# Lawrence Course Catalog

**Please note:** The information displayed here is current as of Sunday, June 20, 2021, but the official Course Catalog should be used for all official planning.

This catalog was created on Sunday, June 20, 2021.

## Mathematics

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

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 tools 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, a statistics track through the 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. In addition to a minor in mathematics, the department supports an interdisciplinary minor in Statistics and Data Science.

The first two years of our program provides an introduction to the areas of calculus, analysis, discrete mathematics, and linear algebra. These courses pave 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 independently 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 support other majors and meet the needs of students wishing to continue their study of mathematics, statistics, or computer science.

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 mathematics and statistics together with their powerful applications.

### Required for the major in mathematics

Students who major in mathematics will develop the ability to learn mathematics and statistics independently, to express their knowledge clearly and cogently, and to understand, critique, and construct mathematical and statistical 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.

In addition to the general mathematics track, we also have a statistics track for students wishing to focus their upper-level work in statistics. The requirements for both tracks refer to the following categories of elective courses:

A) **Algebra & Combinatorics**: MATH 525, 545, 555, 565

B) **Analysis & Topology**: MATH 510, 530, 535, 550, 560

C) **Applied Mathematics**: MATH 340, 350, 400, 420, 435

D) **Statistics**: STAT 255, 445, 450, 455

The major in mathematics (general track) requires the following:

- MATH 140:
*Calculus*, MATH 155:*Multivariable Calculus* - MATH 200:
*Complex Sequences & Series*, MATH 230:*Discrete Mathematics*, Math 250:*Linear Algebra*. - Five 6-unit courses chosen from categories A-D; at most one of these courses may come from category D, and at least 3 of the categories A-D must be represented.
- One six-unit computer science (CMSC) course numbered 150 or above; one of 150, 205, or 210 is recommended.
- Senior Experience: completion of a 6-unit independent study project in at least one term of the senior year.

The statistics track for the mathematics major requires the following:

- MATH 140:
*Calculus*, MATH 155:*Multivariable Calculus*. - MATH 200:
*Complex Sequences & Series*, MATH 230:*Discrete Mathematics*, Math 250:*Linear Algebra*. - MATH 340:
*Probability*. - One additional MATH course from categories A-C.
- STAT 255:
*Statistics for Data Science*. - Two of the following courses: STAT 445:
*Mathematical Statistics*, STAT 450:*Bayesian Statistics*, STAT 455:*Advanced Statistical Modeling.* - Either CMSC/STAT 205:
*Data Scientific Programming*or CMSC/STAT 208:*Machine Learning*. - Senior Experience: completion of a 6-unit independent study project in at least one term of the senior year.

### 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:

- MATH 140:
*Calculus*, MATH 155:*Multivariable Calculus*. - MATH 200:
*Complex Sequences & Series*, MATH 230:*Discrete Mathematics*, Math 250:*Linear Algebra*. - MATH 340:
*Probability.* - One of the following courses: STAT 445:
*Mathematical Statistics*, STAT 450:*Bayesian Statistics*, MATH 510:*Real Analysis.* - ECON 100:
*Introductory Economics.* - ECON 300:
*Microeconomics*, ECON 320:*Macroeconomics*, ECON 380:*Econometrics.* - One 6-unit ECON course numbered between 400 and 580.
- Senior Experience: a 6-unit 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

- MATH 140:
*Calculus*, MATH 155:*Multivariable Calculus*. - MATH 200:
*Complex Sequences & Series*, MATH 230:*Discrete Mathematics*, Math 250:*Linear Algebra*. - Two additional 6-unit courses chosen from categories A-D.

### Required for the minor in data science

- Two core courses:
- CMSC/STAT 205:
*Data-Scientific Programming* - STAT 255:
*Statistics for Data Science*

- CMSC/STAT 205:
- Two statistics and data science electives from the list of:
- CMSC/STAT 208:
*Machine Learning* - CMSC/STAT 405:
*Advanced Data Computing* - STAT 450:
*Bayesian Statistics* - STAT 455:
*Advanced Statistical Modeling*

- CMSC/STAT 208:
- Two additional courses in an area of focus such as economics, biology, psychology, government, computer science, statistical theory:
- ANTH 207:
*Quantitative Analysis in Anthropology* - BIOL 170:
*Integrative Biology: Experimental Design and Statistics* - BIOL 360:
*Introduction to Bioinformatics* - BIOL 375:
*Biostatistics* - BIOL 380:
*Ecological Modeling* - CHEM 210 / CHEM 211
*Analytical Chemistry and Statistical Methods in Analytical Chemistry* - CMSC 210:
*Introduction to Scientific Programming* - CMSC 470:
*Artificial Intelligence* - ECON 223: Q
*uantitative Decision Making* - ECON 380:
*Econometrics* - ECON 481:
*Advanced Econometrics & Modeling* - ENST 235 / GEOS 214:
*Climate and Climate Change* - ETST 302:
*Research Methods in Ethnic Studies* - GOVT 271:
*Research Methods in Political Science* - GOVT 538:
*Outside the Margin of Error: Polling and Quantitative Prediction in Modern Politics* - MATH 340:
*Probability* - PSYCH 280:
*Research Methods I* - PSYCH 281:
*Research Methods II* - STAT 445:
*Mathematical Statistics* - STAT 450:
*Bayesian Statistics*

- ANTH 207:

Additional Stipulations:

- No more than one of ANTH 207 and BIOL 170 may count toward the minor.
- The minor requires a total of six distinct courses. Thus, courses that are listed twice above (such as STAT 450) may count toward one, but not both requirements.
- No more than three courses, counting toward any one major or other minor, may be counted toward the minor.

### 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.

### First-year courses

The department offers two calculus sequences: MATH 140, 155, 200 (*Calculus, Multivariable Calculus, Complex Sequences & Series*) and MATH 120, 130 (*Applied Calculus I, II*). Students intending to major in computer science or chemistry must complete *Calculus I* and *Multivariable Calculus*. Students intending to major in mathematics or physics must take all three courses: *Calculus I, Multivariable Calculus, *and* Complex Sequences & Series*. Sufficiently prepared students should enter the calculus sequence during their first year. Sufficient preparation means strong high school mathematics, including a pre-calculus or elementary functions course. Students who lack this preparation yet need the calculus sequence should consider enrolling in MATH 103: *Preparation for Calculus*. 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.

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.

For students interested in statistics:

- Students
*without*prior study of statistics or calculus should enroll in STAT 107:*Principles of Statistics;* - Students
*with*prior study of statistics (e.g. AP statistics or BIOL 170) or credit for MATH 140 (or equivalent) should enroll in STAT 255:*Statistics for Data Science.*Students with credit for MATH 120 should consult with a member of the math department for placement advice.

### Advanced placement

Advanced placement in the calculus sequence and 6 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. Students with these scores should generally enroll in MATH 155: *Multivariable Calculus* after passing the ALEKS online diagnostic exam.

Six Lawrence units (for STAT 107: *Principles of Statistics*) may be obtained by scoring 4 or 5 on the College Board statistics exam. Students with these scores wishing to study statistics should enroll in STAT 255: *Statistics for Data Science.*

### 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.