Choosing a topic and a supervisor

The two decisions that most affect an honors project are made in its first few weeks: what to work on, and with whom. Students tend to agonize over the first and neglect the second, but experienced supervisors will tell you that the match between student and supervisor predicts the outcome at least as well as the topic does. This lesson treats both decisions systematically. It describes where good questions come from, how to test whether a question is the right size, how to approach potential supervisors, and how to turn a choice into a plan with milestones. It ends with Amara's own decision and a timeline for her year.

Where good topics come from

Most honors topics arrive by one of four routes.

From a supervisor's list. Many departments publish lists of projects offered by faculty members. These have the great advantage of being pre-sized by someone who knows the area: a supervisor who offers a project usually has a reasonable idea of where it can go and what a student might achieve. The disadvantage is that you are working on someone else's question and must take the time to make it your own.

From a course that excited you. A theorem from MATH-351 or MATH-460 that you found beautiful, a proof you did not fully understand, or a remark in a lecture that the general case is still open can all seed a project. Take the idea to the lecturer of that course: they are the natural supervisor and will know whether it leads anywhere.

From your own curiosity, as with Amara's consecutive sums. Self-generated questions are motivating and often surprisingly good, but they need a reality check. The question may be trivial, impossibly hard, or completely solved in a paper from 1925. A potential supervisor can usually tell within a short conversation, which is why you should talk to one early.

From an application. Questions from physics, biology, economics, computer science or industry often contain mathematical problems suitable for a project, especially in modeling, optimization, statistics and numerical analysis. The challenge is to isolate a mathematical question from the applied context and to make sure the project is assessed as mathematics, with theorems and proofs or rigorous analysis, rather than as a report on the application.

Sizing a question

A good honors question has a ladder of versions: a version you are confident you can finish, a version you hope to finish and a version you would be delighted to finish. If the only version of the question is the hardest one, the project is a gamble. If even the hardest version is easy, the project will be thin. Here is the ladder for the model project.

Several tests help to size a question. Can you compute examples within the first week? If not, you will have nothing to look at while you think. Do you understand what an answer would look like: a formula, a classification, an inequality, an algorithm, a counterexample? Does a natural special case exist that you are confident you can do? Is there literature, and if so is the question already completely answered? Does the question connect to mathematics you know, so that you are not learning three new fields before you can start? A question that passes all five tests is probably well sized.

Choosing a supervisor

A supervisor is chosen for expertise, availability and fit. Expertise matters because a supervisor in the right area can point you to the key paper in an afternoon, recognize when a result is known and spot the standard trick you are missing. Availability matters because a brilliant supervisor who meets you twice a term will be less useful than a competent one who meets you every week. Fit concerns working styles: some students want detailed guidance and frequent deadlines, others want freedom and occasional advice, and supervisors vary in the same way.

Before approaching someone, read a little of their work: their course notes, the abstracts of a few recent papers, their list of past honors projects if the department publishes one. Then write a short, specific email. Say who you are and which courses you have taken, say what interests you and why you are writing to them in particular, mention any idea you already have in a sentence or two, and ask whether they would be willing to meet to discuss possible projects. A message of this kind takes ten minutes to write and is far more likely to get a positive response than a generic request for any project.

At the first meeting, ask questions as well as answering them. How often do they usually meet honors students? What do they expect to see at meetings? How do they prefer to receive drafts? Are they on leave or traveling for part of the year? What happened in projects they supervised before? These questions are entirely normal and help both of you decide whether the arrangement will work.

A decision matrix

When choosing between several candidate topics, it can help to make your criteria explicit. Write down the criteria that matter to you, give each a weight, score each candidate from 1 to 5 on each criterion and compute weighted totals. The numbers will not make the decision for you, and should not, but the exercise forces you to state what matters and frequently reveals that you were overweighting something, such as how impressive a topic sounds, relative to things that predict success, such as your background and the supervisor's fit.

The matrix above is Amara's. She was genuinely interested in Ramsey numbers after MATH-460, and the topic scores highly on interest and on open ends, but she lacked the background in probabilistic methods that a serious attack requires and the scope was hard to control: the smallest unknown classical Ramsey number, \(R(5,5)\), has resisted decades of work. The geodesics project scored well on supervisor fit, and the optimization project was solid but less interesting to her. The weighted totals confirmed what she suspected, and she also noticed that the consecutive-sums project had the best floor: she was confident of a result within weeks.

The program below computes such a matrix and also tests its sensitivity. A decision that flips when one weight changes by one is a close call that deserves more thought; a decision that survives all such changes is robust.

criteria = ['interest', 'supervisor fit', 'background', 'scope', 'open ends']
weights  = [3, 3, 2, 2, 1]
scores = {
    'consecutive sums':  [5, 4, 5, 5, 3],
    'Ramsey numbers':    [5, 3, 3, 2, 5],
    'geodesics on tori': [4, 5, 3, 4, 3],
    'sparse LP solver':  [3, 4, 4, 3, 4],
}

def totals(w):
    return {t: sum(a * b for a, b in zip(w, s)) for t, s in scores.items()}

base = totals(weights)
best = max(base, key=base.get)
for t, v in sorted(base.items(), key=lambda kv: -kv[1]):
    print(f"{t:18s} {v:3d}")
print("best:", best)

# sensitivity: change each weight by -1 or +1 and see whether the winner changes
flips = []
for i in range(len(weights)):
    for d in (-1, 1):
        w = weights[:]
        w[i] = max(0, w[i] + d)
        t = totals(w)
        winner = max(t, key=t.get)
        if winner != best:
            flips.append((criteria[i], d, winner))
print("winner changes under single-weight changes:", flips if flips else "none")
margin = base[best] - sorted(base.values())[-2]
print("margin over runner-up:", margin)

consecutive sums    50
geodesics on tori   44
Ramsey numbers      39
sparse LP solver    39
best: consecutive sums
winner changes under single-weight changes: none
margin over runner-up: 6

The project proposal

Many departments ask for a short proposal within the first month. Even when they do not, writing one is valuable. A proposal of one or two pages should state the question in its floor, target and stretch versions; summarize what is known, with two or three key references; describe the planned approach, including any computations; list milestones; and identify risks. Writing it forces you to discover whether you can state the question precisely, and gives your supervisor something concrete to respond to.

A timeline

A plan with dates turns good intentions into a schedule. The chart below is a typical eight-month plan. Observe that reading, experimenting and writing overlap rather than following one another, and that there is a deadline for a full draft well before the final deadline. Students consistently underestimate how long writing and revision take; a common rule is that writing takes at least twice as long as you expect, even after you have applied this rule.

Build in slack. Illness, other courses, a proof that collapses in February: something will go wrong, and a plan with no slack will fail the first time it does. Two weeks of unallocated time before the final deadline is a sensible minimum. The program below takes a list of tasks with estimated durations and dependencies and computes the earliest finish date, which is the simplest version of the critical path method used in project management.

from datetime import date, timedelta

# task: (weeks, prerequisites)
tasks = {
    'choose topic':      (2, []),
    'reading':           (8, ['choose topic']),
    'literature search': (5, ['choose topic']),
    'experiments':       (8, ['choose topic']),
    'main results':      (12, ['reading', 'experiments']),
    'interim talk':      (1, ['experiments', 'literature search']),
    'draft chapters':    (10, ['main results']),
    'revisions':         (3, ['draft chapters']),
    'final talk':        (1, ['revisions']),
}

finish = {}
def earliest_finish(t):
    if t not in finish:
        weeks, pre = tasks[t]
        start = max((earliest_finish(p) for p in pre), default=0)
        finish[t] = start + weeks
    return finish[t]

for t in tasks:
    earliest_finish(t)

start_day = date(2026, 9, 8)
for t, f in sorted(finish.items(), key=lambda kv: kv[1]):
    print(f"{t:18s} finishes end of week {f:2d}  ({start_day + timedelta(weeks=f)})")

# the critical path: walk back from the last task through the latest-finishing prerequisite
path, t = [], max(finish, key=finish.get)
while True:
    path.append(t)
    pre = tasks[t][1]
    if not pre:
        break
    t = max(pre, key=lambda p: finish[p])
print("critical path:", ' -> '.join(reversed(path)))
print("total weeks:", max(finish.values()), "of about 32 available")

choose topic       finishes end of week  2  (2026-09-22)
literature search  finishes end of week  7  (2026-10-27)
reading            finishes end of week 10  (2026-11-17)
experiments        finishes end of week 10  (2026-11-17)
interim talk       finishes end of week 11  (2026-11-24)
main results       finishes end of week 22  (2027-02-09)
draft chapters     finishes end of week 32  (2027-04-20)
revisions          finishes end of week 35  (2027-05-11)
final talk         finishes end of week 36  (2027-05-18)
critical path: choose topic -> reading -> main results -> draft chapters -> revisions -> final talk
total weeks: 36 of about 32 available

The plan fits in about 32 teaching weeks only if the main results take no more than twelve weeks and the drafting starts as soon as they are done. In reality the main results will not finish on a fixed date, which is why lesson 9 recommends drafting chapters in parallel with the research rather than afterwards. The critical path tells you which tasks have no slack: a delay in reading or in the main results delays everything, while a delay in the literature search does not.

Working with your supervisor through the year

Agree on a meeting rhythm at the start, and keep a project notebook, physical or electronic, in which you record what you tried, what worked, what failed and what you plan to do next. Before each meeting, send a short note listing what you have done and what you want to discuss. After each meeting, write down the advice you received and the actions agreed. This habit costs a few minutes and saves hours of repeated discussion; it also produces a record you will be glad of when writing the thesis, because many of the examples and remarks in a good thesis originate in exactly these notes.

If the relationship is not working, because meetings keep being canceled, advice is unclear or you feel unable to ask questions, raise it early, first with the supervisor and then, if necessary, with the honors coordinator. Problems discovered in March are much harder to fix than problems discovered in October.