Math PhD Statement of Purpose: Example and Rules

Math PhD Statement of Purpose: Example and Rules

A math PhD statement of purpose should spend its 2–3 pages (Berkeley's and UT Austin's figure) on mathematics you have proved, read and want to learn next. Harvard says its statement "is not intended to be a biographical sketch", and Harvard math accepts no personal statement at all.

That rules out the opening about how you fell in love with mathematics. At Harvard there is no other document to put it in; at Berkeley and Princeton it belongs in a separate essay. Below: which department wants which essay, how to describe an REU or reading course so a mathematician can check it, and a constructed statement to measure your draft against. When your draft is ready, a free Quick Review against the preparation-first rubric reads it for named coursework, work done and results.

Every department page quoted below was checked on October 30, 2026. Where a page names the entry year, it is Fall 2027. GRE rules change between cycles, so recheck each page before you book a test.

On this page: which essays each department wants · describing an REU · the Subject GRE · a constructed statement · checklist

One essay or two: what five math departments ask for

Read across the rows:

• Two of the five want a second, personal essay. Berkeley and Princeton do. Harvard does not accept one, so the biography you send Berkeley must come out of the Harvard version.
• UT folds the personal material into the statement. "We do not require diversity statements, and encourage you to incorporate such material into your statement of purpose". Its guidance also asks you to "Report 'broader impacts' of your activities on others" and "Explain any special circumstances".
• MIT names what it reads first. "The Department looks for research skills and potential, and focuses primarily on your research statement, recommendations, coursework and grades."
• Applied interests may need a different door. Princeton says: "If your interest is in applied and computational mathematics, you must apply directly to the Program in Applied and Computational Mathematics (PACM)." Berkeley runs a separate Applied Mathematics PhD through the same department page.

Our view: write one mathematical core statement that works at Harvard as it stands, and keep the personal material in a separate file. Then attach it as Berkeley's Personal History or Princeton's Personal Statement, and fold a short version into UT's statement. The basis is the table: Harvard says its statement is not a biographical sketch, Berkeley and Princeton give biography its own essay, and only UT asks for it inside the statement.

The countercase: if your record has a gap that needs explaining (a bad semester, a late switch into mathematics), Harvard gives you no second essay to explain it in. Spend one or two plain sentences on it inside the Harvard statement, then return to the mathematics. UT invites exactly this under "special circumstances".

What goes in the statement and what goes elsewhere

If your interests are really statistics, numerical analysis or modelling, read our statistics and applied math SOP review guide before you settle on a pure mathematics program. If you are coming from an engineering degree, explaining engineering work in an applied math SOP covers that route.

How to describe an REU or reading course so a mathematician can check it

Harvard wants a statement that shows "a deep understanding of mathematics". A sentence a mathematician cannot check does not show that. Four rules: state the problem precisely, with the definition a reader needs; separate your part from the group's; state the status honestly (proved, special case, computed, or open); and name what you read by the result you worked through, not the course title.

Constructed example: what the weak version looks like

Constructed example, written by us, not a real applicant statement.

Last summer I participated in an REU on graph coloring, where our team made significant progress on an open problem. A reading course in algebraic topology then deepened my love of abstract mathematics.

Nothing in it can be checked: no problem, no definition, no part that was yours, no status, no theorem. The second and third paragraphs of the constructed statement below describe the same two experiences: they define the term, separate "we" from "I", say the main question is still open, and name the computation. Every claim there could be checked against a preprint, which UT would let you upload.

Length: department rules beat faculty advice

Use the department's number where it gives one: Berkeley's 1,000 to 1,500 words, UT's 2–3 pages. Harvard, Princeton and MIT state none. For those three, our recommendation is to borrow one faculty reader's published ceiling. Tom Alberts, a mathematics professor at the University of Utah, writes on his personal page: "Don't write more than two pages." He also says "there should be one or two paragraphs that you customize for each school that you apply to." He calls these "my own set of preferences", not department policy.

The Math Subject GRE changes your timeline, not your statement

The table shows the Subject test ranging from required (Berkeley) to optional (Harvard, MIT). Berkeley's page adds: "A score below the 70th percentile would be considered low." UT says it gives "much greater weight to the Mathematics Subject GRE than to the General GRE."

Princeton and UT list the test months as September, October and April. April falls after every deadline in the table, and Berkeley recommends taking the exams "no later than October", so check the testing agency's calendar now for any remaining fall date. Physics applicants face the same General-versus-Subject split; our guide to which GRE tests physics PhD programs require lays it out for Berkeley and Caltech.

Do not spend statement space on a missing or low Subject score unless the department asks. Qualifying exams are a separate, after-entry question: see whether you must pass math PhD quals before entry.

A constructed math PhD statement (560 words)

Constructed example, written by us, not a real applicant statement. Context (invented): a senior with one REU in graph theory and a reading course in algebraic topology, undecided between combinatorics and topology. The bracketed names are placeholders.

My mathematical interests sit between structural graph theory and algebraic topology. I am applying to [University] to decide between them with the training of a PhD program, and this statement describes what I have proved, what I have read, and what I want to learn next.

Last summer, in a ten-week REU at [REU host], a group of four of us studied list coloring. A graph is 4-choosable if, whenever every vertex is given its own list of four colors, a proper coloring can be chosen from the lists. Our mentor asked whether every graph in a class C of planar graphs, defined by two forbidden cycle configurations, is 4-choosable. We did not settle it. My part was the reducibility step. I proved that two of our five candidate configurations cannot occur in a minimal counterexample, using a discharging argument I adapted from the setting of ordinary coloring, and I wrote the computer check for a third configuration. The last two resisted every method we tried. Our preprint proves the case in which no two triangles are within distance two. I learned two things there: how much of a proof is bookkeeping that must be organised before the idea can be seen, and that I enjoy that organising.

In the spring I took a reading course in algebraic topology with [Faculty member at home institution]. I worked through cellular homology and the Mayer–Vietoris sequence, and I computed the homology of real projective space by hand, first with integer and then with mod 2 coefficients. Seeing the answer change with the coefficients was the first time a computation told me something I had not expected. I then read about simplicial complexes built from graphs, where a topological invariant gives a lower bound on a coloring number. I cannot yet follow every step of those arguments. They are the reason I do not want to choose between my two interests before I have studied both properly.

My coursework supports both directions. I have taken two semesters of real analysis, two of abstract algebra, point-set topology, and a graduate course in combinatorics in which I earned an A. I have not yet taken a course in differential geometry and would take one in my first year.

At [University] I would like to work with the combinatorics group, in particular [Professor A], whose recent work on coloring of graphs on surfaces is close to what I did last summer, and [Professor B], whose papers on reducible configurations I read during the REU. On the topology side, [Professor C]'s seminar on applied and combinatorial topology is where I would test whether the topological methods I have begun to read can become my own. I have not decided between these directions, and I would use the first two years of courses and seminars to decide.

After the PhD I intend to stay in research mathematics. I have tutored for the department's calculus help room for two years, and I would welcome teaching as part of graduate study.

In short, I can tell you exactly which part of our REU proof is mine, which parts of algebraic topology I can compute with and which I am still reading, and which three people at [University] I would most like to learn from. I hope to spend the next five years turning those two interests into one research direction.

That example is 560 words inside the quote (our count), under Berkeley's 1,000–1,500-word range on purpose: it is the core. To fit each department:

• Harvard: send it as it stands. There is no personal paragraph to remove.
• Berkeley: expand the REU and reading paragraphs toward 1,000 words, and move the tutoring sentence into the Personal History Essay with anything else about who you are.
• Princeton: keep this as the Statement of Academic Purpose and write the Personal Statement separately.
• UT Austin: add a short "broader impacts" paragraph, which UT's guidance asks for, and upload the preprint as a miscellaneous document.

Math statement checklist before you submit

1. The first paragraph names an area of mathematics, not a childhood memory.
2. Every result is stated precisely enough for a mathematician to check, with its status: proved, partial, computed, or open.
3. Each group project says what was yours.
4. Reading is described by theorems and computations, not course titles.
5. If you are undecided, the statement says so and names the options.
6. Each school's version names faculty with one reason each.
7. Personal history sits where the department wants it: inside UT's statement, in Berkeley's or Princeton's second essay, and at Harvard no more than a sentence or two on any record gap.
8. The length matches the department's rule, or two pages where none is stated.

Questions about the math PhD statement of purpose

Should I write about how I became interested in math?

Not in the statement itself. Harvard says the statement "is not intended to be a biographical sketch or a reflection on one's decision to enter the field." Berkeley and Princeton give you a second essay for that; UT takes it inside its single statement.

How long should a math PhD statement of purpose be?

Berkeley says 1,000 to 1,500 words and UT says 2–3 pages. Harvard, Princeton and MIT state no length on the pages we checked.

Do I need a research result to apply?

No published rule above requires one. MIT says it "looks for research skills and potential". A careful reading course or an exposition you wrote is evidence too, and UT accepts "mathematical expositions" as uploads. Berkeley and UT both ask you to name faculty; see should you name a professor in your SOP for programs that do not.

Get a free review of your math PhD statement of purpose

Get a free Quick Review of your math PhD statement of purpose against GradPilot's preparation-first rubric, written for statements in mathematics, statistics, physics, economics and related fields. It reads for named coursework and methods, the work you did and its results, and whether your stated direction matches the preparation you show. For a whole-statement read, use the general PhD statement of purpose rubric. Applying to a master's in statistics or applied mathematics instead? Use the statistics and applied mathematics master's SOP rubric, or start from the PhD SOP review page.

More math and quantitative PhD guides:

• Math PhD quals before entry: optional early attempts versus real passing deadlines.
• Physics PhD GRE requirements: the General and Subject tests, checked separately.
• Statistics and applied math SOP review: what to check in a quantitative master's statement.
• All PhD application essay guides.

Sources

Retrieved October 30, 2026, except where noted.

• Harvard Griffin GSAS, Mathematics (deadline Dec 15, 2026): statement of purpose wording; personal statement not accepted; GRE General not accepted; Subject test optional.
• UC Berkeley Mathematics, Graduate Admissions (Fall 2027): two essays and their typical lengths; Subject GRE required; 70th-percentile note; deadline.
• Princeton Mathematics, Graduate Admission (no year stated): required materials; Subject GRE optional but highly recommended; PACM routing; test months.
• MIT Mathematics, PhD admission FAQ (no entry year stated; fee-waiver date November 20, 2026): what the department reads; GRE optional; deadline.
• UT Austin Mathematics, Graduate Admissions: How to Apply (General GRE waiver effective Fall 2027): statement length and content; miscellaneous uploads; Subject GRE weight; test months.
• Tom Alberts, University of Utah, Writing a Personal Statement for a Graduate School Application (April 28, 2023; personal faculty page, not department policy): two-page and customisation advice.