News
3 November 2025: Schläfli lecture by Kathlén Kohn.

Teaching

Discrete Mathematics

• 27 January: defined affine and projective planes, proved that you can go back and forth between the two concepts, given two examples of projectieve planes (over the reals, and the Fano plane), defined the order of a projective plane, told about the Prime Power Conjecture: if the order is finite, then it is a prime power, proved Desargues's theorem for the affine plane over the reals, defined the Moulton plane (in which Desargues's theorem does not hold).
• 28 January: Exercises can be found here.
• 3 February: visualised the real projective plane. Some consequences: it is compact, not orientable, not simply connected. Then defined projective spaces over general fields. The projective plane over Z/2Z is the Fano plane. Showed that it is self-dual. Counted points of the projective plane over finite fields.
• 4 February: Exercises can be found here.
• 10 February: counted points on projective (n-1)-space over the field with q elements, and, more generally, counted projective (k-1)-subspaces of projective (n-1)-space (Gaussian q-binomial coefficient). Showed that if k divides n, then projective (n-1)-space can be partitioned into projective (k-1)-spaces. Showed how GL_n(K) acts by automorphisms on projective (n-1)-space, and how ``homogeneous'' this space is under the action. Showed how GL_2(K) acts by M&oulm;bius transformations on the projective line over K. For Möbius transformations of the complex projective line see this Youtube movie.
• 11 February: Exercises can be found here.
• 17 February: More about Möbius, whose transformations send Circles to Circles. Defined and used the cross-ratio. Then dealt with "Circles" over arbitrary field extensions L/K: a "Circle" in P^1 L is a set of the form g P^1 K where g is an element of GL_2(L). These "Circles" form the blocks of a 3-design.
• 18 February: Exercises can be found here.
• 3 March: A bit about designs. Illustrated "double counting" in various instances, among them Katona's proof of Erdös-Ko-Rado and Frankl-Wilson's vectorial analogue (the latter I did only for k dividing n). Then explained Dvir-Alon-Tao's proof of the finite-field Kakeya conjecture.
[The latter will not appear in exams. Double counting might.]
• 4 March: Exercises can be found here.
• 4 March: some extra notes on Circles, Erdös-Ko-Rado, and Frankl-Wilson can be found here.
• I found a very funny lecture on the history of design theory, a bit of which we will see through projective spaces. It can be found here.
• A lot on design theory (more serious stuff) can be found here.