Please note: The information displayed here is current as of Friday, November 22, 2019, but the official Course Catalog should be used for all official planning.

2019-2020 Course Catalog

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This catalog was created on Friday, November 22, 2019.


Mathematics

Professors:S. Corry (chair), K. Krebsbach, A. Parks, B. Pourciau
Associate professors:J. Gregg (on leave term(s) I), R. Sanerib (on leave term(s) I, II, III)
Assistant professors:A. Chakraborty, J. Rana, A. Sage, E. Sattler

Pattern and form surround us—from the branching angles of our blood vessels and the complexity of computer algorithms to inventory scheduling and the four-dimensional geometry of our universe. As the pure expression of pattern and form, mathematics provides the language for science. In the past 100 years, many disciplines have been virtually transformed by the infusion of mathematics, so that alongside the traditional field of mathematical physics, one now finds new disciplines such as mathematical biology, mathematical ecology, mathematical economics, mathematical linguistics and mathematical psychology.

But mathematics is so much more than its applications. As the study of formal structures, mathematics offers a supreme beauty, an abstract forest of pattern and form, at once deep, intricate, logical, and surprising, a forest holding wonders both known and unknown. The search for these wonders is no game, for mathematics bears on eternal truth: Primes—such as 2, 3, 5, 7, 11, 13, ...—cannot be written as the product of two smaller integers. How many primes are there? Infinitely many. This is a well-known wonder proved by Euclid. Twin primes—such as 3 and 5, 5 and 7, 11 and 13, 17 and 19, ...—are “consecutive” primes. How many twin primes are there? No one knows. Mathematicians have unleashed their most sophisticated weapons on this problem, but the question remains unanswered. It is an unknown wonder. Will you be the first to find the answer? Whatever the answer, it is an eternal and universal truth: true for all time, in all places, to every intellect.

To reflect the diversity of modern mathematics and its applications, the department offers a mathematics major and, in conjunction with the economics department, an interdisciplinary major in mathematics-economics. The department's computer science major is described separately under Computer Science.

Our core sophomore sequence provides majors with a firm foundation in two pillars of mathematics (abstract algebra and real analysis), paving the way for exploration of diverse elective offerings at the junior and senior level. We offer courses in many areas of pure and applied mathematics, elementary and advanced statistics, and computer science. Majors engage in a one-term independent study during their senior year, working on a topic of their choice under the guidance of a faculty member. This transforming experience demonstrates a student’s ability to learn mathematics with little supervision and to clearly and cogently express this knowledge both verbally and in writing.

The department offers a number of elementary- and intermediate-level courses designed to meet the needs of students who wish to continue the study of mathematics or to complete required work in another major.

Lawrentians majoring in mathematics or mathematics-economics prepare themselves for a wide variety of interesting careers, but wherever life takes them, they have one thing in common—the logical and precise, yet intuitive and creative, habit of mind instilled by the serious study of abstract mathematics.

Required for the major in mathematics

Students who major in mathematics will develop the ability to learn mathematics independently, to express mathematical knowledge clearly and cogently, and to understand, critique, and construct mathematical arguments. They will apply the principles of careful argumentation—agree on meaning before debating truth, expose all (especially hidden) assumptions, abstract from examples, seek the underlying structure, apply logic pristinely—to critique arguments in other fields.

The major in mathematics requires the following:

  1. Complete or place out of the calculus sequence: MATH 140: Calculus I, MATH 155: Multivariable Calculus, and MATH 200: Complex Sequences and Series
  2. One of the following:
    • MATH 210: Differential Equations with Linear Algebra,
    • MATH 220: Applied Combinatorics, or
    • MATH 240: Probability
  3. One computer science course numbered 110 or above (excluding 170)
  4. MATH 300: Foundations of Algebra and 310: Foundations of Analysis
  5. 24 additional units in mathematics courses numbered 400 or above
  6. Completion of a 6-unit independent study project in at least one term of the senior year.

Course suggestions

In choosing courses beyond the core sequence, students should note that certain advanced courses may be particularly relevant to majors with specific interests or career goals. These lists offer suggestions; students are not expected to take all the courses in a given list.

  • Pure mathematics: 410, 525, 530, 535, 545, 550, 555, 560, 565, and 600
  • Computer science: 420, 435, 525, 555, and 565
  • Operations research: 410, 420, 435, 440, 445, 525, and 550
  • Applied mathematics: 410, 420, 435, 440, 445, 535, and 550
  • Statistics and actuarial science: 410, 420, 435, 440, 445, and 550
  • Engineering: 410, 420, 435, 440, 535, and 550
  • Secondary teaching: 410, 525, 530, 535, 545, 550, and 600

Required for the interdisciplinary major in mathematics-economics

Students who complete the major in mathematics-economics will pursue the outcomes described for the economics and mathematics majors with an explicit focus on economics in constructing and critiquing mathematical arguments. Students pursuing the major must have an advisor in each department.

The major in mathematics-economics requires the following:

  1. The mathematics component of the major is:
    • The following mathematics courses:
      • MATH 140: Calculus I
      • MATH 155: Multivariable Calculus
      • MATH 200: Complex Sequences and Series
      • MATH 240: Probability
      • MATH 300: Foundations of Algebra
      • MATH 310: Foundations of Analysis
    • Either MATH 435: Optimization or MATH 445: Mathematical Statistics
    • Six additional units in a mathematics course numbered 400 or above, with 435, 440, 445, or 560 recommended
  2. The economics component of the major is:
    • ECON 100: Introductory Economics
    • The following theory courses (majors must take all three courses prior to completion of the junior year; the economics department must approve any exception):
      • ECON 300: Microeconomics
      • ECON 320: Macroeconomics
      • ECON 380: Econometrics
    • Any three six-unit courses numbered between 400 and 580
  3. The interdisciplinary component of the major is:
    • Completion of an independent study project that has been approved by both departments.

Senior Experience in mathematics or mathematics-economics

The mathematics department's Senior Experience consists of a 6-unit (typically one-term) independent study project completed in the senior year. The project must demonstrate the capacity to learn mathematics (or statistics) independently or to utilize mathematics or mathematical technique as an innovative or substantive part of a larger project.

Interdisciplinary mathematics-economics majors must demonstrate the ability to combine topics in both disciplines—bringing appropriate techniques of mathematics or statistics to bear on the study of economics, or learning mathematics or statistics suggested by economic models.

For mathematics majors, the project must be approved and supervised by a faculty member in the mathematics department. For mathematics-economics majors, the project must be approved by a faculty member of each department and supervised by a member of one of the departments. Students should consult with departmental members in the spring before their senior year, in order to plan appropriately for their Senior Experience.

Required for the minor in mathematics

  1. The calculus sequence:
    1. MATH 140: Calculus I
    2. MATH 155: Multivariable Calculus
    3. MATH 200: Complex Sequences and Series
  2. One of the following:
    • MATH 210: Differential Equations with Linear Algebra,
    • MATH 220: Applied Combinatorics, or
    • MATH 240: Probability
  3. MATH 300: Foundations of Algebra and MATH 310: Foundations of Analysis
  4. 6 units in any one upper-level mathematics course numbered from 400 to 600

Teacher certification in mathematics

Mathematics majors can seek certification to teach math at the secondary level. Students can add an endorsement in a second area by completing an appropriate minor. Students who plan to seek teacher certification should review the requirements in the Education section of the catalog and meet with the director of teacher education, preferably before the end of the sophomore year.

Course numbering

Typically, courses numbered below 400 are offered each year, while courses numbered 400 or higher are offered every other year.

First-year courses

The department offers two calculus sequences: MATH 140, 155, 200 (Calculus I, Multivariable Calculus, Complex Sequences and Series) and MATH 120, 130 (Applied Calculus I, II). Students intending to major in computer science, physics, or chemistry must complete Calculus I and Multivariable Calculus. Students intending to major in mathematics must take all three courses: Calculus I, Multivariable Calculus, and Complex Sequences and Series. Properly prepared students should enter the calculus sequence their freshman year. Proper preparation means strong high school mathematics, including a pre-calculus or elementary functions course. Strong scores in a standard college preparatory exam offer good evidence as well. Students who lack this preparation yet need the calculus sequence should consult their advisor and the mathematics department as soon as possible. In every case, all students intending to enroll in MATH 140, 155, or 200 must take the ALEKS online diagnostic exam covering topics in pre-calculus, and a score of at least 75% is required for enrollment. Students who score below 75% may receive supplemental instruction through the Center for Academic Success to improve their score.

The Applied Calculus I, II sequence is designed to introduce students to the applied mathematics used in the social and life sciences. This sequence demands less technical proficiency than does the regular calculus sequence. Good performance in high school mathematics through the junior year should be adequate preparation.

Advanced placement

Advanced placement in the calculus sequence and up to 12 Lawrence units may be obtained by presenting a score of 4 or 5 on the AB or BC calculus exams administered by the College Board. Consult the department for proper placement. 

Six Lawrence units (for MATH 107) may be obtained by scoring 4 or 5 on the College Board statistics exam. Consult the department for proper placement.

Tutorials

The department views tutorials as opportunities to enhance its usual course offerings, not duplicate them. In order to reserve tutorials for this purpose, no tutorials or directed studies are given for courses routinely offered, and the department does not normally permit a tutorial to be used to satisfy any requirement for the major.

Off-campus and cooperative programs

Students wishing to combine a liberal arts degree with engineering should consider the 3-2 program in engineering.

The department encourages students to apply to the many Research Experiences for Undergraduates (REU) programs funded by the National Science Foundation; in these summer programs, students receive a stipend and participate in research teams at various campuses throughout the country. Students may also be interested in the Budapest Semester in Mathematics or in one of several other off-campus study options. Department faculty members can provide details.


Courses - Mathematics

MATH 107: Elementary Statistics

For students in all disciplines. Provides the background needed to evaluate statistical arguments found in newspapers, magazines, reports, and journals and the logic and techniques necessary to perform responsible elementary statistical analysis. Topics include basic data analysis, one-variable regression, experimental and sampling design, random variables, sampling distributions, and inference (confidence intervals and significance testing). This course may not be taken on a Satisfactory/Unsatisfactory basis.
Units: 6.
Prerequisite: Completion of 54 units in Lawrence courses or consent of instructor

MATH 120: Applied Calculus I

A course in the applications of mathematics to a wide variety of areas, stressing economics and the biological sciences. Topics may include recursive sequences and their equilibria, the derivative of a function, optimization, fitting abstract models to observed data. Emphasis placed on algebraic and numerical techniques and on understanding the role of mathematical thinking. Mathematics 120 and 130 do not prepare students for more advanced courses in mathematics.
Units: 6.
Prerequisite: Three years of high school mathematics;

MATH 130: Applied Calculus II

A continuation of math 120. Topics may include the indefinite and definite integral, elementary linear algebra including matrix arithmetic and solving linear equations, vectors, partial derivatives, Lagrange multipliers. Both algebraic and numerical computations.
Units: 6.
Prerequisite: MATH 120 or the equivalent

MATH 140: Calculus I

Functions, limits, derivatives, the Mean Value Theorem, definition and properties of integrals, the Fundamental Theorem of Calculus, and applications to related rates, curve sketching, and optimization problems.
Units: 6.
Prerequisite: Four years of high school mathematics and minimum score on ALEKS online diagnostic exam, as set by the department.

MATH 155: Multivariable Calculus

Techniques of integration, vector algebra in the plane and space, matrix algebra, functions of several variables, partial derivatives, double and triple integration, optimization.
Units: 6.
Prerequisite: MATH 140 or equivalent, appropriate score on the departmental placement exam

MATH 191: Directed Study in Mathematics

Directed study follows a syllabus set primarily by the instructor to meet the needs or interests of an individual student or small group of students. The main goal of directed study is knowledge or skill acquisition, not research or creative work.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 200: Complex Sequences and Series

Complex numbers, sequences, convergence, series, power series, additional topics chosen from analysis, geometry, differential equations, and applied mathematics
Units: 6.
Prerequisite: MATH 155

MATH 208: Machine Learning

An overview of techniques used to discover structural patterns and make predictions using complex datasets that are prevalent in today's world. The central machine learning tasks of classification, clustering, and regression will be explored, along with methods for training models and evaluating predictions. This course will be taught in a workshop format. Assignments will involve the use of statistical software.
Units: 6.
Also listed as Linguistics 208, Computer Science 208
Prerequisite: One course in mathematics or computer science, or BIOL 170, or consent of instructor

MATH 210: Differential Equations with Linear Algebra

A study of differential equations and related techniques in linear algebra. Topics include first-order equations and their applications, existence and uniqueness of solutions, second-order linear equations and their applications, series solutions, systems of first-order equations, vector spaces and dimension, linear transformations, and eigenvalues.
Units: 6.
Prerequisite: MATH 160, or MATH 150 and consent of instructor

MATH 217: Applied Statistical Methods

A second course in statistics that covers analyses needed to solve more complicated data-driven problems. Time permitting, topics include multiple regression, analysis of variance, categorical data analysis, nonparametric tests, bootstrap methods, and permutation tests. Class meetings are a mixture of lecture, discussion, and use of statistical software to investigate real data.
Units: 6.
Prerequisite: AP examination credit in statistics or MATH 107, or BIOL 170, or PSYC 170

MATH 220: Applied Combinatorics

An introduction to logic, proofs by mathematical induction, and elementary combinatorics. Additional topics include recurrence relations, generating functions, and the principle of inclusion-exclusion.
Units: 6.
Prerequisite: MATH 150

MATH 223: Quantitative Decision-Making

The students will learn how to develop formal, quantitative approaches to structuring difficult problems, particularly those problems involving probabilistic factors. We will develop and practice the steps of defining a problem, gathering data, formulating a model, performing numerical calculations, evaluating numerical information, refining the model, analyzing the model's alternatives, and communicating the results.
Units: 6.
Also listed as Economics 223
Prerequisite: Sophomore standing

MATH 240: Probability

An introduction to probability and its applications. Topics will include combinatorial and axiomatic probability, conditional probability and Bayes' Theorem, random variables, expectation and variance, discrete and continuous probability distributions, joint and conditional distributions, and limit laws.
Units: 6.
Prerequisite: MATH 160, or MATH 150 and consent of instructor

MATH 300: Foundations of Algebra

An introduction to the rigorous study of mathematics. Topics include elementary theory of sets and mappings, number theory, equivalence relations, finite groups, homomorphisms, quotient groups, and rings.
Units: 6.
Prerequisite: MATH 210, 220, or 240

MATH 310: Foundations of Analysis

A study of the concepts that underlie mathematical analysis: the completeness of the real numbers, convergence, continuity, derivatives, integrals, infinite series, and, if time permits, an introduction to metric spaces or Fourier series.
Units: 6.
Prerequisite: MATH 300

MATH 390: Tutorial Studies in Mathematics

Advanced work in mathematics on topics not covered in regular offerings.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 391: Directed Study in Mathematics

Directed study follows a syllabus set primarily by the instructor to meet the needs or interests of an individual student or small group of students. The main goal of directed study is knowledge or skill acquisition, not research or creative work.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 395: Internship In Mathematics

The academic component of the internship includes readings related to the substance of the internship, discussions with the faculty supervisor, and a written report appropriate to the discipline. Course grades are based on this academic work.
Units: 1 TO 98.

MATH 399: Independent Study in Mathematics

Guided independent study of an advanced topic in undergraduate mathematics or supervised work on an undergraduate research project, generally culminating in a final presentation and/or paper.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 400: Partial Differential Equations

A survey of techniques used in modeling physical systems, with particular emphasis on partial differential equations and methods used to attack problems that do not have clean or simple solutions. Topics include techniques for solving partial differential equations exactly, the Fourier transform, perturbation theory, variational methods, Monte Carlo techniques, and finite difference schemes.
Units: 6.
Prerequisite: MATH 300 or consent of instructor

MATH 410: Linear Algebra

A study of vector spaces, linear transformations, and their representations. The focus will be on algebraic and coordinate-free methods, and topics will include dimension, dual spaces, determinants, canonical forms, inner product spaces, and the spectral theorem.
Units: 6.
Prerequisite: MATH 300

MATH 420: Numerical Analysis

Computer approximated (numerical) solutions to a variety of problems with an emphasis on error analysis. Interpolation, evaluation of polynomials and series, solution of linear and non-linear equations, eigenvectors, quadrature (integration), and differential equations.
Units: 6.
Prerequisite: MATH 300 and CMSC 210 or CMSC 150

MATH 430: Statistical Modeling

An exploration of methods to select, fit, evaluate and compare statistical models, while also providing an introduction to statistical inference. Lectures will develop the necessary theory for regression models while maintaining the focus on applications. Students will complete regular assignments as well as a midterm and final exam.
Units: 6.
Prerequisite: MATH 240

MATH 435: Optimization

The study of local and global maximums and minimums of function, given various sorts of constraints. Linear problems and the simplex algorithm, general non-linear problems and the Kuhn-Tucker conditions, convex problems. Perturbation of problem parameters and duality. Applications to a wide variety of fields, including economics, game theory, and operations research.
Units: 6.
Prerequisite: MATH 310

MATH 440: Probability Theory

The mathematics of chance: probability, discrete and continuous random variables and their distributions, moments, jointly distributed random variables, conditional distributions, the Central Limit Theorem, and weak and strong convergence.
Units: 6.
Prerequisite: MATH 310

MATH 445: Mathematical Statistics

Development of the mathematical theory of statistics and its application to the real world. The course will focus on the principles of estimation and testing from both the frequentist and Bayesian perspectives. Resampling methods (permutation tests and bootstrap intervals) will also be explored.
Units: 6.
Prerequisite: MATH 240

MATH 450: Bayesian Statistics

A study of the Bayesian statistical philosophy, contrasting it with the traditional frequentist approach taught in other statistics courses. Topics include Bayes' Theorem, prior and posterior probability distributions, hierarchical models, and Markov Chain Monte Carlo methods. The course will involve a mixture of lecture, discussion, and use of statistical software. Requirements include exams, a project, and assignments involving the use of statistical software.
Units: 6.
Prerequisite: MATH 240

MATH 525: Graph Theory

A survey of graph theory that balances the abstract theory of graphs with a wide variety of algorithms and applications to “real world” problems. Topics include trees, Euler tours and Hamilton cycles, matchings, colorings, directed graphs, and networks.
Units: 6.
Prerequisite: MATH 300

MATH 530: Topics in Geometry

The axiomatic development of euclidean and non-euclidean geometry, including the historical and philosophical issues raised by the “non-euclidean revolution.” Additional topics, such as projective or differential geometry and convexity, may be included.
Units: 6.
Prerequisite: MATH 300

MATH 535: Complex Analysis

An introduction to functions of a complex variable, the Cauchy-Riemann equations, conformal mappings, Cauchy’s theorem, Cauchy’s integral formula, Taylor and Laurent series, and a sampling, as time and interest permit, of the corollaries to Cauchy’s theorem.
Units: 6.
Prerequisite: MATH 310

MATH 545: Rings and Fields

Modern algebra with topics selected from group theory, ring theory, field theory, classical geometric construction problems, and Galois theory. Emphasis on the use of mathematical abstraction to illuminate underlying relationships and structure.
Units: 6.
Prerequisite: MATH 300

MATH 550: Topics in Analysis

Selected topics in analysis covering a wide variety of spaces and leading to applications of classical importance. In recent years, topics have included fixed point theory, inverse and implicit function theorems, abstract theory of differential equations, Lebesgue measure and integration, Fourier series and transforms.
Units: 6.
Prerequisite: MATH 310

MATH 555: Topics Algebra & Combinatorics

A study of interconnections between abstract algebra (especially finite group theory) and combinatorics (especially graph theory). Topics will include classical results (such as the matrix-tree theorem), as well as recent subjects and advances (such as the abelian sandpile model and the Riemann-Roch theorem for graphs).
Units: 6.
Prerequisite: MATH 300

MATH 560: Topology

A study of metric and topological spaces, including continuity, compactness, connectedness, product and quotient spaces. Additional topics may include Zorn’s Lemma, separation properties, surfaces, the fundamental group, and fixed point theorems.
Units: 6.
Prerequisite: MATH 310

MATH 565: Number Theory

A study of the integers, including unique factorization, congruences, and quadratic reciprocity. Other topics may include finite fields, higher reciprocity laws, and algebraic number theory.
Units: 6.
Prerequisite: MATH 300

MATH 590: Tutorial Studies in Mathematics

Advanced work in mathematics on topics not covered in regular offerings.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 591: Directed Study in Mathematics

Directed study follows a syllabus set primarily by the instructor to meet the needs or interests of an individual student or small group of students. The main goal of directed study is knowledge or skill acquisition, not research or creative work.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 599: Independent Study in Mathematics

Guided independent study of an advanced topic in undergraduate mathematics or supervised work on an undergraduate research project, generally culminating in a final presentation and/or paper.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 600: History of Mathematics

A study of the history of mathematics from the ancient Greeks through the present, emphasizing the role of mathematics in scientific advances, the work of great mathematicians, and the modern branching of the subject into a multitude of specialties.
Units: 6.
Prerequisite: MATH 310

MATH 690: Tutorial Studies in Mathematics

Advanced work in mathematics on topics not covered in regular offerings.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 691: Directed Study in Mathematics

Directed study follows a syllabus set primarily by the instructor to meet the needs or interests of an individual student or small group of students. The main goal of directed study is knowledge or skill acquisition, not research or creative work.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.

MATH 699: Independent Study in Mathematics

Guided independent study of an advanced topic in undergraduate mathematics or supervised work on an undergraduate research project, generally culminating in a final presentation and/or paper.
Units: 1 TO 98.
Prerequisite: Counter Registration Required.