Please note: The information displayed here is current as of Wednesday, June 20, 2018, but the official Course Catalog should be used for all official planning.

2017-2018 Course Catalog

Period: 2017-20182016-20172015-20162014-20152013-2014

This catalog was created on Wednesday, June 20, 2018.


Mathematics

Professors:K. Krebsbach (on leave term(s) III), A. Parks, B. Pourciau (on leave term(s) III)
Associate professors:S. Corry (chair), J. Gregg, R. Sanerib
Assistant professor:J. Rana
Visiting assistant professor:B. Rocks

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, alone or in conjunction with the economics department, offers three separate majors:

  • Mathematics
  • Mathematics-computer science
  • Mathematics-economics

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 and/or computer science 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.

For a full description of Lawrence’s computer facilities and for descriptions of the computer science courses visit the Computer Science website.

Required for the mathematics major

  1. Complete or place out of the calculus sequence: MATH 140, 150, and 160
  2. One of the following:
    • MATH 210
    • MATH 220
    • MATH 240
  3. One computer science course numbered 110 or above (excluding 170)
  4. MATH 300 and 310
  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.
  7. A C average in the major.

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 mathematics minor

  1. Calculus through MATH 160
  2. One of the following:
    • MATH 210
    • MATH 220
    • MATH 240
  3. MATH 300 and MATH 310
  4. 6 units in any one upper-level mathematics course numbered from 400 to 600
  5. C average in the minor.

Teacher Certification in Mathematics or Computer Science

Mathematics or mathematics-computer science majors can seek certification to teach math or computer science 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.

Required for the interdisciplinary mathematics-computer science major

  1. The core sequence: MATH 140, 150, 160 and CMSC 150, 250, 270
  2. MATH 220 and 300
  3. CMSC 460, 510, 515
  4. 6 additional units in mathematics courses selected from:
    • MATH 310
    • MATH 420
    • MATH 525
  5. 6 additional units in a computer science course numbered 400 or above
  6. 6 additional units in a computer science course numbered 400 or above or selected from among MATH 310, 420, 525
  7. Completion of an independent study project prior to the Spring Term of the senior year
  8. CMSC 600 in the senior year

Required for the interdisciplinary mathematics-economics major

  1. The mathematics component of the major is:
    • MATH 140, 150, 160, 240, 300, 310
    • Either MATH 435 or 445
    • 6 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
    • ECON 300, 320, and 380 (majors must take all three courses prior to completion of the junior year. The economics department must approve any exception.)
    • 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.
    • A major must have an advisor in each department.

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 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.

First-year courses

The department offers two calculus sequences: MATH 140, 150, 160 (Calculus I, II, III) and MATH 120, 130 (Applied Calculus I, II). Students intending to major in mathematics, mathematics-computer science, mathematics-economics, physics, or chemistry, or any student intending to take advanced mathematics courses, must complete the Calculus I, II, III sequence. Properly prepared students should enter this 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 three-course sequence should consult their advisor and the mathematics department as soon as possible.

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

Advanced Placement

Advanced placement in the Calculus I, II, III 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. 

Advanced placement and six Lawrence units (for CMSC 150) may be obtained by scoring 4 or 5 on the A or AB College Board computer science exam. 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.

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.

Course numbering

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

Senior Experience in Mathematics

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

Interdisciplinary mathematics-computer science majors must complete their independent study project in two parts: an independent study in the Fall or Winter Term of the senior year (usually 3 units), followed by a presentation of their results in the Winter Term Computer Science Senior Seminar (3 units).

For mathematics and mathematics-computer science 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.


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: Sophomore standing.

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. Placement exam not required.
Units: 6.
Prerequisite: Four years of high school mathematics

MATH 150: Calculus II

Applications of integration, exponential and logarithmic functions, techniques of integration, infinite sequences and series, and Taylor series.
Units: 6.
Prerequisite: Advanced placement, MATH 140, or MATH 120 and consent of instructor

MATH 160: Calculus III

Functions of two or more variables, partial derivatives, chain rules, optimization, vectors, derivatives of vector-valued functions, Lagrange multipliers, multiple integrals, line integrals, and Green’s Theorem.
Units: 6.
Prerequisite: MATH 150 or advanced placement

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 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

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 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 110 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 310 and MATH 440

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.

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