## Mathematics

The mathematics program is broad and flexible, providing the opportunity to study a variety of topics in pure and applied mathematics. Independent work during the junior year usually consists of participating in a junior seminar, which involves students working through some material on their own, after the relevant background is covered, and lecturing on their results. Alternatively some students work on an individual paper one or both terms, under the direction of a faculty member. Senior independent work focuses on the senior thesis, which can be an exposition of some topic in pure or applied mathematics, or the development of a mathematical model with some applications, or an extension of results of papers on current topics in mathematics, or the study of a particular problem in some depth. The focus can be on various topics in pure mathematics or on applications of mathematics in fields such as chemistry, economics, computer science, biology. Some students actually work with faculty members in other departments on applications of mathematics to problems or work in these areas. Most graduating seniors continue with graduate work in pure or applied mathematics, physics, computer science, law, medicine or business; others go directly on to careers in education, business, banking, finance, or industrial research.

### What Students Say

• What is mathematics?

• What can you learn from it?

• What is it like being a mathematics major?

• What are common misconceptions about mathematics majors?

• What kind of internships and international experiences have majors had?

• How will mathematics majors save the world?

• Why would anyone want to date a mathematics major?

• What can you learn from it?

• What is it like being a mathematics major?

• What are common misconceptions about mathematics majors?

• What kind of internships and international experiences have majors had?

• How will mathematics majors save the world?

• Why would anyone want to date a mathematics major?

**What is mathematics?**

Mathematics is the study of the abstract. We all have notions of number or shape—just add two to four or imagine the deformation of a sphere to a point -- but mathematicians seek to understand these concepts at a deeper level. What numbers can be expressed as the sum of two squares? Of four? What is the order of the number of primes from 0 to n? Can a surface that looks locally 7-dimensional and has constant negative curvature be deformed into something more manageable? How can you transmit information over a channel with a fixed error rate with minimal redundancy? Our tools are the definition, theorem, and proof — and reams of paper.

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**What can you learn from it?**You should study mathematics to learn how to think rigorously. Hiring managers at major companies will seek out math majors because of the depth of thought a math major is capable of. In fact, working math graduates are the most highly paid of any group of Princeton graduates (except computer science majors).

That said, your average math major probably makes for boring conversation at a cocktail party. Consider this dialogue:

English major: So, what kind of math are you studying?

That said, your average math major probably makes for boring conversation at a cocktail party. Consider this dialogue:

English major: So, what kind of math are you studying?

Math major: Combinatorics.

English major: What is that?

Math major: It is the study of counting.

English major: Really? Like, how many numbers lie between 1 and 100? That's easy; the answer is 100!

Math major: No, like, given n >= 2r and a family

*A*of distinct subsets of {1, ..., n} all of which are of size

*r*, then the number of sets in

*A*is at most .

English major: Oh.

Math major: You know, you can see that the theorem is really about uniform hypergraphs. That one is the Erdos-Ko-Rado theorem.

English major: Well, it was nice to meet you.

Math major: But wait, there's more...

In reality, there is plenty a math major can add to a conversation. Take the previous one, for instance. The math major could have mentioned Peter Pereira's "Fugue," a poem constructed using a Markov model built from several of ee cummings's poems. In a discussion with an art major, she could have also talked about how mathematicians can test the authenticity of paintings. There is more to mathematics than many believe.

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**What is it like being a mathematics major?**

When most juniors sweat their JPs, those in the math department have the alternative of a seminar, in which each student presents one or two lectures on a subtopic of their choice and write a short final paper summarizing their lectures. The primary goal of the seminars is to learn to teach. Most math majors take at least one.

The average math thesis is far shorter than the average history thesis. Math is dense stuff: a 30 page thesis, if it is mostly technical, is more than sufficient in length. Majors have even written introductory textbooks.

Math majors are known for their camaraderie. As a student in the department, you will face some very daunting problem sets. Most of the time, you will not be able to solve every problem on your own. Fear not! Great friendships come from slaving with your peers at the blackboards in the Fine hall common room — perhaps the ugliest and dingiest (though the most charming) part of the Princeton campus.

Professors, while they may seem aloof, are generally very accessible. For most of the day, there's at least one professor hanging around the Fine common room, and at the daily 3:30 tea time, you are bound to run into someone you know. If not, just help yourself to the free coffee, tea, and cookies.

"Math majors do the same math as physics majors, but without the interesting experiments and real-world applications."

This common misconception is fed by high school math classes, which often focus on what's used in physics. At Princeton, math courses range from a fuzzy boundary with physics (some geometry and analysis) to a fuzzy boundary with computer science (combinatorics) to topics whose applications are often undiscovered for centuries (algebra). Math majors do math for its own beauty, and leave the ugly calculations to physicists.

"Math is hyper-competitive."

Many math students enjoy competitions, but not all; (almost) everyone collaborates to learn in classes and figure out problem sets.

In fact, mathematics is fundamentally less competitive than some fields. In, say, philosophy, you can make a name for yourself by arguing with or tearing apart other peoples' work and arguments. In math, a proven theorem forever follows from its axioms; if other mathematicians return to work on it, they come to build on it, not argue. The only inherently competitive part of math is the race for more knowledge.

Many majors have participated in Research Experiences for Undergraduates (REUs). These summer programs give students the chance to work on math research problems in small teams, usually between 1 and 4 students. There are REUs at scores of universities across the country, and each sponsors projects in different fields of math. Equally many focus on pure math as applied math.

Princeton math majors have also taken internships with the National Security Agency. The Director's Summer Program and the Mathematics Summer Employment Program both take math majors to work on problems of national importance in cryptology and communications technology.

Each year, several majors intern at quantitative hedge funds. Several have gone overseas to study math in England (e.g. Cambridge's Math Tripos), Spain, and Eastern Europe (e.g. Budapest Semesters in Mathematics).

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Mathematics is in nearly everything you use. When you take a picture on your digital camera, your image is compressed using the Fourier transform — a way of breaking up a function into coefficient functions of sinusoids. When you make a call on your cell phone, the algebra of error correcting codes comes into play, allowing you to discern the voice on the other end of the call. Math is used to model everything from nuclear explosions to blood flow in the heart. Math majors may not be easily visible in government or in international affairs, but math is used to drive policy decisions: the NSA is the largest employer of mathematicians in the world.

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Math isn't easy; you need to be passionate about it to study it. If you date someone who can maintain that level of commitment, think about how rewarding your relationship could be.

On a more serious note, math majors also have great pickup lines: "Baby, you're sweeter than 3.14159265..."