**Introductory and advanced courses**

- What courses should I take after integral calculus, if I’m considering a math major?
- Can I skip Math 120, if I took multivariable calculus in high school?
- Can I skip Math 230+231?
- What is the difference between Math 120 and Math 250?
- What is the difference between Math 222 and Math 225?
- What is the difference between Math 230 and Math 250?
- If I enroll in Math 230 and find that it is not the right class for me, can I switch into another course?
- Can I take Math 301 or 300 without Math 231 or 250?
- Can I take Math 310 without Math 301?
- Can I take Math 310 without Math 250/231?
- 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?
- Is there a good math course that fulfills the QR requirements even if I haven’t taken any calculus?

**Requirements of the major**

- Where can I find the details of math major requirements?
- Can a course taken Cr/D/F count toware the math major?
- Can a course be simultaneously counted towards two different categories?
- Can a course be simultaneously counted towards a category and toward a core area?
- Can a course from another department be substituted as a general elective for a math course in the requirements?
- What can be counted as the physical science course for the B.S. requirement?
- Can independent study (Math 470) be counted toward requirements of the major?
- Do I have to take Math 301? 350? 310?
- Can transfer students count math courses taken outside Yale (which are accepted for transfer credit) toward the math major?

**Double major, B.S. / M.S. degree**

- For a double major, how much overlap can there be in terms of courses?
- Where can I find the math B.S./M.S. degree requirements?
- Where can I find technical information about combined B.S./M.S. degrees?
- For the B.S./M.S. combined degree, does it matter whether I have take a particular course with the undergraduate or graduate number?
- What do I need to do to apply for the B.S./M.S. combined degree?
- For the B.S./M.S. combined degree, does it matter whether I have taken a particular course with the undergraduate or graduate number?

**Senior requirement**

- Are all sections of 480 the same? How do I decide which to take?

New: topics of Math 480 for 2019-20 have been added. - Do Math 475 (senior essay) and Math 480 (senior seminar) count as math electives?
- How does Math 475 work?
- If I plan to spend all year working on my senior essay, when should I enroll in Math 475?
- How does senior project work for computer science + mathematics, and mathematics + physics majors?

**Other questions**

- How does advising work in math?
- Can I do research with math department faculty?
- Can I take graduate courses?
- Is there an undergraduate math organization?
- What jobs are available in the Math department?

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

There are two main entry sequences into the major. The first is Math 230+231, a two-term course with an integrated treatment of linear algebra and vector calculus. This is a demanding but rewarding course for students with a very strong interest and background. The second is Math 120+225+250, covering multivariable calculus, linear algebra with proofs, and vector calculus (generalizing calculus to n-dimensions, with proofs). It is possible to replace Math 225 with Math 222 (linear algebra with applications), but it is not recommended for students who wish to continue with proof-based courses, such as Math 250.

**Can I skip Math 120, if I took multivariable calculus in high school? **

If you are considering a math major, you should speak with the math DUS. If they agree that you are ready to take classes beyond multivariable calculus, they can give you permission to substitute a higher level course for Math 120 in requirements of the major (note that this course will then not count towards the required ten courses). If you are considering another major, you must speak with the DUS of that major to ask whether something similar may be possible. If you are interested in taking proof-based math courses, you may also consider taking Math 230+231, as an alternative to calculus and linear algebra. Most students benefit from taking this sequence, even if they have already seen multivariable calculus.

Math 230+231 is a very sophisticated, proof-based course. It is extremely rare that an incoming student cannot benefit from taking it. If you believe that you have all the skills and knowledge that is taught in Math 230+231, please speak with the mathematics DUS.

**What is the difference between Math 120 and Math 250? **

Math 120 is “multivariable calculus” and Math 250 is “vector calculus”. What is the difference? Math 120 focuses on situations where “multi” really means 2 or 3 – that means it leans heavily on geometric intuition, and is strongly grounded in geometric applications in engineering and the sciences. Math 250 generalizes this to the setting of n dimensions, or variables, where n can be 3, 10, or 450,000. In order to do this, linear algebra must be brought in as a tool to manage the computations and to clarify what is conceptually going on – that’s why Math 222 or 225 is a prerequisite to Math 250. The theorems of Green, Gauss and Stokes, the beautiful integral equalities that are the capstone of Math 120, are generalized in Math 250 to a single master theorem (still called Stokes’ theorem) which applies in any dimension. Don’t think that 450,000 dimensions means the subject is not practical: any large system, such as the economy or the weather, can easily involve that many variables.

**What is the difference between Math 222 and Math 225? **

Both courses cover linear algebra, but Math 222 focuses more on computational techniques and applications, while Math 225 emphasizes mathematical proofs and a more conceptual approach. Math 225 is recommended for students who wish to take further proof-based mathematics courses (in particular, all math majors).

**What is the difference between Math 230 and Math 250? **

Math 230+231 is a two-term course with an integrated treatment of linear algebra and vector calculus. This is a demanding but rewarding course for students with a very strong interest and background. Math 250 is the third semester of the sequence 120-225-250 (or 120-222-250), and it covers just the vector calculus.

If it is before Midterm, you can switch from Math 230 into Math 120. You must speak with instructors of both courses to ensure a smooth transition, find out what work you will need to make up in Math 120 etc. Please see the calculus FAQ for further information. If you took multivariable calculus in high school, you may also be able to switch into Math 222 or 225. If your intended / potential major requires Math 120, you will first need to speak with the DUS of that major to see if it is possible to satisfy the requirement in another way (for example by substituting a higher level course).

**Can I take Math 301 or 300 without Math 231 or 250?**** **

This is not recommended, but if your previous background is sufficiently strong you may try to obtain permission from the instructor.

**Can I take Math 310 without Math 301? **

Yes.

**Can I take Math 310 without Math 250/231? **

This is not a good idea unless your background is quite strong.

Not necessarily more calculus. There are several of courses that either do not require or do not emphasize calculus, and offer a window into mathematical thinking, for example: • Math 244, Discrete Math: (Math 115 recommended) the structure of finite sets, graphs, trees, and enumeration. • Math 270, Set Theory: (Math 120 is a prerequisite) The foundational underpinnings of mathematics, through the study of infinite sets.

**Is there a good math course that fulfills the QR requirements even if I haven’t taken any calculus? **

Try Math 107 or Math 108.

**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:

- Math:

http://catalog.yale.edu/ycps/subjects-of-instruction/mathematics/ - Math and Computer science:

http://catalog.yale.edu/ycps/subjects-of-instruction/computer-science-mathematics/ - Math and Economics:

http://catalog.yale.edu/ycps/subjects-of-instruction/economics-mathematics/ - Math and Philosophy:

http://catalog.yale.edu/ycps/subjects-of-instruction/mathematics-philosophy/ - Math and Physics:

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

A roadmap for the math major can be found on the major roadmaps site:

https://registrar.yale.edu/students/major-roadmaps

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

No.

**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.

Any course that has a Math number counts towards the requirement (whether you register for it with the math number or not).

• 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. For joint majors, this is already built into the program, and the mathematics portion of the requirement must be satisfied with courses in the math department.

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 one of 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.

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

- AMTH: 437, 561
- CPSC 267, 365, 366, 427, 440, 468, 662
- ECON 135 (as an alternative to Math 241, we will not count both), same for ECON 136
- ECON 351
- PHIL 267, 427
- S&DS 351, 364

**Currently not approved: **

- CPSC: 201*, 202, 467*
- ECON 531
- EENG 200, 202
- ENAS: 194
- OPRS: 235
- PHIL: 268
- S&DS: 230, 238,

**Special courses that sometimes count:**

- AMTH 364 from Spring 2019 counts for pure math, and for math+cs majors
- CPSC 640 from Spring 2019 counts for pure math, and for math+cs major
- S&DS 669 from Spring 2019 counts for pure math, and for math+cs major

[There are, occasionally, other courses without a math number that we would count for joint majors as well, given the content from a particular semester; feel free to ask the Math DUS about a class that you think might count.]

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

*Another extra note: The DUS of CPSC currently allows S&DS 365 to replace the requirement of taking one of CPSC 440, 462, 465, 468, 469.

**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 (since the requirement is meant to enhance the mathematics major with science courses that are outside of the B.A. portion of the program).

Below are courses that are typically approved to satisfy this requirement. If you wish to find out about a course that is not on the list, please write to the DUS.

- BENG: 475, 476
- CHEM: 328, 333, 470
- PHYS: 342, 344, 401, 402, 410, 420, 428, 430, 440, 441

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

Independent study cannot be counted toward 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.

**Do I have to take Math 301? 350? 310? **

The gateways to the core areas of mathematics are provided by these courses: Real Analysis (301), 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. Intensive majors must take all three.

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.

**Double major, B.S. / M.S. degree**

**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.

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

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

**Where can I find technical information about combined B.S. / M.S. degrees?**

**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.

**What do I need to do to apply for the B.S. / M.S. combined degree? **

Proposals are due at the Dean’s office by the last day of your fifth term at Yale. The proposal is meant to show that you are close to finishing the B.S. requirements, and to provide a tentative plan for completing the M.S. requirements. As of October 2018, there is also a form to fill out, available online at:

https://yalecollege.yale.edu/deans-office/deans-office-forms

The relevant past, current and future courses are to be listed on the form (rather than being listed within the proposal).

You have to enroll with the undergraduate number, until you are accepted into the combined program. After that, you will have to petition to have the relevant courses converted to the graduate version. The petition is called ”Simultaneous Degree Program Course Conversion Request” and it is available on the Registrar’s site https://registrar.yale.edu/forms-petitions .

**Senior requirement**

**Are all sections of 480 the same? How do I decide which to take? **

Every year, one section of the senior seminar, 480, is dedicated to topics of interest to Econ/Math majors. The others are usually on a more pure-math subject, and the precise plan is worked out between the instructor and the participants in the beginning of the semester. Contact the instructor for details, or check the department web page.

Note that all seniors are welcome to choose any section that they like; the section dedicated to Econ+Math topics is open to pure math majors, and likewise Econ+Math majors are free to select one of the other sections if they wish.

NEW: For the academic year 2019-20, the topics will be:

- Fall 2019, taught by Professor Van Vu: probabilistic combinatorics.
*Our main source of material is the book “The probabilistic method” by Alon and Spencer.*

The main content of the course is: probabilistic methods to prove existence of delicate objects, randomized algorithms, and random discrete structures (such as random graphs and random matrices).

Beside the book above, we will also add several research papers and monographs. - Spring 2020, section taught by Professor Van Vu: Applications of harmonic analysis.
*This section will focus on the applications of harmonic analysis in number theory and combinatorics.* - Spring 2020, section taught by Dr. Ofir Lindenbaum (this will be the designated section of special interest to Economics + Mathematics majors).
*This seminar will span themes that are of interest to both economists and mathematicians. The themes center around social networks. Here, we will cover the underlying theory of networks and game theory and examine a variety of applications.**Presentations are mainly based on the textbook:**“Networks, Crowds, and Markets: Reasoning About a Highly Connected World” (2010, Cambridge University Press) by D. Easley and J. Kleinberg, which is available online for free.**We will supplement it by other books and articles as needed.*

List of chapters:*Graph Theory and Social Networks**Game Theory**Markets and Strategic Interaction in Networks**Information Networks and the World Wide Web**Network Dynamics: Population Models**Network Dynamics: Structural Models**Institutions and Aggregate Behavior*

**Do Math 475 (senior essay) and Math 480 (senior seminar) count as math electives? **

Yes, though you may only count one of those.

1. You need to find a 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).

2. Once enrolled, a month before the end of term you need to e-mail the DUS in order to schedule your final presentation, to take place at the end of term.

3. The presentation is given to a committee of two mathematics faculty, who are chosen by the DUS. The presentation itself is 30 minutes long, followed by questions and conversation with the committee. The committee’s evaluation will contribute to your grade for the course.

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

You should enroll during the semester that you are planning to complete your essay and your oral presentation (if you enroll during the Fall and do not complete until the Spring, then you would receive a failing grade for the course in the Fall).

**How does senior project work for computer science + mathematics, and mathematics + physics majors?**

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 must be pre-approved by BOTH departments. You must obtain an explicit approval of the mathematics DUS early in the semester, before you submit your project proposal for the course in computer science / physics.

3. A month before the end of the semester, you need to e-mail the mathematics DUS in order to schedule an oral presentation to a committee of two mathematics faculty, who are chosen by the DUS. The presentation will take place at the end of term, and must focus on the mathematical aspects of your senior project. It 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.

**Other questions**

**How does advising work in math?**

When you declare a math major, the DUS (of both departments, for joint majors) automatically become your official advisers. We encourage you to declare early (it is not binding) so that we know you are interested in the major, can assist you with your progress, contact you with relevant information (job opportunities, summer programs, new courses, etc.)

Before each semester begins, adviser lottery is run for all newly declared majors. Thorugh the lottery, an additional adviser is assigned to you - an experienced faculty member who will be available to you until graduation. In particular, they will meet with you at the beginning of term to talk about courses and to sign your schedule.

This additional-adviser-system was put in place so that each adviser would have a smaller group of advisees, which gives more opportunity for individual conversations about classes, summer programs, applying for graduate school, or anything else you wish to discuss.

Note that the advisers do not necessarily have full knowledge of every detail of math major requirements, and are not meant to - with such questions, students are still meant to see the DUS.

Our office hours are posted on the math DUS site. As noted above, you should still see us with technical questions about requirements of the major, and you are welcome to see us with other questions as well, any time that you wish - or if you cannot reach your adviser, or if you are interested in math courses but haven’t declared math as a major, or declared recently and have no adviser assigned through the lottery yet, etc. We are here to assist you, don’t hesitate to come see us :)

**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 stipends for independent work during the summer. Contact the DUS, or check the website for more information.

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, 381, 320, 325).

**Is there an undergraduate math organization? **

The Yale Undergraduate Math Society 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.

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.

**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, 230, and 350. In Math 110, ULA’s run weekly workshops for groups of students, with content prepared by the instructor. In Math 230, they run weekly study group sessions. In Math 350, each ULA runs a weekly proof writing workshop with prepared content, and a weekly study group session. The position is for a maximum of 10 hours a week.

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.

Our main hiring process for both of the above positions typically takes place around Spring break, for the following academic year. When the application opens, we send an e-mail to all math and joint math majors, and we post a notice here, and on the Math DUS site, at

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

3. Graders. These positions are available in nearly all math courses. Application for Fall positions is open right now, at

https://yalesurvey.ca1.qualtrics.com/jfe/form/SV_ewByvEHi26smYMR

These positions are for a maximum of 10 hours a week. For more information, send a message to John Hall at first.last@yale.edu .

4. Private tutors. Students who need additional assistance with a course can request a private tutor, who meets with them once or twice a week for the rest of the semester, at a time that they arrange with each other. The application for calculus private tutoring positions is posted on SEO before the start of every semester. For more information, send a message to John Hall at first.last@yale.edu . There is no set number of hours for these positions, it depends on how many students request a tutor, and how many students a particular tutor wants to work with.

Private tutors for courses level 200 and above are hired through the CTL. You can find more information about the position, and an application form, at

https://poorvucenter.yale.edu/tutoring/quantitative-reasoning-science/small-group-and-1-1-tutoring .

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