Mathematics Honours Talks

Fri Oct 23. Physics Lecture Theatre 2

• 12 Noon

  Josh Deprez
  Paradoxes and monsters, and the fuzzy area between geometry and group theory

  
  

  A pinnacle of theoretical mathematics over the last 80 years has been the Banach-Tarski paradox, which has led to marvellous research in such diverse topics as group theory, geometry, measure theory and the foundations of mathematics.

  It is possible to transfer the paradoxical nature of the free group of rank 2 into ordinary three-dimensional space, with an intuitive dilemma resulting: a sphere can be sliced into pieces and reassembled using only rotations and translations into two copies of the original.

  Elements from the proof of this startling result are reviewed, and some responses discussed. A famous conjecture of von Neumann, formulated not long after the publication of the paradox, held that only free groups of rank at least 2 could create this mayhem. This conjecture was disproved as recently as 1980 by Ol'shanskii using geometric group theoretic techniques.

  Fundamental groups of surfaces and graphs are presented, as well as van Kempen's Lemma and diagrams, leading into the small cancellation theory underpinning Ol'shanskii's monsters.

• 2:10 pm

  Simon Arneaud

  Are the cards really shuffled?

  
  

  A popular belief is that card shuffling methods are highly effective or at least too complicated to study. In recent decades, mathematicians have attacked the topic with powerful and elegant techniques and found surprising answers and links to other fields.

  My own work has involved a combination of abstract algebra, Fourier theory and computer science. I will be giving an overview of the analytic methods and some results comparing the effectiveness of shuffles.