# Math major and advanced courses FAQ

Welcome to the Math major FAQ!

- For information about placement exam and calculus, please see our Calculus FAQ.
- For assistance with choosing your first math course(s) at Yale, see our intro courses guide.
- For questions not addressed on these three sites, e-mail math.dus@yale.edu .

### Introductory courses

#### What courses should I take after integral calculus, if I’m considering a math major?

After integral calculus (such as Math 115 or Calculus BC), students wishing to pursue study of mathematics typically enroll in Math 120 or 225. For more guidance, please visit our intro courses guide, where you can find detailed notes in the section about choosing your first math class at Yale.

#### Do I have to complete Math 120 / 302 as soon as possible?

If you have not seen multivariable calculus before and need it for your major (e.g. all the combined Math majors), or for other courses you plan to take (e.g. in Economics or Physics), then you should consider taking Math 120 as soon as you can.

For the pure math major, you have a choice to complete either Math 120 or 302. Most students either complete Math 120 in their first year (before Math 225 and 255), or complete Math 225 and 255 first and then take 302 in their second or third year.

#### Can I substitute another course for Math 120, if I took multivariable calculus in high school?

Yale does not allow transfer credit for courses taken during high school, even if they were taken at a local college. Beyond that, the answer and its details depend on why you need Math 120:

1. **If you need it as a pre-requisite for another Yale course**, it should not be an issue: instructors typically accept a high school course in place of Math 120. If you’re not able to reach the instructor, contact their program’s DUS.

2. **If you need it for the pure math major: **You can complete the multivariable requirement with Math 302. This option is available to all students, whether or not they have already completed a course similar to Math 120.

3. **If you need it for a**** combined math major: ** You can request an evaluation of the course that you have completed. If it is equivalent to Math 120, then you will be able to substitute another course in the same area for Math 120, in requirements of the major. Please visit our intro courses guide, where you can find detailed notes about the evaluation process (including a list of materials we need) in the section about choosing your first math class at Yale.

4. **If you need it for another major,** please contact the major’s DUS to see what options there may be, as they are in charge of all requirements and exceptions for their major.

#### Can I substitute ENAS 151 for Math 120 in requirements of the math / joint-math major?

No (but if you passed ENAS 151 before deciding on the math major, you can complete the multivariable calculus requirement with Math 302).

#### Can I substitute another course for Math 222 / 225 /226 if I took linear algebra during high school?

Yale does not allow transfer credit for courses taken during high school, even if they were taken at a local college. Beyond that, the answer and its details depend on why you need linear algebra:

1. **If you need it as a pre-requisite for another Yale course**, it should not be an issue: instructors typically accept a high school course, if it is equivalent to the pre-requisite. If you’re not able to reach the instructor, contact their program’s DUS.

2. **If you need it for the pure math or combined math major: ** Please visit our intro courses guide, where you can find detailed notes in the section about choosing your first math class at Yale. Note that pure math, math+econ and math+CPSC require proof-based linear algebra: in all three majors, students who completed linear algebra in high school typically still enroll in Math 225 or 226.

3. **If you need it for another major,** please contact the major’s DUS to see what options there may be.

#### Can you tell me more about the test to place out of Math 225?

First, please note that the test does not award course credit or fulfill any requirements at Yale. It can only grant permission to substitute a higher level course in the same area for Math 225, in requirements of the math and joint math majors.

The test is available to incoming first-year students who learned the material of a fully proof-based linear algebra course (such as Math 225) prior to enrolling at Yale, but have not taken an actual class that can be evaluated for the same purpose. (See above for how to have a course evaluated.)

If you wish to take the test, you must let us (math.dus@yale.edu) know by August 15, during the summer before you first enroll at Yale. (The exam is not available to continuing students.)

We will offer the test in person, at the end of August, just before classes start (usually that Monday or Tuesday). We will set the time according to everyone’s availability: you will hear from us after August 15 about scheduling, if your name is on our list.

Students who pass will receive permission to substitute a higher level course in the same area for Math 225, in requirements of the math or combined math major (Math + CPSC, Econ, Phil, Phys)

If you need linear algebra for another major, please check in with their DUS - our process only works for math majors, we have no say in requirements of other majors (though we would be happy to assist, if the DUS wishes).

#### Can I substitute for other required courses?

As a rule, the answer is “no”. Courses required for the major must be completed. In rare cases where an incoming student has learned in high school the material and skills taught in a particular required course, the DUS may grant an option to substitute a higher level course in the same area. Please be sure to discuss this with the DUS prior to your first semester at Yale.

#### What is the difference between Math 222 and the proof-based linear algebra courses?

All three courses cover linear algebra. Math 222 focuses more on computational techniques and applications, while 225 and 226 emphasize mathematical proofs and a more conceptual approach.

Math 225 (linear algebra) or 226 (intensive linear algebra) is recommended for students who wish to take further proof-based mathematics courses. **Students majoring in mathematics, Math + CPSC or Math + Econ are required to complete either Math 225 or Math 226. **

In more detail: For math majors, and students who wish or need to continue with proof-based math courses, the theory and proof writing skills learned in Math 225 or 226 provide essential preparation for further study. That is why the math major does not accept Math 222.

Outside of the major: generally speaking, proofs are useful in many fields. They help one to become a better problem solver, evaluate different approaches to a question, consider options along the way, keep track of important details, check solutions for correctness, and present them in a way that is accessible to others. These skills are very helpful, in math and outside of it.

For that reason, many students find it beneficial to take Math 225 or 226, even if they do not continue with further math courses. If you try the course and enjoy it, the skills you learn will not be wasted.

If you enjoy applications more, and want to get lots of practice solving concrete problems, rather then focus on theory, then Math 222 is a better choice. It provides great preparation on the practical side of linear algebra, and it is preferred by the majority of students in engineering, social sciences, and many other fields.

Moving from Math 226 to Math 225 is permitted until Midterm. Moving from Math 226 or 225 to Math 222 is permitted until the end of the third week of classes (this has an earlier deadline because the courses are quite different, and it is too difficult to catch up later on). You can read more about how moving works in our Calculus FAQ.

#### I don’t want to be a math major but I want to know what mathematics is about in some deeper way. What should I take?

Not necessarily more calculus. There are several level 200 courses that may be of interest to you. For a current list, please visit our First year student resources site.

### Requirements of the major

#### Where can I find the details of math major requirements?

Our requirements are listed in the Yale College Programs of Study:

For detailed notes about choosing which courses to start with, and a path through the introductory sequences, please visit our first-year student resources site.

#### For a double major, how much overlap can there be in terms of courses?

At most two courses can be counted simultaneously toward both majors.

Extra note: Pre-requisites do not count toward the overlap. If your second major has Math 120 as a pre-requisite, for example, you can still use Math 120 to fulfill the multivariable calculus requirement in math, even if you have two other overlapping courses.

The rules get more complicated if you also want to earn a B.S./M.S. degree in one of your two majors. In this case, you are either allowed to overlap 1-2 courses between the two majors, xor 1-2 courses between your B.S. and M.S. in the same major, but not both. That is to say, one set of overlaps must be empty (even if the other only has one course in it).

#### Can a course taken Cr/D/F count toward the math major?

No.

#### Can I take a graduate course - and what number should I use, if it’s cross-listed?

To get the most out of math graduate classes, we recommend that students should first take at least three or four regular level 300 courses.

For most students, the most accessible graduate courses to start with are those cross-listed with both an undergraduate (level 300) number and a graduate (level 500) number (for example, Modern algebra is listed as Math 380 and Math 500). These course also carry attributes that can be counted toward the math major (whereas pure graduate courses do not, with the exception of Math 544).

Generally speaking, when taking a cross-listed course, undergraduates should use the undergraduate number, unless they are planning to apply the graduate version toward the graduate course requirement in the intensive major or the B.S. / M.S. degree.

If you completed a course with the “wrong” number: (a) If you enrolled with the undergraduate number but need the graduate one for your intensive major or M.S. degree, you will have to convert it (see the section on M.S. degrees below for more information). (b) If you enrolled with the graduate number but need the course to apply to undergraduate requirements, degree audit will not recognize it - e-mail Miki and we will sort it out.

#### Can I take a graduate course for Cr/D/F?

No. (This is a university-wide rule: Cr/D/F is only available for Yale College courses.)

#### Can a graduate math course count toward the undergraduate degree?

Yes, though for cross-listed courses, you should use the undergraduate number, if you wish to apply the course toward undergraduate requirements.

Note 1: If you are in the B.S. / M.S. program, there is a maximum overlap (see the B.S./M.S. degree section for details).

Note 2: As a rule, graduate courses do not carry attributes, either category or core area. The one and only exception is Math 544, which counts toward Geometry / Topology category.

Undergraduate courses that are cross-listed as graduate do carry attributes.

#### Can a course be simultaneously counted towards two different categories?

No.

#### Can a course be simultaneously counted towards a category and toward a core area?

Yes.

#### Can a course from another department be substituted as a general elective for a math course in the requirements?

Any course that has a Math number counts towards the requirements. If you wish to count it toward math, we recommend enrolling with the math number (e.g. Math 241, as opposed to S&DS 241). If you register with another number and later change your mind, and need to count the course toward math, please write to math.dus@yale.edu and we will help you sort it out.

About courses that do not carry a math number:

• Math majors can substitute up to two courses from another department, provided that the math DUS approves the selection.

• Joint math majors may not substitute courses from another department for the mathematics portion of their major requirements. The idea behind allowing substitutions is to allow pure math majors to explore applications of mathematics by taking up to two math-intensive courses in other departments, while still taking a minimum of eight math courses. For joint majors, the mathematics requirement is smaller, and cross-discipline exploration is already built into the program. That being so, the mathematics portion of the requirement must be satisfied with courses that have a math number. No exceptions will be granted.

Below is a list of courses that the DUS will normally approve for substitution towards the math (not joint math) major, and a list of courses that we do not currently approve. **Please note that these courses do not carry core area or category designations**, with the exception of PHIL 267 and 427 (logic).

If you have a question about a course that is not on the list, please e-mail the DUS, and we will evaluate it for this purpose. Please note that the evaluation is not instant; but we can usually do it within a couple of business days.

As a general rule, **graduate courses from other departments cannot be counted.**

Extra note: These lists are regularly revised, but some of the courses may not be offered anymore. We need to keep each discontinued course on the list for four years, so that it remains available to everyone who may have completed it during their time at Yale.

**Typically approved, for pure math (not joint math) majors:**

- CPSC 365 = Econ 365,
- CPSC 366 = ECON 366 = AMTH 366
- CPSC 440
- CPSC 468
- CPSC 486
- ECON 135 (cannot be counted in parallel with Math 241 or 242)
- ECON 136 (cannot be counted in parallel with Math 241 or 242)
- ECON 351
- LING 224
- PHIL 267
- PHIL 427
- S&DS 364 = AMTH 364 = EENG 454
- S&DS 410
- S&DS 431 = ECON 431 = AMTH 431
- S&DS 432

**Currently not approved: **

- AMTH 361, 364, 428, 553
- APHY 470
- CPSC: 201, 202, 447, 455, 467, 469, 481
- ECON 350, 361, 530, 531
- EENG 200, 202
- ENAS: 194
- OPRS: 235
- PHIL: 268
- PHYS 460
- S&DS: 230, 238, 240, 317, 363, 365, 411

#### What can be counted as the physical science course for the B.S. requirement?

If a course has a Math number, it cannot be counted towards this requirement (the requirement is meant to enhance the mathematics major with science courses that are outside of the B.A. portion of the program). The same applies to courses that do not have a math number, but are essentially mathematics courses.

Note that the courses on this list are not introductory, as the degree is meant to certify that students reached an advanced level in their physical science education. This is to say that if you wish to complete the B.S. degree, you should plan ahead for the science courses, as you will first need to complete their pre-requisites first. (Courses without physical science pre-requisites typically do notcount toward this requirement.)

Below are courses that are typically approved to satisfy this requirement. If you wish for us to evaluate a course that is not on either list, please write to the DUS.

As a general rule, graduate courses cannot be counted.

- ASTR 418, 430
- CHEM: 333, 470
- PHYS: 342, 344, 401 or 410 (but not both), 402 or 430 (but not both), 420, 440, 441

##### Currently NOT approved

- APHY 470
- ASTR 465
- CHEM: 328*, 332
- PHYS 341, 343, 345, 356, 412*, 428, 439, 448, 460
- All courses numbered strictly below 300
- All courses that are not physical science (e.g. we cannot count courses in life sciences, engineering, computer science, etc.)

*Chem 328 counts for everyone who completed it before Spring 2020.

*Phys 412 counts in semesters when it is a full credit course (not half credit).

#### How do I know which courses will be offered next year?

Next year’s schedule will be posted prior to April registration. Until then, it will not be possible for us to give a definite answer as to which courses will be offered.

The list of undergraduate courses usually does not change that much from year to year, but there are several courses we typically offer only ever other year, such as Math 353, 354, 373, 421, 430, 435, 440, 447. If they are offered in the current year, it is unlikely (though possible) that they will be offered again the following year.

Graduate courses (that are not cross-listed as undergraduate) vary from year to year. Our intention is to offer Math 526, 536, and 544 every year. Topic courses (= all courses level 600+) change every year.

#### How does Math 470 work?

Math 470 is independent study, intended for learning a full course worth of math topics that are not regularly offered. It is pass / fail, and cannot be counted toward the major, but it does receive graduation credit. (Please note the “not regularly offered” condition: it means that there cannot be a significant overlap with any course that we typically offer.)

To sign up, you will need to find a math faculty adviser who will supervise the study. It is recommended that you start looking during the previous semester, so that everything is ready to start at the beginning of term.

You and your supervisor should agree on a tentatively list of topics that will be covered, and how the class will work. Typically, this includes a weekly meeting, the student either writes or presents about what they have been learning and / or solutions to selected problems. The course usually ends with a project. The student is required to write a short paper (10 - 15 pages).

Before you can get permission to enroll, you will need to submit a proposal to the DUS. We recommend submitting it before the semester starts, or first day of class at the latest, to leave time for any adjustments, if needed. The final deadline is three days before Add / Drop ends.

The proposal should be 1-2 pages long, approved by your adviser, and contain the following:

- Your name, and your adviser’s name
- Tentative list of topics that will be covered
- Main book(s) or other sources that will be used
- Brief note about plan for the course (such as weekly meetings with homework / presentations, etc.)

Note that writing a paper at the end is required (but you can choose the topic later on).

If you have any questions, please don’t hesitate to contact the Math DUS at math.dus@yale.edu .

#### Can independent study (Math 470) be counted toward requirements of the major?

Independent study cannot be counted toward undergraduate requirements of the B.S. or B.A. major. A sufficiently advanced independent study (taken for credit) might receive permission to be substituted for a graduate course required for the intensive major or for the M.S. portion of the B.S. / M.S. combined program. This must be approved by the DUS prior to the beginning of the independent study.

#### Do I have to take Math 305? 350? 310?

The gateways to the core areas of mathematics are provided by these courses: Analysis 2 (305), Algebra (350) and Complex Analysis (310). They are beautiful subjects and every math major is strongly encouraged to take them and continue upwards with the sequences that they begin. All math majors are required to take courses in 2 out of 3 of the core areas (real analysis, complex analysis, algebra). Intensive majors must take all three.

#### Can I transfer courses from another university?

Technical note: Yale does not allow transfer credit for courses taken during high school (even college courses). You can only transfer college courses that you completed after graduating from high school.

Yale has a limit of two transfer courses from other universities (except for transfer students, for whom the rules differ).

Courses may be transferred for graduation credit, for university requirements such as QR, and / or for major credit. Your College Dean would be the best person to assist with transfer procedures.

#### Can transfer students count math courses taken outside Yale toward the math major?

This depends on the situation. Typically, we require at least half the courses toward the major to be taken at Yale, but the DUS must decide individually in each case.

To see which of your courses may count, please contact the DUS at math.dus@yale.edu . We will need a copy of your transcript (unofficial is fine), the syllabus and detailed list of topics for each course, as well as the final exam. (If exams were not returned, and the instructor wishes to keep the test confidential, a copy can be sent directly to math.dus@yale.edu .)

### Intensive major, B.S. / M.S. degree

#### For the intensive major, do the two graduate courses also count toward the required ten?

Yes. In other words, the total is still 10 math courses level 200+, same as the regular major, but two of the ten have to be graduate courses.

#### Do cross-listed courses count as graduate, for purpose of the intensive major?

Course cross-listed as both undergraduate and graduate can be counted toward the graduate portion of the requirement. For example, if you take Math 380 = Math 500, it can count as a graduate course.

Technical note: You should use the undergraduate number if you wish to apply the course toward undergraduate requirements, and the graduate number if you wish to apply the course toward the graduate portion of the degree.

If you enroll with the undergraduate number, and then decide to count the course as a graduate one, you must petition to have the number converted afterward. This is done via a form called “Simultaneous Degree Program Course Conversion Request”, which can be found on the Registrar form site. The form will require a signature from the instructor and from the math DGS.

#### Where can I find general information about combined B.S. / M.S. degrees?

Please note that the **minimum eligibility criteria in mathematics include at least 75% A/A- within the major as well as 75% A/A- overall**. It must be satisfied at the time of application, and remain satisfied after fifth-semester grades are included.

#### Where can I find the math B.S./M.S. degree requirements?

http://catalog.yale.edu/ycps/subjects-of-instruction/mathematics/

Extra note: In order to earn the combined B.S./M.S. degree in math, you must complete the requirements of a B.S. in math, rather than B.A. Earning a B.S. in another department (on account of double-majoring in CPSC, for example) will not count toward the math B.S./M.S. degree requirements.

#### Where can I find detailed information about the qualifying exam required for the M.S. degree?

One of the requirements of the M.S. degree is passing a written qualifying examination in algebra, analysis, or topology. Examinations in all three subjects are offered toward the end of the Fall semester.

You will need to choose one (and only one) of the three subjects, and sign up for the examination. If you do not pass, you may try one more time, in the same subject, the next time the examination is offered (typically, this would be the following Fall semester, though some subjects may occasionally be offered in the Spring as well).

Syllabus for each of the exam is posted at https://math.yale.edu/grad-programs/syllabi-qualifying-examinations .

For dates of the examinations, and to see copies of past papers, please see the Math Registrar in DL 438.

#### For the B.S. / M.S. combined degree, how many courses can be counted toward both?

At most two courses can be counted simultaneously toward the B.S. and the M.S. portions of the requirements. Extra note: if you are also doing a double major, you are only allowed one set of overlaps. For example, if you are double majoring and math and physics, and also doing the B.S. / M.S. in math, you can either (a) overlap one or two courses between math and phyics, or (b) overlap one or two courses between B.S. and M.S. in math, but not both - one of these overlaps must be empty.

#### What (and when) do I need to do to apply for the B.S. / M.S. combined degree?

The application is now online. It must be endorsed by the math department, both the DUS and the DGS. That being so, you must submit the form **at least two weeks before the last day of classees of your fifth semester at Yale. **You should then schedule a follow-up meeting with the DUS to talk about the program.

The application must reach the Dean’s office by the last day of classes of your fifth term. The Dean’s office does not accept late applications. Decisions are usually sent out about a month after the following semester starts.

At the beginning of your last semester, you must submit a petition for receiving a Master’s degree. This also has a firm deadline, typically about three weeks after classes start. The form can be found on the Registrar’s website, under forms and petitions, and it is called “Degree petition (en route and terminal)”.

Note that for this petition, you will need the M.S. courses to be listed on your transcript with graduate numbers (see below).

#### For the B.S. / M.S. combined degree, does it matter whether I have taken a particular course with the undergraduate or graduate number?

If a course has two numbers (such as Math 380 = Math 500), and you wish to count it toward the M.S. degree, you should enroll with its graduate number. In math, the coursework does not depend on the number, so if you want to count a course that you already completed with the undergraduate number, you can do it - but you must first submit a form called “Simultaneous Degree Program Course Conversion Request” to convert the number to the graduate one. The form can be found on the Registrar form site, and will require a signature from the instructor and from the math DGS.

### Senior requirement

#### What are the senior seminar options?** [Includes new option for 2024-25]**

A brief summary of the pure math options could be like this:

**Math 475, senior essay.** It is a semester-long project done independently, with a math faculty adviser. At the end, students write an essay, and give a 30-minute presentation. The essay must contain some original work; some students incorporate original research, but this is not required. An insightful expositions into existing results also works, so long as the essay provides something new (e.g. some applications of the topics, or thoughtfully constructed examples).

**Math 480 is the pure math senior seminar.** In the seminar, students explore topics in an area chosen by the instructor, they read papers, present within the seminar, and attend presentations of the other students. The number of presentations, and whether they are individual or group projects, depend on the instructor and on enrollment. Pre-requisites likewise depend on the instructor; for some sections, Math 225 + 255 are sufficient, some require level 300 course work. Topics of the senior seminar are typically announced below in March, for the upcoming year.

**Math 481**description is very similar to Math 480, except the topics are

**math + economics**. Pre-requisites are Math 225 + 255. It is offered every Spring.

**Math 482 is a new math education section,**with 225 + 255 as pre-requisites. The main focus will be on developing presentation and teaching skills. Students will also read pedagogy papers, discuss classroom case studies, there will be sessions on active learning, inclusive teaching, public speaking, etc. It will be offered for the first time in Fall 2024 - during the pilot run, enrollment will be limited to 12 students, selected through an application process during April registration.

#### What are the choices for math, Econ + Math, and Math + Philosophy? **[NEW for Fall 2024]**

**Pure math major **allows all of the above options to count toward the senior requirement. The course will count toward the required number of math courses level 200+. Math majors may only enroll in these courses for the purpose of fulfilling the senior requirement.

**Math + Economics major **allows Math 480 or Math 481. Students are only permitted to enroll in one of these if they are using it for the purpose of fulfilling the senior requirement.

**NEW** As of Fall 2024, **Math + Economics majors** can also complete their senior requirement with an Econ theory seminar - these will carry the designation “YC ECON: Theory Seminar” on YCS.

Technical note 1: The major requires seven economics and five mathematics courses. If the senior requirement is completed with Math 480 or Math 481, this will also count as one of the five math courses. If the senior requirement is completed with an Econ theory seminar, this will also count as one of the seven Economics courses.

**Math + Philosophy major **allows Math 480, Math 481, Math 482, or a senior seminar in Philosophy approved for the student by the Philosophy DUS prior to the start of term.

#### What option should I choose toward the pure math major?

For description of the options, see the first question in this section.

All of the options develop skills that are valued by graduate schools, and that can be useful outside of Academia as well. The best choice to make is whichever one you find most interesting.

**Math 480 and 481 **run as a combination of a reading project and guided exploration experience, with presentations to other students in the seminar. Either is a great option for students who prefer a structured program where a mentor and reading projects to work on are already identified. It requires no research experience, and is accessible to any senior majoring in mathematics.

**Math 475 **requires more independent work. To enroll, the student must find a faculty adviser to sponsor the project. It is recommended to start looking for an adviser at least a semester in advance. Typically, the student will be expected to have an idea of a project they wish to work on, or an area of interest that is aligned with the adviser’s field of study, and allows the adviser to help the student identify a suitable problem to work on.

It is recommended that students who wish to complete the senior essay should get a bit of research or independent study experience prior to senior year. Some options include a summer REU, our own summer program SUMRY, applying for the Lewis summer research award, the Yale math directed reading program run by graduate students every semester, independent study through Math 470, or working on an unofficial project with a faculty member. These will help you get the experience and research project ideas you will need to successfully complete a senior essay.

**Math 482 **is a pilot in Fall 2024, focused on mathematics education at the college level, and includes a lot of practice with teaching. It is a good option for students who are interested in education and would like to develop or sharpen their teaching skills.

#### What are the senior seminar topics for the upcoming year?

Topics for 2024-25 can be found below.

**Math 480 Fall 2024 topics, Prof. Richard Kenyon: Discrete geometry**

Discrete geometry deals with a number of subjects including tiling and packing problems, polyhedra, rigidity, scissors congruence, triangulations. Standard tools are combinatorics, linear algebra, group theory, and some Euclidean geometry.

Pre-requisites: Math 225 + 255

**Math 482 Fall 2024 topics, Miki**

Math 482 focuses on math education - see description above. For the pilot year, enrollment will be limited to 12 students. Application will be available on the Canvas course site starting in mid-March, and due by April 24.

Pre-requisites: Math 225 + 255

**Math 480 Spring 2025 topics, Prof. Ebru Toprak**

- Fourier analysis on R: as a linear map Fourier transform, convergence of Fourier series, Hilbert Transform, Singular integrals
- Abstract Fourier analysis: Fourier analysis on finite/compact groups, representation and reducibility
- As a PDE tool: solutions of some PDE’s
- Harmonic analysis (for more advanced students): there will be some projects such as kakeya sets

Pre-requisites: Math 225 + 255

**Math 481 Spring 2025, Prof. Andrea Ottolini (Economics) and Prof. Pei-Chu Su (Math)**

Optimal transport is a powerful mathematical theory at the interface between optimization and probability theory with far reaching applications. It defines a natural tool to study probability distributions in the many situations where they appear: data science, partial differential equations, statistics and economics. In this course we will read and present the classical theory of optimal transport, efficient algorithms to compute it and economic applications such as matching, optimal pricing, auctions and others.

Pre-requisites: Math 225 + 255

#### How does Math 475 work?

0. You will need a project to work on, keeping in mind that the final essay must contain some original work. This could be a new result, an extension of an old result, an application of an old result to some new mathematics, or a new proof of something known. An insightful exposition of existing results also works, provided that you explore some aspects of the topic on your own, and add some elements to what is already available (thoughtfully constructed examples, for instance).

1. You need to find a math faculty adviser who will sponsor the essay. It is recommended that you start looking during the previous semester, so that everything is ready to start at the beginning of term (some students even start their project a semester early, though you can only enroll in Math 475 once, for the semester when you wish to complete the essay). You may also have a secondary adviser in another field, if relevant.

2. At least three days before the end of add / drop period, you need to send a brief project proposal to math.dus@yale.edu . Submitting the proposal earlier is recommended. Please be sure to discuss the proposal with your project adviser, as their approval will be essential - and to list your adviser’s name on the proposal.

The proposal should be 1-2 pages long, approved by your adviser, and contain the following:

- Your name, and your adviser’s name
- Background reading and topics that you plan to learn
- Topics you plan to investigate for the project
- Books, papers, and / or other sources you anticipate using

3. During the semester, you will be working on your project, and meeting with your adviser for an hour every week. During each meeting, you will need to present to the adviser the work that you have done the previous week. (Presenting your work is one of the skills that Math 475 helps to (further) develop, and it helps the adviser to guide your project, suggest resources to look at, etc.)

4. By the last day of classes, you will need to submit your essay, with length of 20-25 double space pages, to your adviser, with cc to the Math Registrar. The essay will be shared with the committee (and must be shared with them at least a week before your presentation takes place).

(If your essay is turning out to be longer than 25 pages, you could move some parts to an appendix, for example background information that does not contain your own work, or detailed notes from calculations. The total length, counting this appendix, should not exceed 50 pages.)

5. Two weeks before the end of term, you should work with your math adviser to schedule your final presentation, to take place at the end of term (typically during reading or finals week). The presentation is given to a committee of two mathematics faculty, one of whom is your project adviser. The other committee member is selected by your adviser and the DUS’es. The presentation itself is 30 minutes long, followed by up to 30 minutes of questions and conversation with the committee.

6.. The grade for Math 475 is based on (a) Your work during the semester (as reported by your adviser), (b) your paper submission at the end, (c) your presentation and Q&A with the committee.

For the talk, you can use any format you wish: you can give a blackboard talk, or use slides (beamer, powerpoint, etc.), or a combination of these. Recently, most presentations have used slides to some extent, because the time is relatively short, and the audience doesn’t have to take detailed notes. But your format is up to you.

##### tips for using slides:

- The typical purpose of slides is to supplement your presentation. They do not have to deliver the entire detailed content (nor would the audience be able to read it that fast).
- The recommended slide rate is about 1 per minute.
- Present only one idea per slide.
- Keep the slide relatively sparse, with main phrases / bullet points only, so your audience has a chance to absorb it real time while listening to your presentation. (Slides full of text are impossible for the audience to read, and so they are rarely helpful.)
- As noted above, it’s good to try your presentation on someone who doesn’t know your project, and see if they are able to follow the ideas and absorb the slides with your current content and rate of going through them.

#### If I plan to spend all year working on my senior essay, when should I enroll in Math 475?

You cannot enroll in Math 475 twice. You should enroll during the semester that you are planning to complete your essay and your oral presentation (otherwise you will need to apply for a temporary incomplete, and finish your work the following semester).

#### Is any information available about past Math 475 projects?

Here are descriptions of a few sample projects, posted with permission from the students:

- Sonali Durham ‘20: Nesting rules for political redistricting: methods for sampling matchings and tripling.
- Sanelma Heinonen ‘20: Discrete convexity with applications to maximizing user satisfaction in bike sharing systems.
- Noah Kravitz ‘20: Lonely runner conjecture, and Noah’s advice for writing a research-based senior essay.
- Claire Thompson ‘19: Binary irreducible quadratic-residue codes and the Good code problem.

#### How does senior project work for computer science + mathematics, and mathematics + physics majors? **[New version, from Fall 2024 onward]**

Computer science + mathematics majors can enroll either in MATH 475 or in CPSC 490. Mathematics + physics majors must enroll in PHYS 471 or 472. For joint majors, the project must be on a topic acceptable to both departments.

For students in the joint major, the additional requirement for their chosen course is that the project proposal must include a specific description of how the projects uses topics from both majors, and it must be approved by DUS’es in both departments early in the semester. The deadline in CPSC 490 is typically on the third Thursday of classes. For students in Phys 471 or 472, we set the same deadline for receiving the proposal in math. You can send the proposal to math.dus@yale.edu .

As for Fall 2024, students are no longer required to give an oral presentation to the Math department on the mathematical aspect of their project.

#### How does senior project work for computer science + mathematics, and mathematics + physics majors? **[Version for Spring 2024]**

0. E-mail math.dus@yale.edu at the beginning of the semester that you will be completing your thesis. This is to get on Miki’s list of students who need to give an oral presentation in math at the end of term, and to have a math reader assigned (or confirmed, if you have found one yourself).

1. Computer science + mathematics majors typically enroll in CPSC 490, and mathematics + physics majors enroll in PHYS 471 or 472, in order to complete their senior project. For joint majors, the project must be on a topic acceptable to both departments; in particular, it must have sufficient mathematical content to satisfy the mathematics portion of the senior requirement.

2. The project proposal must contain an explicit description of the mathematical component, and it must be approved by both departments early in the semester. The deadline in CPSC 490 is typically on the fourth Thursday of classes. For students in Phys 471 or 472, we set the same deadline for receiving the proposal in math. You can send the proposal to math.dus@yale.edu .

When you submit your proposal, please be sure to list the name of your project adviser, as well as your math reader (see point 0. above).

3. At the end of the semester, you will need to give an oral presentation on the mathematical aspects of your project, to a committee of two mathematics faculty chosen by the DUS. One of the two faculty will typically be your math consultant.

The presentation will be 15 minutes long, followed by 10 minutes of questions and conversation with the committee. A passing grade from the committee will satisfy the mathematics portion of the senior requirement.

The presentation has a prescribed format: you will have five minutes to introduce the physics / computer science background of your project, and then **you must spend ten minutes explicitly discussing the mathematics you have learned and used during your project**. The committee’s evaluation will be based entirely on the mathematics portion.

You may invite up to ten guests to your presentation (friends, your project adviser, etc.)

*Presentations are typically scheduled for Monday of reading week.*

For the talk, you can use any format you wish: you can give a blackboard talk, or use slides (beamer, powerpoint, etc.), or a combination of these. Recently, most presentations have used slides to some extent, because the time is relatively short, and the audience doesn’t have to take detailed notes. But your format is up to you.

##### General tips:

- Keep in mind that audience members are typically not familiar with the setting of your project. Your presentation should be understandable to them.
- As noted above, you must spend the majority of the time discussing the mathematics in your project, because that is what the committee will grade. Even the best project will not pass if there is insufficient mathematical component to your work as you present it.
- Try your presentation on someone who is not familiar with your project, to see if they can follow everything.
- Be sure to do a dry run of your presentation in advance, just the same as you plan to give it on presentation day. Your math reader would be happy to serve as your audience for the practice run. This is partly to get feedback from the adviser, and partly to make sure that your presentation fits into 15 minutes. (During the practice run, most students find that the actual presentation time is far longer than they thought it would be.)

##### Tips for using slides:

- The typical purpose of slides is to supplement your presentation. They do not have to deliver the entire detailed content (nor would the audience be able to read it).
- The recommended slide rate is about 1 per minute. (It could be a little bit more if each slide present only a very small amount of information.)
- Present only one idea per slide.
- Keep the slide relatively sparse, with main phrases / bullet points only, so your audience has a chance to absorb it real time while listening to your presentation. (Slides full of text are impossible for the audience to read, and so they are rarely helpful.)
- As noted above, it’s good to try your presentation on someone who doesn’t know your project, and see if they are able to follow the ideas and absorb the slides with your current content and rate of going through them.

#### Are joint senior projects / theses allowed?

No. Senior essay must be independent, you cannot do a joint project with others. Your essay may fit into a bigger project that other people are working on, but you must perform and write up work that you have done yourself.

#### How does calculation for distinction in the major work?

Nomination for distinction in any major requires 1. Grade of A or A- on senior project / essay, and 2. Grades of A / A- in three quarters of courses within the major,

For distinction in Mathematics, students are also required to complete at least one course in each of the three core areas.

For distinction in the Econ + Math major, students must also write the optional essay in Economics, with a grade of A or A-.

Note that the percentage-of-A’s calculation is based on all courses you have taken in the major, as well as all courses that routinely count toward the major. In other words, if you have taken more than the necessary number of courses, we are required to count them all, rather than selecting the ones with best grades.

### Other questions

#### Degree audit is not displaying my courses correctly, what should I do?

Here are some of the common situations with degree audit:

1. For some of the requirements, it only recognizes undergraduate course numbers.

Solution: E-mail math.dus@yale.edu .

2. Sometimes, degree audit will display half-completed categories as empty. (E.g. if you took exactly one course in “logic and set theory”, it might not show up until you complete the whole category.)

Solution: I [Miki] don’t know a way to fix this, but be assured that if you took a course with a category attribute, it does carry the attribute, whether degree audit displays it or not. To see which courses carry attributes in a particular semester, you can search for attributes using Yale Course Search (or open a particular course there, and see what attributes it displays).

3. Degree audit will sometimes list in-progress courses for requirements that are already completed. (E.g. if you have already completed the algebra core area with Math 350, and then you enroll in Math 370, degree audit might list the core area as “in progress” with Math 370, instead of listing it as “done” with Math 350.)

Solution: According to the Registrar, there isn’t any way to make degree audit prioritize completed courses. If you’d like confirmation as to the requirements your existing courses satisfy, e-mail math.dus@yale.edu , we’d be happy to check for you.

General note: These peculiarities may be a bit confusing. If you have questions about what degree audit says, or encounter some issue with it that isn’t mentioned above, e-mail math.dus@yale.edu to ask about it.

#### How does advising work in math?

The default system in STEM is that the DUS advises all the majors. In our case, that is typically well over 200 math (including joint-math) majors.

To make sure that all math majors have a chance for meaningful conversations with their advisers, we assign individual faculty advisers to everyone. The main benefit is that each adviser has only 15-20 students, as opposed to 200.

Most of our advisers are dedicated to a single major (e.g. pure math, or Econ + Math).

**Extra note: **The DUS will continue to be available to everyone. Our office hours are posted on the math DUS site. You should still see us with technical questions about requirements of the major, for transfer credit or other types of exceptions, and for anything else that you wish or need, for example if you have trouble reaching your designated adviser for any reason. We are here to assist you, don’t hesitate to come see us :)

Timing: Adviser lottery is usually run before the start of each semester; that is, late December / early January, and early August. Assignments are given to all students who are newly on the math major list at that time.

Extra note: First year advising is done through the colleges, and we do not assign faculty advisers until just before the second year. In the meantime, you are very welcome to consult the DUS about the mathematics major - we will be happy to assist.

#### Is there an unofficial math mailing list?

We do have an unofficial mailing list, created for students who are interested in getting messages from the math DUS (about events, job openings within and outside of the department, summer programs, news from the department, and other similar information). All students are welcome to sign up, whether or not they are majoring (or thinking about majoring) in math. The link is

https://subscribe.yale.edu/browse?search=math+dus ,

and you can use it to subscribe or unsubscribe any time you wish. Be sure to use the “Yale NetID login” (guest e-mails are not permitted for this list).

Official math major information is sent to all current majors, independently of the DUS newslist. The newslist was partly created for reaching students who are interested in news from us but are majoring in something else or have not declared a major yet. Partly because we try to keep the official list for essential information only, and avoid sending to everyone unofficial messages that many math majors might not be interested in receiving. The unofficial list is there so you can sign up to receive them if you like :)

(Extra note: Only undergraduate students will receive messages that are sent to the DUS newslist.)

#### Can I do research with math department faculty?

Undergraduate research and independent study opportunities do exist, depending on your interest and that of available faculty. During the summer the department organizes REU opportunities for interested students, and runs its own summer research program called SUMRY. The department also awards the John Alan Lewis prize each spring, which provides a stipend for independent work during the summer.

During the academic year, our graduate students organize a directed reading program, where students can explore topics and work on projects with a graduate student mentor. Applications to the DRP are typically due at the beginning of each semester.

We also offer Math 470, independent study, which allows for one opportunity to do a more extensive directed reading project with a faculty member. More information about Math 470 is available above, under “How does Math 470 work?”

Students can also approach a faculty member to ask about mentoring an informal research project during the academic year. This tends to be more independent work, on a project that’s proposed by the student, and students usually select it after they have some research experience from a more structured program such as a summer REU or the Yale DRP. If you have a project in mind, with or without prior research experience, don’t hesitate to ask!

#### Does Yale have a summer research program?

Our summer research program is called SUMRY, and you can find its website at https://sumry.yale.edu/ . We also award the John Alan Lewis prize each spring, which provides a stipend for independent work during the summer.

#### Is there an undergraduate math organization?

The **Yale Undergraduate Math Society** (YUMS) hosts a number of events for undergraduate students, including colloquia, study sessions, game nights, and panels about the math major and summer opportunities. Check out https://yums.sites.yale.edu for more information. You can also take a look at their recent seminar recordings, and visit our contact information site to sign up for their weekly newsletter.

**Dimensions** seeks to inspire, celebrate and empower women in mathematics at Yale. To help facilitate a community of Yale women in math, Dimensions offers a mentorship program, pairing graduate students, upper and lower class people based off interests, as well as workshops, speaker events and organized meetings with professors. We hope to create an encouraging space for women and other gendered minorities to pursue their interests in the under represented fields of mathematics. All students are welcome to join Dimensions as well as attend all events sponsored by Dimensions. Please visit us at our website or our Istagram page.

#### What jobs are available in the Math department?

There are four regular positions that we have available:

1.** Undergraduate learning assistants (ULA’s). **Currently, we have ULA’s in Math 110 / 111, 225, 226, 255, 256, 305, 350, 370. The position is for a fixed salary at 112.5 hours per semester (which translates to approximately 8.5 hours per class week). ULA’s typically run sessions with prepared content, as well as walk-in study groups. They may also help with grading quizzes or exams to some extent, or perform other duties. As of Spring 2024, the pay is $2,081 per semester.

2. **Peer tutors.** These positions are available in the larger courses numbered between 112 and 244. Peer tutors hold four walk-in office hours a week, in blocks of two hours at a time. The position is for maximum of 6 hours a week (this includes time for prepararation, any extra time spent in the sessions, and time for a small administrative component that takes about 15 minutes a week). As of Spring 2024, the pay is $18.50 / hr.

Our **applications for peer tutors ULA’s **are typically due before Spring break, for the following academic year. When the application opens, we send an e-mail to “Math DUS news” - if you would like to sign up for the list, the link is listed in the “unofficial mailing list” question earlier in this FAQ.

We rarely have open positions in the middle of the year (that is, for the Spring term); if it happens, a note will be sent to the DUS newslits.

Tutors who are hired will go through **training sessions** just before the beginning of the Fall semester (one session is with the department, for ULA’s there is a second training session with the CTL).

**A few rules to mention:**

(a) Yale does not permit first-year students to work as peer tutors or ULA’s (but you can apply during the Spring of your first year, for Fall positions).

(b) Only students who have already completed at least one math course at Yale are eligible to apply. That is, you may apply in March only if you have completed a Yale math class the previous Fall or earlier.

(c) The following coursework is required, by the time the job is to start:

- Tutoring Math 110/111 requires completion of Math 115 or higher.
- Tutoring Math 112 or 115 requires completion of Math 120.
- Tutoring Math 116 or 121 requires completion of those specific courses (as they have specialized curriculum that is not taught in other math classes).
- Tutoring Math 118 requires completion of Math 120 and linear algebra (Math 222/225/226).
- Whenever feasible, tutoring for courses numbered 120 and above requires completion of at least one course beyond the one being tutored, preferably in the same area. For example, with Math 225, preference is given to students who have completed Math 240 or 350. For tutoring Math 255, we prefer at least one level 300 analysis course. For tutoring Math 350, we ask for Math 370. (These are needed by the time the job starts; e.g. if you are enrolled in Math 370 when applications open in the Spring, you can apply for the Math 350 position to start in the Fall.)

3.** Graders.** These positions are available in nearly all math courses.

Applications for Fall 2024 are now open - the deadline to apply is May 12, at this link.

Math needs fewer graders in the Spring, so most of the Spring positions are filled with continuing graders, and there are rarely any openings in the middle of the year.

Grading positions are for a fixed salary at 75 hours a semester. In most classes, this translates to approximately 6.5 - 7.5 hours per problem set. As of Spring 2024, the pay is $1,125 per semester.

For more information, please reach out to Meghan Anderson at first.last@yale.edu .

Rules: Every student in their second year or above is eligible to grade for any math course that they have completed (at Yale or elsewhere).