Markov numbers and the free group on two generators, Caroline Series | LMS Summer School

Markov numbers are the integer solutions of the equation x2+y2+z2 = 3xyz. Markov proved that the solutions can be arranged around the vertices of a trivalent tree, which leads to a tantalising unsolved problem called the uniqueness conjecture. Another well-known classical theorem says that the possible sets of generators of the free group on two generators can be arranged in a similar way. It turns out that these two results are closely related. After explaining how all this works, we will go on to some modern work involving complex number solutions of the Markov equation where there are some very nice results but also many unanswered questions.
==========
The London Mathematical Society has, since 1865, been the UK's learned society for the advancement, dissemination and promotion of mathematical knowledge. Our mission is to advance mathematics through our members and the broader scientific community worldwide.
For further information:
► Website: www.lms.ac.uk
► Events: www.lms.ac.uk/...
► Grants and Prizes: www.lms.ac.uk/...
► Publications: www.lms.ac.uk/...
► Membership: www.lms.ac.uk/...
Follow us:
► Twitter: / londmathsoc
► Facebook: / londonmathematicalsociety
► LinkedIn: / the-london-mathematica...
► KZbin: ‪@LondonMathematicalSociety‬