Mathematics

Associate Professor Russo, Chair; Professors Cruz-Uribe, Georges^†, Mauro, Seabury Professor of Mathematics and Natural Philosophy Robbins, and Professor Stein; Associate Professors Sandoval and Wyshinski∙; Visiting Professor Panik; Harold L. Dorwart Visiting Assistant Professors Melanson and Wang; Visiting Assistant Professors Kelsey and Miller; Aetna Quantitative Center Senior Lecturer Gregory, Interim Director

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 post-baccalaureate 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 11 courses, including MATH 126 or 131, 132, 231, 228, 307, 331, and 400. No course with a grade of less than C- may be counted toward the major. Of the four electives, one must be a 300-level mathematics course, two must be mathematics courses at the 200 level, and the fourth can either be another 200+ mathematics course, or may be chosen from the courses listed below, which are offered by other departments.

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 and the department chair will together select an honors adviser (usually the candidate’s academic adviser) who will supervise the honors process.

The honors thesis 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 draft to the honors adviser no later than two weeks before the last day of classes of the spring semester. An informal talk will be given by the candidate prior to the day on which senior grades are due.

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 Web site.

Many study-abroad programs in English-speaking countries offer a wide range of mathematics courses that will count towards 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.

101. Contemporary Applications: Mathematics for the 21st Century— This course offers students new insights into fundamental mathematical concepts as they apply to a variety of current local and national issues. Areas of concentration are numerical, statistical, algebraic, and logical relationships. Three hours of lecture and one hour of laboratory per week. (Enrollment limited)-Russo, Staff

[117. Visually Displaying Data: Graphical Literacy]— This course will examine the efficient communication of complex quantitative ideas in many formats: data maps, time-series, space-time narrative designes, charts, and graphs. Students will learn what properties make a graphic display coherent and compelling and what practices introduce distortions and confusion and should be avoided. Theories will be illustrated by historical examples such as Florence Nightingale’s statistical diagrams, Snow’s data maps of the cholera epidemics in 19th-century London, and the charts used by engineers and project managers in their decision to launch the Challenger spacecraft. As part of this course each student will complete a project involving the analysis and effective display of information from Trinity’s City Data Center. Readings will include Tufte’s Visual Display of Quantitative Information, and selections from the “Visual Revelations” column of the Chance journal. Computer software used: Excel, PowerPoint, and GIS. (Enrollment limited)

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. Students having a mathematical background which includes Mathematics 231 should consider the Mathematics 305, 306 sequence for work in probability and statistics. (Enrollment limited)-Kelsey, Miller

[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. (Enrollment limited)

[123. Mathematical Gems]— An introduction to mathematical topics from number theory, geometry, game theory, infinity, chaos, and more. (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. (Enrollment limited)-Miller

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”). (1.5 course credits) (Enrollment limited)-Cruz-Uribe, Kelsey, Robbins, Wang

[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 Mathematics 131, or an appropriate score on the AP Examination or Trinity’s Mathematics Qualifying Examination. (1.5 course credits) (Enrollment limited)

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 Mathematics 131, or an appropriate score on the AP Examination or Trinity’s Mathematics Qualifying Examination. (1.5 course credits) (Enrollment limited)-Mauro

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 distibution credit only for non-majors with a strong mathematical background. (Enrollment limited)-Mauro

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: C- or better in Mathematics 107. (Enrollment limited)-Russo, Wang

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 Mathematics 132 or a 200-level Mathematics course, or permission of the instructor. (Enrollment limited)-Sandoval

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. (1.5 course credits) (Enrollment limited)-Melanson

[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 132 or 142. (Enrollment limited)

[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. (Enrollment limited)

[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: Computer Science 115, either MATH 132 or MATH 142, and any mathematics course numbered 200 or higher.

[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.

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 Mathematics 231. (Enrollment limited)-Sandoval

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. (Enrollment limited)-Melanson

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. (Enrollment limited)-Robbins

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 credit) -Staff

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

[102. Newsmath: Logic and Statistics in the Media]— Can you believe everything you read? This course will examine the basic principles of quantitative argument and reasoning, including statistics and statistical inference, comparisons of sizes and rates, and graphical displays of all kinds. We will make extensive use of media such as newspapers, magazines, articles on the Web, advertisements, letters to the editor, and policy statements of government officials to become active and critical consumers and presenters of data and argument. Throughout the semester each student will create an annotated notebook of current examples of fallacies, invalid arguments, misleading uses of data, and graphical distortions, along with critiques, and for some graphs, a corrected version. Computer software used: Excel. (Enrollment limited)

116. Fair Division: Quantitative Approaches— Fair division problems involve the allocation of people, goods or power among the members of a group. This course will examine algorithms for allocating both divisible and indivisible assets, and, especially, the notion of fairness as a quantifiable property and as the subject of several important theorems. Theories will be illustrated by historical and contemporary examples, such as original quantitative arguments by Thomas Jefferson and Daniel Webster, a 1999 patent application for a division algorithm, the Law of the Sea Convention’s regulations pertaining to sea-bed mining, and the mechanism of the U.S. Electoral College. Computer software tools will be MapInfo and Excel. This course does satisfy the Numerical and Symbolic Reasoning Requirement. (Enrollment limited)-Staff

[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. (Enrollment limited)

123. Mathematical Gems— An introduction to mathematical topics from number theory, geometry, game theory, infinity, chaos, and more. (Enrollment limited)-Wyshinski

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: Mathematics 125 with a grade of C- or better. (Enrollment limited)-Staff

[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”). (1.5 course credits) (Enrollment limited)

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 Mathematics 131, or an appropriate score on the AP Examination or Trinity’s Mathematics Qualifying Examination. (1.5 course credits) (Enrollment limited)-Robbins, Sandoval, Staff

201. Problem Solving in Mathematics— Problems appear in every part of mathematics and often have an intrinsic beauty and appeal. Mathematical problem solving is not a distinct branch of mathematics, but rather is a “mindset” which combines results from all branches of mathematics with a collection of useful techniques and strategies. Attempts have been made to develop “systems” for problem solving, but for the most part facility is gained through experience. The purpose of this course is to develop skills in and foster an appreciation of mathematical problem solving. It will not be a “cookbook” course which teaches students to match stereotypical problems with canned solutions. Rather, the course will be a hands-on experience, and students will be expected to explore and present solutions to a wide variety of non-routine and challenging problems, both individually and in groups. Since the range of problems which a student can solve expands as a student masters more branches of mathematics, students can profitably repeat this course. This course may only be taken Pass/Fail and may be retaken for credit with permission of the department. Prerequisite: C- or better in Mathematics 126 or Mathematics 131, or an appropriate score on the AP Examination or Trinity’s Mathematics Qualifying Examination. (0.5 course credit) (Enrollment limited)-Staff

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. (Enrollment limited)-Melanson

[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: Computer Science 115L and a C- or better in either Mathematics 132 or 142. (Enrollment limited)

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. (Enrollment limited)-Wyshinski

[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. (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: Mathematics 305 with a grade of C- or better. (Enrollment limited)

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. (Enrollment limited)-Sandoval

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: Computer Science 115, either MATH 132 or MATH 142, and any mathematics course numbered 200 or higher. (Enrollment limited)-Melanson

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. -Mauro

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. (Enrollment limited)-Staff

[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. (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. (Enrollment limited)

332. Analysis II— Further topics which may include Fourier analysis, general integration theory, and complex analysis. Prerequisite: C- or better in Mathematics 331. -Cruz-Uribe

400. Senior Exercise— One of the most important equations in all of mathematics involves just five numbers. It relates these five numbers in such a fundamental fashion that a fair case can be made that these are the numbers on which hangs all of mathematics. Known as Euler’s equation, it is very simple: eip + 1 = 0. We are generally pretty comfortable with 1 and 0. But what about the others? What are they, and how does the relation among the five of them obtain? The goal of the course is to investigate e, i, and p: to find out something of their history and their properties, and the relationship among them. This is a writing- and proof-intensive course. (Enrollment limited)-Mauro

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

[Computer Science 219. Theory of Computation]— View course description in department listing on p. 345. Prerequisite: C- or better in Computer Science 115L and either Computer Science 203 or Mathematics 205.