# Department of Mathematics and Statistics

## Academic Programs

##### Major Requirements* - Updated April 2023*

Any student who matriculated **before April 2023** can use the old or new requirements for the Mathematics major. The prior major requirements are available in the 2022-23 Academic Catalog (page 331)

**Department Website:**

*https://www.haverford.edu/mathematics-and-statistics*

The courses in the Department of Mathematics and Statistics aim to:

- promote rigorous thinking in a systematic, deductive, intellectual discipline.
- help students identify and articulate mathematical and statistical problems that they encounter, both in formal academic work and elsewhere.
- foster technical competence in mathematics and statistics as an aid to the better comprehension of the physical, biological, and social sciences.
- guide and direct majors toward an interest in research in the mathematical and statistical sciences.

## Learning Goals

The Mathematics and Statistics curriculum is designed so that, after completing course requirements, focusing on at least one of the subdisciplines of pure mathematics, applied mathematics, or statistics, and producing their senior thesis, students will gain the following:

**Mastery**of mathematical and/or statistical methods and concepts that allow students to solve non-routine and non-algorithmic problems, to apply mathematical or statistical techniques to new contexts, and to achieve ownership of mathematical or statistical ideas. Mastery includes acquiring skills such as processing technical material, understanding and creating logical arguments, formulating models and structures, and being able to deploy appropriate computational techniques.**Communication**of mathematical and statistical ideas in a clear, precise, and structured manner to audiences both expert and general. The core skill is to define terms and assumptions carefully and to build on those definitions using precise arguments and appropriate evidence.**Breadth**of knowledge across the mathematical sciences, including an understanding of links between subfields, connections to other disciplines both in STEM and in the broader liberal arts curriculum, and, in the spirit of Haverford’s motto Non doctior, sed meliore doctrina imbutus ("Not more learned, but imbued with better learning"), the social context in which mathematics is developed and used.**Independence, resilience, and persistence**in learning and using mathematics and statistics as a foundation for life-long learning and problem solving, balanced with a capacity to work collaboratively and to create and join communities of mathematical scientists.

Haverford’s Institutional Learning Goals are available on the President’s website, at *http://hav.to/learninggoals*.

## Curriculum

Majors take a core sequence of courses in calculus and linear algebra designed to provide a foundation for further study in pure math, applied math, or statistics. Apart from these core courses, each student chooses a "focus" (pure math, applied math, or statistics) and takes five courses to fulfill requirements for that focus (see "Major Requirements"). Finally, all majors take a junior seminar and write a senior thesis (supported by a senior seminar).

## Majors Offered by the Department

The Department of Mathematics and Statistics offers a major that gives students a choice among three focuses that each address the above learning goals in a distinctive way:

- Major: Mathematics (focus in pure math)
- Major: Mathematics (focus in statistics)
- Major: Mathematics (focus in applied math)

## Major Requirements

- Calculus II (MATH H118)
- Multivariable Calculus (MATH H121 or MATH H216)
- Linear Algebra (MATH H215)
- Junior Seminar (MATH H299)
- Senior Seminar (MATH H399)
- Senior Research (MATH H400), including a senior paper and oral presentation
- Five courses in the student’s chosen focus (see below)

### Focus Requirements in Pure Math

- Two of the following courses: Analysis (MATH H317), Algebra (MATH H333), Topology (MATH H335)
- Three additional math courses 200-level or higher (excluding Junior or Senior Seminar, Senior Research, Teaching Assistant, Independent Study), of which:
- at least one must be in pure math, and
- at least one must be outside of pure math, and
- at least one must be 300-level, and
- at most one can be cross-listed with another department.

### Focus Requirements in Statistics

- A course in introductory statistics (STAT H203, ECON H203, ECON H204, PSYC H200, or SOCL H215)
- Applied Multivariate Statistical Analysis (STAT H361)
- Probability (MATH H218)
- Two additional courses from the following list: Mathematical Statistics (STAT H328), Advanced Topics in Statistics (STAT H396), Analysis (MATH H317)

### Focus Requirements in Applied Math

- Differential Equations (MATH H204)
- Analysis (MATH H317)
- Two courses, of which at least one is 300-level, from the list below:
- Linear Optimization (MATH H210)
- Scientific Computing: Continuous Systems (MATH H222)
- Modeling and Differential Equations (MATH H382)
- Partial Differential Equations (MATH H383)
- Constrained and Combinatorial Optimization (MATH H384)
- Advanced Topics in Applied Math ( MATH H397 and/or MATH B325); can be taken more than once if topics are sufficiently different)

- One additional applied math course, which must be 300-level if a 200-level course was used as part of fulfilling the previous requirement. This additional course could be, but does not have to be, a third course from the list above. We have posted on our webpage a list of courses that are pre-approved as counting towards this requirement.

## Senior Project

### Senior Project Learning Goals

Our students will engage with advanced content and techniques in pure mathematics, applied mathematics or statistics. They will gain ownership of the process and material through understanding the content and the details of the problem they are investigating, constructing illustrative examples, carrying out novel computations or carefully analyzing a data set. Our students will write clear, careful and correct mathematics/statistics, from precise definition or description of a model to rigorous proofs or well-supported analyses. They will develop an oral presentation that highlights the central ideas of their thesis work at a level appropriate for an audience in the mathematical/statistical sciences.

### Senior Project Assessment

The grade for the senior thesis is determined by the following:

- Level of engagement with advanced mathematics or statistics.
- Level of ownership of the material and of the writing process.
- Adherence to professional standards of written mathematics and statistics.

The grade for the senior seminar is determined by the following:

- Completing all the assignments in accordance with the assignment description.
- Meeting deadlines for each assignment.
- Quality of intermediate drafts, including whether easily discernible progress has been made from one assignment to another.
- Engaged participation in seminar meetings.
- Quality of the thesis presentation.

## Policies Related to Major Requirements

- In reviewing student thesis topic proposals, the department will prioritize proposals where the topic is in the student's focus, to the extent possible given thesis advisor availability in that focus.
- If a student places past a course listed as required, that requirement is waived, but the student must take an additional course that counts for the major at the 200-level or higher (excluding Junior or Senior Seminar, Senior Research, Teaching Assistant, Independent Study). For example, a student placed past MATH 118 would need to take 1 additional course; a student placed past MATH 121 would need to take 2 additional courses; a student placed past STAT 203 who chooses statistics as their focus would need to take an additional course along with any additional courses implied by their calculus placement.
- In order to develop a connection between a student and the pool of potential thesis advisors, a major must take at least one course in their chosen focus at Haverford; exceptions to this rule are granted if, at the time of major declaration, the student gets approval from the department chair of a course plan that develops that connection in a different way.
- A student’s transcript can only show one focus, but can minor in a second focus area in certain combinations (see below).
- In the atypical circumstance that a student elects to graduate in seven semesters, they must inform the department of this choice when they declare their major in the spring of their sophomore year and complete the senior seminar and senior paper during the fall and spring of their junior year, i.e., during their fifth and sixth semesters. (This means that students have to complete Math 299 (Junior Seminar) before the 5th semester.)
- Students who graduate nine or more semesters after matriculating should make every effort to complete the Senior Seminar and senior paper during a single academic year (fall then spring), but students in this circumstance can complete these requirements “off-cycle” (spring then fall) with permission of the department chair. This permission should be sought as soon as the student knows of their interest in an off-cycle thesis, and no later than the spring pre-registration immediately preceding their proposed spring-fall thesis year. Off-cycle thesis students may have limited options for thesis advisors, since their spring-fall thesis year will overlap with two different fall-spring faculty sabbatical periods.
- If a student elects to spend the whole junior year abroad, they must complete Math 299 (Junior Seminar) before the 5
^{th}semester and inform the department of this choice when they declare their major in the spring of their sophomore year.

## Minors Offered by the Department

- Minor: Pure Math
- Minor: Statistics
- Minor: Applied Math

### Pure Math Minor Requirements

- Calculus II (MATH H118)
- Multivariable Calculus (MATH H121 or MATH H216)
- Linear Algebra (MATH H215)
- Two of the following courses: Analysis (MATH H317), Algebra (MATH H333), Topology (MATH H335)
- One additional course in pure math (a third course from the previous requirement, or another 200- or 300-level course in pure math)

### Statistics Minor Requirements

- Multivariable Calculus (MATH H121 or MATH H216)
- Linear Algebra (MATH H215)
- A course in introductory statistics (STAT H203, ECON H204, PSYC H200, or SOCL H215)
- Probability (MATH H218)
- Advanced Multivariate Statistical Analysis (STAT H361)
- One additional course in statistics from the following list: Mathematical Statistics (STAT H328), Advanced Topics in Statistics (STAT H396), or Advanced Econometrics (ECON H324)

**Applied Math Minor Requirements**

- Calculus II (MATH H118)
- Multivariable Calculus (MATH H121 or MATH H216)
- Linear Algebra (MATH H215)
- Differential Equations (MATH H204)
- Two additional courses in applied math, of which at least one must be at the 300-level; these two courses should come from the following list (or be approved as a substitution):
- Modeling and Differential Equations (MATH H383)
- Partial Differential Equations (MATH H383)
- Scientific Computing: Continuous Systems (MATH H222)
- Linear Optimization (MATH H210)
- Constrained and Combinatorial Optimization (MATH H384)
- Advanced Topics in Applied Math (MATH H397 and/or MATH B325); can be taken more than once if topics are sufficiently different)

## Policies Related to Minor Requirements

- If a student places past a course listed as required, that requirement is waived, but the student must take an additional course that counts for the minor at the 200-level or higher (excluding Junior or Senior Seminar, Senior Research, Teaching Assistant, Independent Study). For example, a student placed past MATH 118 would need to take 1 additional course; a student placed past MATH 121 would need to take 2 additional courses; a statistics minor placed past STAT 203 would need to take an additional course along with any additional courses implied by their calculus placement.
- A pure math minor
**cannot**be combined with a math major (focus in pure math) or with a math major (focus in applied math). - An applied math minor
**cannot**be combined with a math major (focus in pure math) or with a math major (focus in applied math). - A statistics minor
**cannot**be combined with a math major (focus in statistics). - Other than the case discussed in the previous bullet point, a minor in our department can be combined with a major, minor, or concentration (inside or outside of our department) as long as three or fewer courses are “double-counted”. For example:
- A student could complete a “Minor in statistics” and a “Concentration in Mathematical Economics” (as an economics major) if they use MATH 215, 216, and 218 to fulfill minor and concentration requirements, and then have no overlap in the courses they use to satisfy the remaining requirements.
- A student who places past MATH 118 would replace the MATH 118 major requirement by an additional 200-level course of their choosing (as per a major requirement policy above). Such a student could complete a "Major in Math (with focus in applied math)" and a "Minor in statistics" if they use MATH 121 and 215 to satisfy requirements for both the major and the minor, and MATH 218 to satisfy a minor requirement and as their “additional course” (replacing 118) for the major, and then have no overlap in the courses they use to satisfy the remaining requirements.

For further information about the statistics minor, please see the PDF supplement on the mathematics website, or contact the minor coordinator.

## Concentrations and Interdisciplinary Minors

Mathematics majors can pursue four areas of concentration:

### Computer Science (more theoretical)

It may come as a surprise to some that many of the fundamental questions in computer science (including the famous P versus NP problem) are in essence mathematical questions. Conversely, some of the deepest foundational questions about the nature of mathematics (such as: what constitutes a proof?) are inherently computational in nature. Computers have also become a powerful tool in mathematical research and its applications, both theoretical and experimental. A full understanding of their capability and potential can only be realized by formal coursework in computer science. The concentration is open to math or physics majors.

### Scientific Computing (more applied)

Many disciplines in the natural and social sciences include a significant sub-discipline that is explicitly computational. Examples include astronomy, biology, chemistry, economics, and physics. In some fields, such as biology, the use of computation has become so widespread that basic literacy in computation is increasingly important and may soon become required. The Concentration in Scientific Computing gives students an opportunity to develop a basic facility with the tools and concepts involved in applying computation to a scientific problem, and to explore the specific computational aspects of their own major disciplines.

### Mathematical Economics (for majors interested in applying their skills to economic problems)

Mathematics and economics are complementary disciplines. Most branches of modern economics use mathematics and statistics extensively, and some important areas of mathematical research have been motivated by economic problems. Economists and mathematicians have made important contributions to each other’s disciplines. Economist Kenneth Arrow, for example, did path-breaking work in the field of mathematical optimization; and in 1994 mathematician John Nash was awarded the Nobel Prize in economics for introducing a theory of equilibrium in non-cooperative games that has become central to contemporary economic theory. Haverford’s Area of Concentration in Mathematical Economics enables students in each of the disciplines not only to gain proficiency in the other, but also to understand the ways in which they are related and complementary.

### Mathematics Education (for majors interested in teaching mathematics)

The Bryn Mawr-Haverford Education Program invites students to study the discipline of education; explore the interdisciplinary field of educational studies; begin the path of teacher preparation for traditional classrooms; and participate in teaching experiences in a range of classroom and extra-classroom settings. Focused on teaching and learning as social, political, and cultural activities, the Education Program challenges students to explore the relationships among schooling, human development, and society as they gain knowledge and skills of educational theory and practice. Students who complete one of the Education Program options are prepared to become lifelong learners, educators, researchers, leaders and agents of change.

For the requirements for these concentrations, see those headings in this catalog or visit the departmental website.

## Affiliated Programs

Many of our graduates have pursued successful and interesting careers in various engineering disciplines. Our Accelerated Masters program with the University of Pennsylvania, 3/2 engineering program with CalTech, and the Master’s degree course exchange agreements with Swarthmore and the University of Pennsylvania offer robust—and unique—opportunities. For more information on these options, visit the Engineering website: *https://www.haverford.edu/engineering/*