The Honors Track

The Honors Track is the suggested track for students who plan to pursue math-heavy graduate study (especially a Ph.D. in mathematics). Completing the Honors Track will grant a Bachelor of Science degree and is deemed credit-intensive (meaning that SAS students who successfully complete the track do not need a minor or second major to graduate); this is unlike the standard option, which awards a Bachelor of Arts and is not credit-intensive.

Note that it is possible to graduate with Honors in Mathematics without completing the full Honors Track by completing two honors credits (typically, each course from 411/412/451/452/501/502/503/551/552 completed with a grade of B or higher counts as an honors credit, though alternate ways of obtaining honors credits do exist). For more details, see the Math Department’s page on Graduation with Honors in Mathematics.

Core courses for the Honors Track

The core of the honors track lies in the honors analysis sequence 411–412 (Mathematical Analysis I–II) and the honors algebra sequence 451–452 (Abstract Algebra I–II). Foundational coursework in analysis and algebra at this level is an essential as preparation for graduate study, providing students with a necessary background to approach many graduate-level topics. The usual prerequisites to getting permission for these courses are as follows:

411: 311 with a grade of A, 311H preferable (B+ possibly allowed);
451: 350 with a grade of A, 350H preferable (B+ possibly allowed); OR 351 with a grade of A;
412 and 452: a grade of B or higher in the previous course (an SPN is not necessary for registration in this case).

The first courses in each sequence always run in the fall semester, and are followed by the second course in the sequence running in the spring semester.

The seminar requirement

There are three seminar courses: 196 (freshman-sophomore seminar), 491 (Putnam seminar), and 492 (junior-senior seminar). The honors track requires two seminars, at least one of which must be 492. However, graduate courses have been used as a substitute for seminars (including 492) in previous years. Check with the honors advisor to see if graduate courses will satisfy the honors track requirement before planning on using them towards requirements.

What to take

In addition to these four core courses and seminars or substitutes, your major in mathematics will consist of other courses.

Courses before the 400-level honors sequences

At first, you will be taking calculus courses in the usual sequence (from calculus 1 to differential equations), starting from whichever course you placed into. If this lines up with multivariable calculus in the fall and differential equations in the spring, then consider the 291–292 sequence (Honors Calculus III and IV). These courses are entirely different from 251(H) and 252, giving a taste of basic proof-writing and some more abstract topics you will see in later courses combined with multivariable calculus and differential equations material. If you have to wait a semester to take 291 or 292, then move on to 251 or 252 instead of waiting—scheduling while doing the honors track is very sensitive to these delays, so the most important thing to prioritize (after your grades) is progressing through courses as fast as you comfortably can!

Additionally, you will need to satisfy the introductory linear algebra requirement. You can satisfy this requirement by getting a grade of A in 291 or 292, or, with a B/B+ in these courses, can satisfy it by proficiency examination. Without these courses or a linear algebra course which did not transfer for credit, it is not likely that you will be able to successfully petition the department to let you take the proficiency exam, so it is safer to take the class. Satisfying this requirement early on is essential, as Math 250 is an important prerequisite; moreover, for students who did not place into at least multivariable calculus in their first semester, taking Math 250 early on unlocks Math 300.

Math 300, the starting place for writing proofs in your courses, is the next essential prerequisite to the core of the honors track; taking it early is preferable, as the prerequisite chains for honors track scheduling can be quite lengthy (especially if your goal is to take certain graduate courses). The prerequisite for the course is at least one of 250, 251, and 291. The honors section of 300 may be preferred, but is not required—good performance in any section of 300 is what matters.

To get to the 400-level analysis sequence 411–412, most students first take 311 and have a good performance (typically at the level of an A) in that course to get approval. An honors section is not necessary, but may help in the event of borderline performance. There is also precedent for getting approval for 411 without 311, but the process for this has varied depending on the Honors Advisor. The prerequisites for 311 and 300 and differential equations, but some people have been able to waive the latter requirement for 311 (in cases where they already have background in proofs, especially from other courses).

To get to the 400-level algebra sequence 451–452, good performance in 350 (honors section again not necessary but possibly helpful) or 351 is needed; most students choose 350. A previous Honors Advisor suggested that students take 350H over 351 for a pre-451 course (see their recommendation below), though with new honors advisors things are subject to change. Additionally, for students intending to attend graduate school, a proof-based linear algebra course is typically expected, which is another reason why 350 may be a better choice. The prerequisites for 350 and 351 are 250 and 300.

Former honors advisor’s recommendation: 350H or 351?

It is MUCH better to do 350H. Really, 351-352 is designed for people who don’t intend to do 451-452.

Here is the reason: At the foundations of modern algebra, there are four fundamental abstract structures: groups, rings, fields, and vector spaces. You have familiarity with examples of all: if you ignore multiplication, the integers are the fundamental example of an infinite group (and the integers mod 12, i.e. clock arithmetic, are the fundamental example of a finite group). The integers WITH the full multiplicative structure as well are the fundamental example of a ring; the rational numbers are the fundamental example of a field, and Euclidean space of any dimension is the fundamental example of a vector space. When you strip everything else away and isolate the key items that run these structures, you have the four I listed in the abstract above, and there are countless different example of them, and the key it to understand the fundamental properties that follow JUST from the definition of such a structure. And in ALL four cases, the key items are,

what are the maps between such spaces that preserve all of some of the structure (the morphisms)

how do you build more complicated examples out of simple examples

how do you find substructures that help you understand the main structure

when can you relate substructures to particular useful forms

In 350H we treat vector spaces in the abstract - and unlike 250, the idea is not to consider just finite dimensional vector spaces over the reals and then have a key method of computing answers to the questions above, using matrices and Gaussian elimination. Instead, is general vector spaces (e.g. over the reals or complex numbers of another field) and all of what can be done independent of the particular choice of a basis (which in the finite dimensional case corresponds to choosing coordinates in R^n to represent your space). Linear algebra in this sense is used in a fundamental way in virtually all areas of mathematics (analysis, geometry/topology, etc.) not just algebra. And this is the top level course in which we do the material at the undergrad level.

And that honors approach to it gets you ready perfectly for 451-452, because it is the same pattern of questions and approached, but now for groups, rings and fields. Much better to do the linear algebra as preparation (because it is SO important in its own right) than to see the stuff on groups at a relatively pedestrian level and then do it again in 451.

So that’s my recommendation!

Other undergraduate electives

Some other undergraduate courses that honors track students may find useful, especially if they intend on applying to graduate school in mathematics, are the following:

Probability (477). While not focused on pure math like the other courses in the honors track and typically not having a comparable difficulty to them, this is a common course that many students applying to graduate school have taken.
Topology I (441). This is a class that looks great for graduate school applicants, and is frequently taken alongside honors classes (especially 411, which has some related content). Arguably the most important class for graduate school applications outside of the honors track core. Note that this course only runs in fall semesters.
Topology II (442). This class follows 441, but is not as important for a generic applicant. It has a particular focus on algebraic topology. It may not be offered every year due to low interest. Students interested in algebraic topology may instead opt for the graduate algebraic topology courses 540–541. Note that this course only runs in spring semesters.
Complex Variables (403). This course introduces some of the methods of complex analysis, which is also helpful to have on your transcript before applying to graduate school. The prerequisite for this course is only differential equations, so it may not be as proof-based as the honors track courses. For this reason, a number of honors track students opt for the graduate complex analysis course 503. Note that this course only runs in spring semesters.

Graduate courses

Some of the most common mathematics graduate courses taken by undergraduate students are the courses which prepare students for the PhD qualifying exams at Rutgers (all of which run in fall semesters):

Real Analysis/Measure Theory I (501), usually taken after 411–412.
Complex Analysis (503), usually taken after 411–412.
Abstract Algebra I (551), usually taken after 451–452.

These courses follow the 400-level honors sequences. Because of how they are situated as a bridge from undergraduate to graduate level topics, these courses tend to have heavy workloads.

Some other graduate courses also rely on an honors sequence for prerequisites, while others (such as the ones in logic, combinatorics, and applied math) tend to have more flexible prerequisites.

Declaring the Honors Track

Contrary to what you may read on outdated pages, there is no longer a requirement to submit an application to the Honors Track; all that is needed is to register for the usual major and take honors courses (apply for permission to take these through the Honors SPN process).

Instead of receiving applications for the Honors Track, the department will keep track of students who complete honors coursework, and should approve the conferral of the B.S. degree to any student who completes all Honors Track requirements. If you are unsure if you are going to receive a B.S. or B.A. in Mathematics, you can contact the Honors Advisor.

Sample schedules

In this section, some examples on how to progress through the honors track are provided. These sample schedules will assume no summer classes and no skipping out of classes (such as 311), but these may be possible options to accelerate your studies and have more time for advanced courses.

The sample schedules here assume a four-year timeline. Only math courses are included; other required courses or courses taken for other reasons can be placed wherever there is space available.

Do not feel pressured to match these samples exactly! Find a pace which works for you.

Sample schedule 1 (starting in multivariable calculus)

First year. Fall: 291; Spring: 292, 300, possibly an elective (in this case, 477).
- You can also do 250+251 first semester, then 252 or 292 in the second semester.
- You can also drop down from 291 to 252; take 250 with 252 after getting a prerequisite override if you did not get proficiency credit from 291.
Second year. 311 and 350 spread across the year, possibly with a few math electives spread across the year as well.
- It is unlikely that you will be able to take graduate courses at this point, though it may be possible.
Third year. Fall: 411, 451; Spring: 412, 452; possibly a few math electives spread across the year as well.
- 411, 451, and a third math course such as 441 (and likewise in the spring semester) could be a heavy load, so if one of these sequences seems less necessary for you, then you may find it worthwhile to delay it by a year. However, this is a common combination of courses, it is possible to find the necessary time to dedicate to each of them.
- You may be able to start taking some graduate courses this year, particularly in the spring (when you will have the first course from one or both honors sequences completed).
Fourth year. At this stage, you have flexibility to take courses of your choosing, some of which may reflect a specialization, or take more “standard” courses (501, 503, 551). Ensure that you will satisfy the seminar requirement, either by taking the seminar classes (if you did not do so before) or by taking graduate courses.

Sample schedule 2 (starting in Calculus II)

First year. Fall: 152, 250; Spring: 251, 300.
- You can also move off 250 to the spring semester, in which case you take 300 in the following fall semester.
Second year. Fall: 252, 350; Spring: 311; possibly with a few math electives spread across the year as well.
- If you take 300 in the fall, then you will do 311 and 350 together in the spring.
Third and fourth years. Similar to the above sample.

Sample schedule 3 (starting in Calculus I)

First year. Fall: 151; Spring: 152; 250 during one of these semesters.
- If you take 250 in the fall, then you can take 300 in the spring.
Second year. Fall: 291, 300; Spring: 292, 350; possibly an elective or two spread across the year.
- If you take 300 in the previous year, then 350 can be taken in the fall semester. With that background, it may be possible to get a prerequisite override and take 311 in the spring of this year.
Third year. Fall: 451; Spring: 452; 311 sometime this year and possibly with a few math electives spread across the year as well.
- You may be able to take 411–412 this year by (a) getting approval to take 311 in the previous year, (b) taking 311 during the summer before, or (c) getting approval to skip the 311 prerequisite, which may be possible with the background from the previous year in this plan.
Fourth year. Fall: 411; Spring: 412 (if 411–412 not yet completed); you may have some flexibility to take some courses reflecting a specialization, or take more “standard” courses (551). Ensure that you will satisfy the seminar requirement, either by taking the seminar classes (if you did not do so before) or by taking graduate courses.

Sample schedule 4 (a real example from the page’s writer)

This schedule is a bit accelerated, and reflects an interest in analysis. Note that this is what worked for one person, and there is no need for readers to try something similar.

First year. Fall: 291; Spring: 196, 292, 300H, 477.
- 196 is the freshman-sophomore seminar.
Second year. Fall: 411, 454; Spring: 350H, 412.
Third year. Fall: 451, 501, 503, 573; Spring: 452, 502.
- 573, Special Topics in Number Theory, is no longer offered. Grading-wise, it was an “automatic A,” so the workload was essentially zero outside of attending class. When I took it, the course topic was Automorphic Forms and L-functions.
- I regret not taking one more graduate course in the spring semester.
Fourth year. Fall: 441, 507, 517; Spring: 508, 518.
- 507–508 is the functional analysis sequence, and 517–518 is the partial differential equations sequence.