[SNU Number Theory Seminar 30 June] Do Tropical Markov Cubics dream of Hyperbolic Origami?

by qsms posted Jun 26, 2023
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일정시작 2023-06-30
배경색상 #FF5733
  • Date:  June 30 (Fri) 16:00 - 18:00 
  • Place:  27-325 (SNU)

  • Speaker:  Seung uk Jang (University of Chicago)
  • Title:  Do Tropical Markov Cubics dream of Hyperbolic Origami?
  • Abstract:  

    Non-archimedean fields and varieties over them admit the operation of tropicalizations, which provides a piecewise-linear approximate sketch of  varieties that encapsulates many key aspects. For Markov surfaces $x^2+y^2+z^2+xyz=D$, this viewpoint was initiated by works of Spalding and Veselov,  who focused on its tropical and dynamical aspects.

              In this talk, we will be working on a more general family of Markov surfaces and discover that, for any parameters, we have a copy or a shadow of the  hyperbolic plane with the $(\infty,\infty,\infty)$-triangle reflection group action. Such a viewpoint easily yields corollaries in Fatou domains (dynamical side) or  the finiteness of orbits of rational points with prime power denominators (number theory side). Some interesting number-theoretic aspects of this system  may be introduced, such as Farey triples.


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