# What ought to be in any pure mathematics degree?

With the recent cut-backs in mathematics and statistics programmes around Australia, we are now being asked to rationalise our offerings and provide less choice for our undergraduates. So what are the essential threshold concepts that someone intending to major in pure mathematics ought to know? For example, “linear independence” is such a concept, but “group theory” is not a good answer as there are extraneous areas of group theory which are not essental (e.g., Grigorchuk groups). Nor is “linear algebra” a good answer as it is debatable whether everyone should learn what a tensor product is (but you are allowed to disagree with me on this one). So should every pure-maths graduate know…

(1) that every integer has a unique factorisation into primes?

(2) the epsilon-delta definition of a limit?

(3) the Prime Number Theorem?

(4) proof by induction?

(5) an axiomatic notion (e.g., vector space, group, ring)?

(6) finite Galois fields?

(7) a theorem by Euler?

(8) the basic ideas of cardinalities (e.g., rationals are countable)?

So please contribute so we can cut out courses!

Maybe we can do the same thing to our undergraduate students as we do to our raw materials; export them, pay others to add value and then reimport the finished product.

I knew 7.5 of these when I graduated, knowing the statement of (3) and enough complex analysis to understand the proof, but never having seen a proof. I think that every pure maths student should know at least six of these eight things by the end of their second year.

Hey John, I am a pure math student here at uwa. At some universities, to do a PhD, one would require to sit for an exam. I think the materials that are on the exams covers the very basics of what everyone needs to know for research and further study.

Algebra

Sets: Cardinals, ordinals. Countability. Zorn’s Lemma.

Linear algebra: Finite-dimensional vector spaces, bases. Tensor product. Isomorphism of Mn(F) and End(Fn). Orthogonality, examples of classical groups. Diagonalization, Cayley-Hamilton theorem, spectral theorem, Jordan canonical form.

Group theory: Significance of classification of finite simple groups. Central and derived series. Structure of finitely generated abelian groups. Group presentations. Representations of finite groups.

Ring and module theory: Euclidean domain, principal ideal domain, unique factorization domain. Polynomial rings, reducibility. Noetherian rings, Hilbert basis theorem. Some noncommutative rings, e.g. matrix rings. Free and projective modules. Exactness.

Field theory: Fundamental theorem of algebra, algebraic closure. Classification of finite fields. Examples of Galois groups.

Category theory: Examples of categories, functors, natural transformations, adjoint functors.

Analysis

Part I: Advanced Calculus

Properties of the reals such as Bolzano-Weierstrass, Heine-Borel and equivalent of norms in Rn.

Differential calculus of Rm valued functions on subsets of Rn. Continuity and uniform continuity, differentiability, partial derivatives, Jacobians, implicit and inverse function theorems.

Differential equations: existence and uniqueness theorem of initial value problems.

Infinite sequences and series of numbers and functions. Absolute and uniform convergence, equi-continuity, Arzela-Ascoli theorem, Weierstrass approximation theorem.

Riemann and Riemann-Stieltjes integrals, fundamental theorem of Calculus.

Line integrals, surface integrals, differential forms. The theorems of Stokes and Green and the divergence theorem. Change of variables in multiple integrals.

Metric spaces, completeness, limit and continuity.

Part II: Real Analysis

Functions of bounded variation and absolutely continuous function.

Definition and elementary properties of Lebesgue measure.

Borel measures, measurable functions and simple functions.

Lebesgue integral and its elementary properties.

Convergence theorems.

Various types of convergence such as almost everywhere, in measure, in mean.

Multiple integrals and changing the order of integration (Fubini’s theorem).

Lebesgue’s differentiation theorem, Vitali’s covering lemma.

Basic properties of Lp spaces, such as density of C¥ functions, approximation identities, Riesz representation theorem.

Hilbert space and its basic properties.

Part III: Complex Variables

Cauchy-Riemann equations. Analytic functions. Contour integration. Cauchy integral formula. Taylor series. Residues and poles. Laurent series. Isolated singular points, removable/essential singularities, poles, residues, residue theorem, improper real integrals and their evaluation using the residue theorem. The argument principle. The open mapping theorem and the maximum modulus principle. Conformal mapping and linear fractional transformations. Harmonic functions.

I hope this is detailed enough. The material mentioned is covered in the pure math PhD qualifying exam from National University of Singapore.

Thanks Daniel,

In attempting to write down the fundamental concepts of pure mathematics that we believe are necessary in any pure mathematics bachelor’s degree, we often fall into the trap of stating what we learnt or what we know. I spent some time at a Belgian university, and if I asked the average student from there, they might have projective geometry and formal logic at the top of their list, because they are the subjects which they teach early on in their pure maths degree.

Do you think that everyone would agree with having category theory in any pure mathematics degree?

Cheers,

John.

I am sad to say that the university undergraduate environment does not provide us with the atmosphere that fosters creativity and a love for learning, things which are important in any education especially in pure math. Doing exams are just about everything in uni. Perhaps that is the only way to learn something, although quite superficially, in a very short amount of time. This is why I placed an exam up on my post.

This is further compounded by the fact that many of us undergraduates in pure math are also taking other majors to pay the bills. I myself am studying engineering and had to overload to take and additional pure math major for most of my semesters here at uni. Regretfully, an education in pure math can be at best a crash course given such circumstances.

I would love an undergraduate education like what they do in Europe but such an education would require that our entire energies be spent just in pure math. Unless one is exceptionally talented which I am far from, I do not expect my professional career to be in mathematics. Hence it is important to have an additional degree, a skill if you like.

It would be nice to have exposure to basic set theory, formal logic and projective geometry. If numbers and funds permit, I would love to have undergraduate seminars during the semester holidays for second year pure math majors on these topics. Even better, if we could coax the government to have these basic foundations taught in high school.

Sadly I caught the pure math bug very recently, only last semester as a matter of fact. The lecturers at UWA are fantastic in forcing the concepts down in such a short period of time, especially for people like myself who haven’t had that history of math olympiads, talent or even good grades etc.

Category theory, I think is very useful in the more abstract proofs. Especially when objects are not very define and the only way to characterize them is via the relationships to other objects. I faced the usual bafflement when I came across a few of them in Andrew Wiles proof for the Taniyman-Shimurra conjecture that indirectly proved Fermat’s Last theorem. It will be good to touch briefly on that subject especially during the honors year (if I make it that far).

Thanks

Daniel

At the risk of bringing up a very old post, as a third year pure mathematics student at UWA I’d like to offer my brief and disappointingly less detailed thoughts on this topic. Though it will probably end up being more a rant about the structure of the UWA pure mathematics course.

I will be completing the pure mathematics component of my major at the end of this year. By the end of this year I will have completed all six third year pure mathematics units the university offers, compromising introductory courses in group theory, ring theory, complex and linear analysis, topology, and a course on codes and ciphers.

I was pleased to note that I have learned, or will learn in the coming months, most of what was on the list Daniel supplied. Although at no point in my degree will I have been taught or even encountered tensors or formal logic.

Of the 8 things outlined in the initial post, I have seen all of them at a second year level. I would add to it

9. the proof that every non-empty open interval on the real number line contains a rational number

10. the proof that the square root of 2 is irrational

11. the construction of the rational and real numbers

12. modular arithmetic

13. introductory topology, mainly from a geometric point of view (constructing basic manifolds from “gluing”, as well as a more formal definition of gluing as equivalence relations, which would already be covered in the construction of the rationals)

9 and 10 are very simple proofs, so if you’re including the Prime Number Theorem, you may as well include them. 11 is a good exercise in pure mathematics, and is a good way to introduce students to the concept of equivalence relations. 12 is there just because.

The concepts in 12 are so simple that any reasonably intelligent high school student should be able to understand that they can make a torus from gluing the sides of a square in the right way. It would mainly be an exercise in making students think in a way that they most likely never have, and, unless they pursue topology, probably never will again.

All up I find it very depressing business, the notion that pure mathematics courses are being cut down. Much of the first few weeks of the third year pure mathematics units are spent rehashing what was covered in MATH2300. It is unfortunate that the unit is not required study for many of the third year pure mathematics units. In an ideal world, much of the 3P5 and 3P0 courses would be second year material, and we would be introduced to graph theory in third year, rather than having to wait until fourth year to see it.

If there has to be some course-cutting at UWA, I would say that 3CC Codes and Ciphers is the only expendable pure mathematics course. This is an unfortunate casualty of the War Against Mathematics (the WAM for short) as it is a very interesting unit. But it does not contribute much to a student’s knowledge of pure mathematics, but more serves as an example of the great power of pure mathematics. I think there’s good argument that 3CC, much like 3M2, should be considered both an applied and pure mathematics unit.

A bold suggestion of mine is to replace the linear algebra of MATH1010 with the linear algebra of MATH2020. Much of MATH2020 is just relearning more abstractly the linear algebra of MATH1010 anyway. Why waste time learning properties about matrices when we’ll just relearn all of those properties again a year later but more abstractly? Take that to the maths department and see what they think of it!

Dave