Published on *Department of Mathematics* (https://math.yale.edu)

For information about placement exam and calculus, please see our Calculus FAQ [1].

Notes for continuing students about curriculum revision [2], and how to transition to the new system (starting in Fall 2021).

For questions not addressed in the FAQ, e-mail math.dus@yale.edu [3] .

After integral calculus (such as Math 115 or Calculus BC), students wishing to pursue study of mathematics typically enroll in Math 225 (linear algebra and introduction to proofs), and Math 255 (real analysis and introduction to proofs). Math 225 and Math 255 can be taken in either order, though it is recommended to take Math 225 first.

Most students complete multivariable calculus before enrolling in Math 225, however, prospective mathematics majors and students with interest in abstract mathematics may consider enrolling in Math 225 directly after Math 115 or equivalent, and complete their vector analysis/multivariable calculus requirement with Math 302.

Students with a strong mathematical background that includes exposure to mathematical proofs are encouraged to consider the intensive version of the introductory sequence, Math 226 and Math 256.

No, that is not necessary. Multivariable calculus / analysis is required for the major, but you do not have to take it right away. Once you complete Math 225/226 and Math 255/256, you will have several options among the level 300 courses, including real and complex analysis, and abstract algebra. Some of the more advanced courses do require Math 302, but you do not have to rush to take it.

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**, please contact the instructor, to see if they would accept your high school course instead. If you’re not able to reach the instructor, contact their program’s DUS. (This is usually not a problem, but you should ask, to be safe.)

2. **If you need it for the pure math major**, you may instead complete the multivariable requirement with Math 302.

3. **If you need it for a joint math major**, you can either complete the requirement with Math 302, or contact the Math DUS to see if there are other options for courses you could use in place of Math 120.

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

No.

If you passed ENAS 151 before deciding on the math major, you may complete the multivariable calculus requirement with Math 302.

If you passed ENAS 151 before deciding on a joint-math major, please contact the DUS to see what the options may be.

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**, please contact the instructor, to see if they would accept your high school course instead. If you’re not able to reach the instructor, contact their program’s DUS.

2. **If you need it for the pure math major: ** The major requires proof-based linear algebra that provides extensive introduction to proof writing, and will not accept a course equivalent to Math 222. If you have taken a course similar to Math 222, you may enjoy the extra challenge in Math 226, or you can complete the requirement with Math 225.

If you have had a full proof-based course on linear algebra, we would encourage you to consult with the Math DUS. Upon evaluating the course you have taken, we can grant permission for you to substitute a higher level algebra course for Math 225 in requirements of the major, and you can enroll directly in Math 255 or 256.

To evaluate the course, we will need a copy of your transcript, the syllabus (with detailed list of topics covered), and a copy of the final exam (or a sample final exam for the course). If the instructor wishes to keep the test secret, it can be sent directly to the math DUS.

If you did not take an official course in proof-based linear algebra, but have learned it another way, please consult the Math DUS during the summer before you start at Yale. If you pass an examination before your first semester (typically offered in late August), you will get permission to substitute a higher level course for Math 225.

3. **If you need it for a joint-math major: **Please consult the math DUS - we will evaluate the course and may grant you permission to substitute a higher level course in the requirements. See the notes just above for what the evaluation needs.

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

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.

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. **Math majors are required to complete either Math 225 or Math 226. **

In more detail: For math majors, and students wishing 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.

The Registrar permits moving from Math 226 to Math 225 or Math 222, and from Math 225 to Math 222, until Midterm. You can read more about how moving works in our Calculus FAQ [4].

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 [5].

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

- Math [6]
- Math and Computer science [7]
- Math and Economics [8]
- Math and Philosophy [9]
- Math and Physics [10]

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.

No.

If you have the pre-requisites for a graduate course, you may certainly enroll. You should consider starting with courses that are 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). The cross-listed courses tend to be more accessible for undergraduates, and they carry attributes (which other graduate courses typically do not - see below).

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.

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

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.

No.

Yes.

Any course that has a Math number counts towards the requirement, whether you register for it with the math number or not. Note that degree audit may not recognize the course under a non-math number, but you can contact the Registrar or the Math DUS and they can add it to the math requirements for you.

• 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.

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.

- AMTH: 431, 437, 561
- CPSC 267, 365, 366, 440, 460, 468, 562, 662
- ECON 135, 136 (neither of these can be counted in parallel with Math 241 or 242)
- ECON 351
- PHIL 267, 427
- S&DS 364, 410, 432, 631

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

*Extra note: CPSC 201 and 467 count for everyone who took them before or during Fall 2018.

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.

- ASTR 418, 430
- CHEM: 333, 470
- PHYS: 342, 344, 401, 402, 410,420, 430, 440, 441

- 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).

Independent study cannot be counted toward undergraduate requirements of the B.S. or B.A. major. A sufficiently advanced independent study (taken for credit) can, in some cases and with DUS permission, 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.

The gateways to the core areas of mathematics are provided by these courses: Multivariable analysis (302), 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.

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. They 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.

For counting a transfer course toward the math major, the course has to be reasonably equivalent to an existing Yale course, as far as topics taught and level of rigor.

For the final approval, after the course is completed, the DUS will need the syllabus, detailed list of topics taught in the course, and copies of the exams. (The blank test is enough, if the graded exams are not returned.)

Students are strongly advised to consult the DUS before enrolling in the course, with all materials available (including a list of topics, and a sample final exam). That way, the DUS can provide an estimate as to whether the course is likely to count or not.

Generally speaking, the higher the Yale course number, the more strict the evaluation. In particular, it is rare that we would accept a level 300 core class from another university, outside of official Yale study abroad programs such as the Budapest semesters is mathematics or the Exeter programme.

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.

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.

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, you can 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 [11]. The form will require a signature from the instructor and from the math DGS.

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**. This is a higher percentage than the two third required by the university in general.

http://catalog.yale.edu/ycps/subjects-of-instruction/mathematics/ [6]

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.

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 [13] .

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

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.

The application is now online [14]. 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, and no late applications are accepted.

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 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 [11], and will require a signature from the instructor and from the math DGS.

Starting in 2021-22, the section dedicated to topics of interest to Econ+Math majors has been renamed to Math 481.It will continue to run during the Spring semester. Math 480 sections typically focus on more pure-math topics, and there is one in each semester.

Both Math 480 and 481 will satisfy the senior requirement in math as well as in math + econ. Enrollment in both is limited to 25 students. Preference in Math 480 will be given to pure math majors, and in Math 481 to math + econ majors; but all seniors can sign up for the remaining spots.

The topic of the seminar will be chosen from the following two, depending on the interests of the class: “Groups in Mathematics”, or “Number Theory”.

———————–

I plan to give a brief review of history, general facts, and various applications of the theory to different areas of Mathematics. Students will be able to choose a topic related to their interests and knowledge, and prepare 1-3 lectures (depending on the size of the class) under my supervision. I will provide the students with the literature as well as with my personal guidance during preparation of their lectures. Below are a few examples of topics:

———————–

1. Groups in Mathematics:

Since their introduction in the XIXth century, groups play increasingly important role in all areas of Mathematics: algebra, geometry, analysis. Group theory provides a unified picture of seemingly isolated concepts of Mathematics. It also has fundamental significance in Physics.

1) Regular solids and subgroups of SO(3),

2) Symmetric functions and permutation groups,

3) Representations of SU(2) and the notion of spin,

4) Representations of SU(3) and classification of elementary particles,

5) Reflection groups,

6) Hyperbolic geometry in dimensions 2 and 3 and their symmetry groups,

7) Space-time and Lorentz group,

8) Harmonic oscillator and Heisenberg group,

9) Quaternions and octonions and their symmetry groups,

10) Codes and lattices and their symmetry groups,

11) Fourier series and their analogues on 2 and 3 dimensional spheres,

12) Problems of linear algebra and representations of quivers,

13) Haar measure.

———————–

2. Number Theory:

Number theory is one of the oldest areas of Mathematics. It is still an important field of modern research, with connections to Applied Mathematics and Theoretical Physics. In the senior seminar, we will study some famous achievements in analytic, algebraic and geometric number theory.

1) Modular group,

2) Modular functions,

3) Theta-functions,

4) Riemann zeta-function,

5) Eisenstein series,

6) P-adic numbers, adeles,

7) Tate’s approach to zeta-functions,

8) Roots of unity, abelian extensions,

9) Elliptic curves,

10) Fermat’s theorem, descend method,

11) Fermat’s theorem and elliptic curves.

————————

Plus) Any other topic within the chosen general theme that students would like to propose. (Since essentially all results in pure mathematics and mathematical physics are related to certain types of groups or appear in number theory, in one way or another, students are welcome to suggest their favorite topics).

These are samples of possible topics, and we’ll certainly not be able to cover all of them, but just a few, and not necessarily from this list. Each topic can be developed into a sequence of lectures by several students.

[Pre-requisites: Linear algebra (Math 222, 225, or 226), real analysis (Math 255, 256, 300, 301, or 305). Having seen one or more math courses level 200+ (in any area) will be helpful.]

The topic for this section of MATH 480 is “The Laplacian and its Eigenfunctions.” You have likely heard the multivariable calculus definition of the Laplacian as the divergence of the gradient. What you may be less familiar with is that some form of the Laplacian is present in many different areas of mathematics and that a lot can be learned from studying its eigenfunctions. In this seminar we will cover a few of these areas and the connections between them. (Fun fact: 2 and 3 below are equivalent, in a sense.) My hope is that each student will be able to find at least one area particularly interesting (and relevant to their background), explaining it (and the necessary background) to their classmates.

Specifically, the main topics will be:

- Fourier series, Fourier transform, connections with PDE and complex analysis (hopefully of interest to those with a background in analysis)
- The graph Laplacian: graph partitioning, etc. (graph theory and applied math)
- The Laplace-Beltrami operator on manifolds, spherical harmonics (Riemannian geometry)
- The Laplacian on domains of different shapes, can you hear the shape of a drum? (partial differential equations)

I will emphasize that I do not expect you to have a background in all of these subjects, or even any of them. I just list them to help you determine your interest. We will go over the necessary background as a class and I will help you find something you can present given what you already know.

Yes, though you may only count one of those.

Both the seminar and the senior essay will fulfill the senior requirement.

Math 480 and 481 runs as a combination of a reading project and guided research experience, with presentations to other students in the seminar. It is a great option for students who prefer a structured program where a mentor and projects to work on are already identified. It requires no prior 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 experience prior to senior year. Some options include a summer REU, our own summer program SUMRY, 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.

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 (both) the DUS (up to one page long). Please be sure to list your project adviser’s name. [We recommend sending it even earlier, in case some adjustments need to be made.]

3. By the last day of classes, you will need to submit a 20 - 25 page double-spaced summary of your work, to your adviser, with cc to the Math Registrar.

4. 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 presentation itself is 30 minutes long, followed by questions and conversation with the committee.

5. 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.

Extra note: some of the projectors in LOM are ancient, and you may need an adapter to make it work with your laptop. We strongly adviser you to test your setup at least a day ahead of time, so that you can be sure everything will work on the day of your presentation.

- 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.

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).

Here are descriptions of a few sample projects, posted with permission from the students: (This part is being prepared and the links will be available soon.)

- Sonali Durham ‘20: Nesting rules for political redistricting: methods for sampling matchings and tripling [15].
- Sanelma Heinonen ‘20: Discrete convexity with applications to maximizing user satisfaction in bike sharing systems [16].
- Noah Kravitz ‘20: Lonely runner conjecture [17], and Noah’s advice [18] for writing a research-based senior essay.
- Claire Thompson ‘19: Binary irreducible quadratic-residue codes and the Good code problem [19].

0. E-mail Miki at the beginning of the semester that you will be completing your thesis. (This is to get on my list of students who need to give an oral presentation in math at the end of term.)

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. As of Spring 2021, the deadline in CPSC 490 is noon on the fourth Thursday of classes. For students in Phys 471 or 472, we set the same deadline for receiving the proposal in math.

When you submit your proposal, please be sure to list the name of your project adviser, as well as your consultant in the math department (if you don’t have one, typically your assigned math faculty adviser will take on this role).

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.

Extra note 1 (Spring 2022): You may invite guests to your presentation (friends, your project adviser, etc.) Given the current restrictions, there is a limit of five guests at most, they must all be members of the Yale community. During the presentation, everyone must follow current classroom guidelines.

Extra note 2: the projectors in LOM are ancient, and you may need an adapter to make it work with your laptop. We strongly encourage you to test your setup at least a day ahead of time (this is particularly important because on presentation day, there will be talks back to back, and you will not be able to get into the room in advance).

*Presentations are typically scheduled for Monday of reading week. In Spring 2022, this date falls on Monday, May 2.*

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.

- 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 in your presentation.
- Try your presentation on someone who is not familiar with your project, to see if they can follow everything.

- 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.

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.

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.

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 [3] .

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 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).

(Extra note: Attributes have to be added to the courses by hand every semester, and it’s possible that one was missed. If you see a course that normally carries an attribute, but it doesn’t have it during one semester, e-mail the Math Registrar to see if that might have been in error.)

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: As far as I know, there isn’t one, other than to keep track yourself. If you’d like confirmation as to the requirements your existing courses satisfy, e-mail math.dus@yale.edu [3] , 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 solved by the above, e-mail math.dus@yale.edu [3] to ask about it.

**Old system**: The default system in STEM is that the DUS advises all the majors. In our case, that is nearly 200 math (including joint-math) majors.

**New system: **Math and joint-math majors are assigned a faculty adviser, through a lottery. We changed to this system so that students would have more opportunities for personal conversations with their adviser (as the advising groups are 15-20 students, rather than 200).

Within this system, each adviser is focused on a particular group of majors:

- Professor Moncrief is the designated adviser for Math + Phys.
- Professors Coifman and Gilbert advise Math + CPSC majors.
- Professors Jones, Schlag, and Wang advise Math + Econ majors.
- Professor Loseu advises Math + Phil majors.
- Professors Frenkel, Goncharov, Neitzke, Oh, Schotland advise pure math majors (the intent was to do it by class year, but many class years changed recently due to leaves, so it’s now more-less random).
- All other advisers work with students across several majors at the moment, and we will transition most of them to a designated role in August 2021.

**Extra note: **The DUS will continue to be available to everyone. Our office hours are posted on the math DUS site [20]. 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 very happy to assist.

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 [21] ,

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 have not declared a math major yet, or are majoring in something else. 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 :)

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 [22] each spring, which provides a stipend for independent work during the summer.

During the academic year, our graduate students organize a directed reading program [23], where students can explore topics and work on projects with a graduate student mentor. Applications to the program are typically due at the beginning of each semester.

Our summer research program is called SUMRY, and you can find its website at https://sumry.yale.edu/ [24] .

Yes, if you have the appropriate background. You should obtain the approval of the instructor and of the Director of Graduate Studies, and make sure you have agreed ahead of time on what you will be required to do and how you will be graded. This is not necessary for those graduate courses that are cross-listed with undergraduate courses (315, 380, 320, 325).

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 [25] for more information. You can also take a look at their recent seminar recordings [26].

**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 are welcome to public events sponsored by Dimensions. To subscribe to our panlist and stay updated on events, please email Dimensionsatyale@gmail.com [27].

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 maximum of 10 hours a 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. They meet weekly with the instructor.

2. **Peer tutors.** These positions are available in the larger courses numbered between 112 and 370. 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).

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.

Tutors who are hired will go through two **training sessions** just before the beginning of the Fall semester (one session is with the department, one is with the CTL).

As of the academic year 2021-22, the pay for ULA and peer tutor positions is $18.50 / hr.

**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 next year’s positions).

(b) Only students who have already completed at least one math course numbered 120 or above are eligible to apply. (In particular, students currently enrolled in Math 120 are not eligible to tutor Math 112 or 115.) The only exception is the ULA position in Math 110-111, which may be held by students who completed integral calculus.

(c) Where possible, we require that tutors complete (by the time the job starts) at least one course above the one they are tutoring. For example, tutoring single-variable calculus requires Math 120, the ULA position in Math 350 requires completion of Math 370, etc.

3.** Graders.** These positions are available in nearly all math courses. Application is usually available on SEO before the start of each semester. These positions are for a maximum of 10 hours a week. For more information, send a message to Ian Adelstein at first.last@yale.edu [28] .

Rules: Every student in their second year or above is eligible to grade any math course that they have completed.

**Links:**

[1] https://math.yale.edu/calculus-and-placement-exam-faq

[2] https://math.yale.edu/curriculum-revision

[3] mailto:math.dus@yale.edu

[4] https://math.yale.edu/undergraduate/calculus-and-placement-exam-faq#switch

[5] https://math.yale.edu/undergraduate/first-year-student-resources

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

[7] http://catalog.yale.edu/ycps/subjects-of-instruction/computer-science-mathematics/

[8] http://catalog.yale.edu/ycps/subjects-of-instruction/economics-mathematics/

[9] http://catalog.yale.edu/ycps/subjects-of-instruction/mathematics-philosophy/

[10] http://catalog.yale.edu/ycps/subjects-of-instruction/mathematics-physics/

[11] https://registrar.yale.edu/forms-petitions

[12] http://catalog.yale.edu/ycps/academic-regulations/special-arrangements/#Simultaneous_Bach_and_Masters

[13] https://math.yale.edu/grad-programs/syllabi-qualifying-examinations

[14] https://forms.sis.yale.edu/url/YCSimDegree

[15] https://gauss.math.yale.edu/~cav7/dept-pdfs/sonali_durham.pdf

[16] https://gauss.math.yale.edu/~cav7/dept-pdfs/sanelma_heinonen.pdf

[17] https://gauss.math.yale.edu/~cav7/dept-pdfs/noah_kravitz.pdf

[18] https://gauss.math.yale.edu/~cav7/dept-pdfs/noah_tips.pdf

[19] https://gauss.math.yale.edu/~cav7/dept-pdfs/claire_thompson.pdf

[20] https://math.yale.edu/director-undergraduate-studies#overlay-context=undergrad-programs

[21] https://subscribe.yale.edu/browse?search=math+dus

[22] https://math.yale.edu/undergraduate/competitions-and-prizes

[23] https://sites.google.com/yale.edu/yaledrp/project-suggestions?authuser=0

[24] https://sumry.yale.edu/

[25] https://yums.sites.yale.edu

[26] https://drive.google.com/drive/u/1/folders/1OazElzTAjlesznH3bY6PiKrArDEVw3fV

[27] mailto:Dimensionsatyale@gmail.com

[28] mailto:first.last@yale.edu