— Algebra of complex numbers, analytic functions and conformal mappings, integrals of analytic functions and Cauchy’s theorem, expansion of analytic functions in series, calculus of residues. Prerequisite: C- or better in Mathematics 231. (NUM) (Enrollment limited)

399. Independent Study— Submission of the special registration form, available in the Registrar’s Office, and the approval of the instructor and chairperson are required for enrollment. (0.5 - 2 course credits) –Staff

400. Senior Exercise— A capstone course for senior math majors. Prerequisites: permission of instructor. (NUM) (Enrollment limited) –Wyshinski

466. Teaching Assistant— Submission of the special registration form, available in the Registrar’s Office, and the approval of the instructor and chairperson are required for enrollment. (0.5 course credit) –Staff

497. Senior Thesis— Required of, but not limited to, honors candidates. –Staff

Courses Originating in Other Departments

[Computer Science 219. Theory of Computation]— View course description in department listing on p. 355. Prerequisite: C- or better in Computer Science 115L and Computer Science 203

Spring Term

Courses offered by the Aetna Quantitative Center

[101. Foundational Techniques for Quantitative Reasoning]— This course offers students new insights into important and widely used mathematical concepts, with a strong focus on numerical and algebraic relationships. (Enrollment limited)

Courses offered through the Mathematics Department

107. Elements of Statistics— A course designed primarily for students in the social and natural sciences. Topics include graphical methods, measures of central tendency and dispersion, basic probability, random variables, sampling, confidence intervals, and hypothesis testing. This course is not open to students with credit for Mathematics 131 or above, or who have placed into Mathematics 207 on the Mathematic Placement Examination. Prerequisite: A satisfactory score on the Mathematics Placement Examination or a C- or better in Quantitative Literacy 101. Students who qualify for Mathematics 131 or 207 will not be eligible to enroll in this course. (NUM) (Enrollment limited) –Nu’Man, Pellico, Ray

118. Mathematics of Games and Gambling— We introduce at an elementary level the mathematics necessary to analyze and understand games of strategy and chance, including: lotteries, poker, craps, tournaments, the prisoner’s dilemma, and the Monte Hall problem. Prerequisite: A satisfactory score on the Mathematics Placement Exam (NUM) (Enrollment limited) –Georges

[126. Calculus with Algebra and Trigonometry]— A continuation of Mathematics 125. Topics will include: the analytic geometry of lines, circles, and parabolas; functions and graphs; continuity; derivatives; and applications. Not open to students who have received credit for Mathematics 131. This course completes the sequence started in Mathematics 125. Together, Mathematics 125 and 126 combine a study of the differential calculus of functions of one variable with the necessary algebraic and trigonometric background. Prerequisite: C- or better in Mathematics 125. (NUM) (Enrollment limited)

131. Calculus I— The real number system, functions and graphs, continuity, derivatives and their applications, antiderivatives, definite integrals, and the fundamental theorem of calculus. Mathematics, natural science, and computer science majors should begin the Mathematics 131, 132 sequence as soon as possible. Not open to students who have received credit for Mathematics 126 or who have received credit by successful performance on the Advanced Placement Examination of the CEEB (see Catalogue section “Advanced Placement for First-Year Students”). Prerequiste: A satisfactory score on the Mathematics Placement Examination. (1.25 course credits) (NUM) (Enrollment limited) –Pellico

132. Calculus II— Topics concerning the Riemann integral and its applications, techniques of integration, first-order ordinary differential equations, and sequences and series. Prerequisite: C- or better in Mathematics 126 or 131, or an appropriate score on the AP Examination or Trinity’s Mathematics Qualifying Examination. (1.25 course credits) (NUM) (Enrollment limited) –Mauro, Ray, Staff, Wash

[205. Abstraction and Argument]— This course deals with methods of proof and the nature of mathematical argument and abstraction. With a variety of results from modern and classical mathematics as a backdrop, we will study the roles of definition, example, and counterexample, as well as mathematical argument by induction, deduction, construction, and contradiction. This course is recommended for distribution credit only for non-majors with a strong mathematical background. (NUM) (Enrollment limited)

207. Statistical Data Analysis— An introductory course in statistics emphasizing modern techniques of data analysis: exploratory data analysis and graphical methods; random variables, statistical distributions, and linear models; classical, robust, and nonparametric methods for estimation and hypothesis testing; analysis of variance and introduction to modern multivariate methods. Students with a strong mathematical background are advised to take Math 207 in place of Math 107. Those who successfully complete Math 107 may take Math 207 for credit due to its increased depth of coverage and breadth of topics. Prerequisite: A suitable score on the Mathematics Placement Examination or a grade of C- or better in Mathematics 107.. (NUM) (Enrollment limited) –Adelstein, Goldwyn, Skardal

228. Linear Algebra— A proof-based course in linear algebra, covering systems of linear equations, matrices, determinants, finite dimensional vector spaces, linear transformations, eigenvalues, and eigenvectors. Prerequisite: C- or better in Mathematics 142 or 132, or a 200-level mathematics course, or permission of instructor. (NUM) (Enrollment limited) –Emerick

231. Calculus III: Multivariable Calculus— Vector-valued functions, partial derivatives, multiple integrals, conic sections, polar coordinates, Green’s Theorem, Stokes’ Theorem, and Divergence Theorem. Prerequisite: C- or better in Mathematics 132 or 142. (1.25 course credits) (NUM) (Enrollment limited) –Adelstein

234. Differential Equations— An introduction to techniques for solving ordinary differential equations. Series solutions, initial value problems, and Laplace transforms. Prerequisite: C- or better in Mathematics 132 or 142. (NUM) (Enrollment limited) –Emerick, Skardal

252. Introduction to Mathematical Modeling, I— Application of elementary mathematics through first-year calculus to the construction and analysis of mathematical models. Applications will be selected from the natural sciences and social sciences, with an emphasis on the natural sciences. Several models will be analyzed in detail, and the computer will be used as necessary. The analysis will consider the basic steps in mathematical modeling: recognition of the non-mathematical problem, construction of the mathematical model, solution of the resulting mathematical problems, and analysis and application of the results. Both Mathematics 252 and 254 may be taken for credit. Prerequisite: C- or better in Computer Science 115L and Mathematics 132 or 142. (NUM) (Enrollment limited) –Goldwyn

[253. Number Theory and Its Application]— An introduction to the standard topics in number theory. Topics will include congruences, representation of integers, number theoretic functions, primitive roots, continued fractions and Pythagorean triples. Applications may include cryptology, primality testing, and pseudorandom numbers. Prerequisite: C- or better in Mathematics 132 or 142. (NUM) (Enrollment limited)

[254. Introduction to Mathematical Modeling, II]— A companion to Mathematics 252, with an alternate set of topics and an emphasis on applications selected from the social sciences, especially economics. See description of Mathematics 252. Both Mathematics 252 and 254 may be taken for credit. Prerequisite: C- or better in Computer Science 115 and one year of calculus, or permission of instructor. (NUM) (Enrollment limited)

[305. Probability]— Discrete and continuous probability, combinatorial analysis, random variables, random vectors, density and distribution functions, moment generating functions, and particular probability distributions including the binomial, hypergeometric, and normal. Prerequisite: C- or better in Mathematics 231. (NUM) (Enrollment limited)

306. Mathematical Statistics— We consider confidence intervals and hypothesis testing from a theoretical viewpoint, with emphasis on sufficiency, completeness, minimum variance, the Cramer-Rao lower bound, the Rao-Blackwell theorem, and the Neyman-Pearson theorem. Other topics as time permits. Prerequisite: C- or better in Mathematics 305. (NUM) (Enrollment limited) –Mauro

[307. Abstract Algebra I]— An introduction to group theory, including symmetric groups, homomorphism and isomorphisms, normal subgroups, quotient groups, the classification of finite abelian groups, the Sylow theorems. Prerequisite: C- or better in Mathematics 228 or permission of instructor. (WEB) (Enrollment limited)

308. Abstract Algebra II— A continuation of Mathematics 307. Further topics from group, ring, and field theory. Prerequisite: C- or better in Mathematics 307. (NUM) (Enrollment limited) –Wash

[309. Numerical Analysis]— Theory, development, and evaluation of algorithms for mathematical problem solving by computation. Topics will be chosen from the following: interpolation, function approximation, numerical integration and differentiation, numerical solution of nonlinear equations, systems of linear equations, and differential equations. Treatment of each topic will involve error analysis. Prerequisite: C- or better in Computer Science 115, either MATH 132 or MATH 142, and any mathematics course numbered 200 or higher. (NUM) (Enrollment limited)

[314. Combinatorics and Computing]— Introduction to combinatorics. Topics may include, but will not necessarily be limited to, computer representation of mathematical objects, enumeration techniques, sorting and searching methods, generation of elementary configurations such as sets, permutations and graphs, and matrix methods. Prerequisite: C- or better in Mathematics 228 or permission of instructor. (NUM) (Enrollment limited)

325. Special Topics in Graph Theory— (NUM) (Enrollment limited) –Georges

[325. Special Topics in Analysis]— A course which will be offered from time to time to meet the special needs and interests of mathematics students. (NUM) (Enrollment limited)

331. Analysis I— Properties of the real number system, elementary topology, limits, continuity, uniform convergence, differentiation and integration of real-valued functions, sequences, and series of functions. Prerequisite: C- or better in Mathematics 228 or permission of instructor. (WEB) (Enrollment limited) –Russo

[332. Analysis II]— Further topics which may include Fourier analysis, general integration theory, and complex analysis. Prerequisite: C- or better in Mathematics 331. (NUM) (Enrollment limited)

341. Complex Analysis— Algebra of complex numbers, analytic functions and conformal mappings, integrals of analytic functions and Cauchy’s theorem, expansion of analytic functions in series, calculus of residues. Prerequisite: C- or better in Mathematics 231. (NUM) (Enrollment limited) –Wyshinski

399. Independent Study— Submission of the special registration form, available in the Registrar’s Office, and the approval of the instructor and chairperson are required for enrollment. (0.5 - 2 course credits) –Staff

400. Senior Exercise— A capstone course for senior math majors. Prerequisites: permission of instructor. (NUM) (Enrollment limited) –Mauro

497. Senior Thesis— Required of, but not limited to, honors candidates. –Staff

Courses Originating in Other Departments

Computer Science 219. Theory of Computation— View course description in department listing on p. 358. Prerequisite: C- or better in Computer Science 115L and Computer Science 203 –Miyazaki