Since I have had the experience to give two talks about maths topics which I found interesting, and in both cases I spent about a day researching them, I thought it would be a good idea to collect a list of things which one can spend a day or two and (I think) end up with a decent understanding of them (at least of the statement).1 These topics aren’t really in any particular order, and the content here might reflect my personal interests.

  • Resultants and resolvents (algebra, algebraic geometry)
  • Groebner bases (algebra, algebraic geometry)
  • Isospectral surfaces (differential geometry, analysis)
  • Veblen-Young (projective geometry) and Tits buildings (algebraic topology)
  • Aperiodic monotile, also see Einstein tile (plane geometry, combinatorics)
  • Weyl’s criterion for equidistribution (analysis, measure theory)
  • Peter-Weyl theorem (representation theory, harmonic analysis)
  • Non-Desarguesian planes (projective geometry)
  • Finite projective geometry (projective geometry, combinatorics)
  • Spin groups (differential geometry, topology, group theory)
  • Galois connections (algebraic topology, Galois theory, order theory)
  • Eilenberg-Mazur swindle (separately, algebra and geometric topology, or knot theory)
  • Morse theory (differential geometry)
  • Hilbert’s tenth problem, MRDP theorem (number theory, computability theory)
  • Dehn invariants (geometry)
  • Gelfond-Schneider (number theory)
  • Rademeister’s theorem (knot theory)
  • Rational tangles (knot theory)
  • Wreath products (group theory)
  • Inverse limits and profinite groups (category theory, group theory, Galois theory)
  • Nielsen-Schreier (group theory, algebraic topology)
  • Kaplansky’s unit conjecture (algebra)
  • Kakeya sets over finite fields (geometry, combinatorics)
  • Eckmann-Hilton
  • Frobenius and Hurwitz theorems
  • Nonstandard analysis
  • On Numbers and Games
  • Borel’s normal number theorem
  • Mapping class groups
  • Belyi’s theorem
  • Peauceullier-Lipkin

This list is (almost by definition) incomplete, so I might add things to this list from time to time.

  1. For a suitably prepared student, which will depend on the topic of choice. ↩︎