# Lawrence Course Catalog

**Please note:** The information displayed here is current as of Monday, July 13, 2020, but the official Course Catalog should be used for all official planning.

This catalog was created on Monday, July 13, 2020.

## Mathematics

Professors: | S. Corry, K. Krebsbach (on leave term(s) I), A. Parks, B. Pourciau (on leave term(s) I, II, III) |
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Associate professors: | J. Gregg, R. Sanerib |

Assistant professors: | A. Chakraborty, J. Rana (on leave term(s) III), 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:

- Complete or place out of the calculus sequence: MATH 140:
*Calculus I*, MATH 155:*Multivariable Calculus*, and MATH 200:*Complex Sequences and Series* - One of the following:
- MATH 210:
*Differential Equations with Linear Algebra*, - MATH 220:
*Applied Combinatorics*, or - MATH 240:
*Probability*

- MATH 210:
- One computer science course numbered 110 or above (excluding 170)
- MATH 300:
*Foundations of Algebra*and 310:*Foundations of Analysis* - 24 additional units in mathematics courses numbered 400 or above
- 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:

- 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*

- MATH 140:
- 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

- The following mathematics courses:
- 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*

- ECON 300:
- Any three six-unit courses numbered between 400 and 580

- ECON 100:
- 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

- The calculus sequence:
- MATH 140:
*Calculus I* - MATH 155:
*Multivariable Calculus* - MATH 200:
*Complex Sequences and Series*

- MATH 140:
- One of the following:
- MATH 210:
*Differential Equations with Linear Algebra*, - MATH 220:
*Applied Combinatorics*, or - MATH 240:
*Probability*

- MATH 210:
- MATH 300:
*Foundations of Algebra*and MATH 310:*Foundations of Analysis* - 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.