Is there any alignment on what a maths grad should know?

Question

This more specific question relates to a more general question of what is a maths degree aiming for.

• Do any universities define a high level goal for pure mathematics degrees at all?
• If so, are there specific differences in the goals of higher degrees?

Also interesting:

• Whether related institutions of bodies have something along this line
• Whether the concept is seen as too narrowing and consciously avoided

@james-s-cook 's comments below draw out that any answer is likely country specific. An honours degree from the UK is much more specialized than typical for most of the US. Perhaps equivalent to a research specialization.

Answer 1

(Addressing the second half of the question)

The UK QAA has a subject benchmark for what it means to have an undegraduate degree in maths/stats/OR (and other ones for other subjects). In terms of content (explicitly or implicitly listed), it specifies:

• some linear algebra (ie vectors/matrices)
• some calculus (ie differentiation/integration)
• some modelling
• some estimation
• a mathematical computing package

Answer 2

This may be largely a matter of opinion, but:

The goal of a Ph.D. program is to train researchers. The expectation is that, as a Ph.D. student, you will be guided by a mentor to produce some original, publishable work, called a thesis (or dissertation). The expectation is that by the time you finish your Ph.D., you should be able to continue to work and write papers on your own, and perhaps to mentor students of your own (either at the doctoral level, or at the undergraduate level). A Ph.D. program is, in essence, the end of an apprenticeship program in academia.

Historically, the job of a bachelor's degree program was to prepare younger students for advanced degrees—the bachelor's program was, more or less, the first part of an apprenticeship in academia. The goal of such a program was (and ideally, still is) to give students the background necessary to start their own research. Depending on how specialized the program is, this might mean specializing in mathematics very early (this seems to be the European model), or acquiring a broad background for a couple of years, then specializing (this is a more American model).

Speaking more specifically of the United States, someone with a bachelor's degree in mathematics should, ideally, have the background required in order to start a Ph.D. program (since the underlying goal of a bachelor's degree, in my opinion, is to provide training to nascent researchers). In as much as there is a "high level" goal, I would say that this is it.

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As you also seem to be interested in curriculum, an undergraduate degree should give a student enough background to start a Ph.D. program. The usual expectation is that students entering a Ph.D. program (in "pure" mathematics, rather than applied mathematics, statistics, or mathematical education; in the US) should know at least:

1. computational techniques, which are typically taught in lower division calculus, linear algebra, and differential equations courses;
2. some basic mathematical logic—not necessarily anything foundational, but enough to argue mathematically using (for example) contraposition, induction, exhaustion, and so on;
3. some basic real analysis (the theory of differentiation and Riemann integration over $\mathbb{R}^d$);
4. some basic modern algebra (the study of groups, rings, and fields; maybe some Galois theory);
5. point-set topology; and
6. something a little more specialized.

Basically, a student should be able to perform computations, write a coherent proof, have familiarity with the essential pillars of modern mathematics (analysis and algebra, IMO), and have been exposed to something a bit more specialized which they might—or might not—follow up on in a Ph.D. program (complex analysis, knot theory, fractal geometry, graph theory, mathematical modeling, number theory, etc.).

If you are interested in specifics of what Ph.D. programs expect students to know, you might look at the GRE Mathematics Subject Test. Many (most?) US Ph.D. programs in mathematics require GRE math subject scores for admission.