Previous Oxford teaching
Previous Oxford teaching
Embedded Files
Prelims Material
Prelims Material
Self Explanation Training from the Department of Mathematics Education at the University of Loughborough, and the associated:
The Xena Project (Kevin Buzzard at Imperial, formalising mathematics in Lean), and the "Natural Number Game" -- verifying basic properties of the natural numbers coming from Peano's axioms!
Proof in the Pudding: Robert C. Moore, Mathematics Professors' Evaluation of Students' Proofs: A Complex Teaching Practice. Int. J. Res. Undergrad. Math. Ed. 2 pp. 246–278, (2016).
The Wikipedia page for the Circles of Apollonius - an alternate (but equivalent to the usual centre-and-radius) definition of a circle, due to Apollonius of Perga.
Cards on the Table: Matthew Inglis and Adrian Simpson, Mathematicians and the Selection Task. In M. Johnsen Hoines & A. B. Fuglestad (Eds.), Proc. 28th Int. Conf. on the Psychology of Mathematics Education, Vol. 3 pp. 89-96, (2004).
Tyres in the Library: Ed Dubinsky, Flor Elterman, and Cathy Gong, The Student's Construction of Quantification. For the Learning of Mathematics 8(2) pp. 44-51, (1988).
Belt in space: in class, I showed you the belt trick (also know as the plate trick) which is ultimately a demonstration that "the fundamental group of SO(3) is (cyclic and) order 2"! You know quite well what the elements of SO(3) are; rotations about the origin in \R^3, i.e. 3x3 orthogonal matrices with determinant 1.
Groups: I was asked a very nice question about finite groups, that boils down to: if I know the order of every element in a group, do I know the group? The answer is no! Here is a counterexample: set G = \Z/4\Z x \Z/4\Z and H = Q_8 x \Z/2\Z (where Q_8 is the quaternion group). The former is abelian, the latter is not, but both groups have exactly 1 element of order 1, 3 elements of order 2 and 12 elements of order 4.
These groups have the same order statistics but are not isomorphic, because H is nonabelian. What if we removed this obstruction? The answer becomes yes! This follows from a very powerful theorem about the structure of finite abelian groups.
When we discussed the "Chinese Remainder Theorem", we mentioned a bit about its history: the earliest appearance of a method to solve multiple congruence relations appears in work of Sun Zi, around the third century. The complete theorem and its proof first appears in a treatise of Qin Jiushao in 1247 (I recommend reading his biography, linked). The historical development of the CRT has been well-documented by Shen Kangsheng.
In answering questions about the number of "essentially different" edge colouring of a tetrahedron (colouring up to rotations) we found it hard to visualize precisely the action the group of symmetries of the regular tetrahedron has on the set of vertices. Thankfully, there is a GeoGebra tool for that.
A5 Topology
A5 Topology
We examined the topology of the "topological dunce hat", obtained by gluing three sides of a triangle together. This was the subject of Zeeman's 1963 paper, where a certain topological question about contractible spaces is addressed -- Zeeman's conjecture in turn implies the Poincaré Conjecture, one of the Millenium Prize Problems, proven in 2002 by Grigori Perelman. Zeeman's conjecture remains open. (Below; sketch from Topology and Geometry by Glen Bredon.)
C3.7 Elliptic Curves
C3.7 Elliptic Curves
Here is a blog on the Ax-Grothendieck Theorem; any injective polynomial function \C^n --> \C^n is necessarily surjective. It explains in more detail what I mentioned in class -- that a counterexample can be descended (using model theory) to a finite field, where we know the statement is true from a simple counting argument.
I was asked very good questions about the genus of a curve, and the connection between an elliptic curve and a torus. It goes beyond the scope of C3.7, but here (part 1) and here (part 2) a precise correspondence is drawn. It can further been shown the isomorphism \Phi is a homeomorphism, meaning the topology is preserved.
We discussed an alternate formulation of Hensel's Lemma ("simple zeros lift uniquely") -- here is a handout by Keith Conrad on the two formulations (2.1 is "simple zeros lift uniquely", 4.1 is the formulation familiar to you from class) which contains the proof of their equivalence. I also gave a handout on "visualising the 3-adic numbers", based on this Quanta article and this image from Daniel Litt.
In our final class, we discussed computing the rank of an elliptic curve -- this is a hard problem in general, and several basic facts about rank are still unknown. It is "widely believed" there is no maximum rank for an elliptic curve (though see this Quanta article) but the best that is currently known is an elliptic curve of rank at least 28, due to Elkies in 2006. It is also conjectured that "in a suitable asymptotic sense" (using heights) the average rank of all elliptic curves is one half; i.e 50% are rank 0, 50% are rank 1, and 0% are of higher rank.
B8.5 Graph Theory
B8.5 Graph Theory
Tien Chih, Generalizing Cantor-Schroeder-Bernstein: Counterexamples in Standard Settings. The Mathematics Enthusiast 11(3) Art. 3, (2014).
This paper has an example that, if G has H as an induced subgraph, and H has G as an induced subgraph, then G and H are not necessarily isomorphic!
A TikTok by Haley Joy about the discharging method in discrete mathematics: "central" in the proof of the Four Colour Theorem.
What about a generalisation of the Four Colour Theorem, to surfaces of arbitrary genus? E.g.: how many colours are needed to colour a map on a torus? This is somehow easier than the traditional map-colouring problem! Read more on this blog.
This generalisation has its roots with P.J. Heawood and the Heawood conjecture. This was resolved in 1968 by Ringel and Youngs -- 8 years before the four colour theorem was proven -- with two exceptions: the sphere (hence this did not solve the traditional map-colouring problem) and the Klein bottle.
Since we're talking about planar graphs and colourings: Kuratowski's theorem states that a finite graph is planar if and only if it does not contain a subdivision of K_5 or K_{3,3} -- in a sense, nonplanar graphs are descendents of these two!
(Straight from Wikipedia: "A subdivision of a graph is a graph formed by subdividing its edges into paths of one or more edges. Kuratowski's theorem states that a finite graph G is planar if it is not possible to subdivide the edges of K_5 or K_{3,3}, and then possibly add additional edges and vertices, to form a graph isomorphic to G.")
Kazimierz Kuratowski published a proof of this theorem in 1930. His paper (in French) can be found here.
Something similar (and a bit sharper) is the Kelmans–Seymour conjecture: every 5 connected graph that is not planar contains a subdivision of K_5. The proof of this was published only recently (in 2020) across 4 papers! (By Xingxing Yu of the Georgia Institute of Technology, and his Ph.D. students Dawei He and Yan Wang; the first paper can be found here.)
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