Sullivan’s dictionary on geometric prime number theorems

Seminar: 
Junior Colloquium
Event time: 
Friday, April 8, 2022 - 12:00pm
Location: 
LOM 206
Speaker: 
Hee Oh
Speaker affiliation: 
Yale
Event description: 

The prime number theorem states that the number of primes of size at most T grows llike T/log⁡T, proved by Hadamard and de la Vallee Poussin in 1896. For Gaussian primes, that is, prime ideals in Z[i], not only does the number of Gaussian primes of norm at most T grow like T/log⁡T but also the angular components of Gaussian primes are equidistributed in all directions, as proved by Hecke in 1920. Geometric analogues of these profound facts have been of great interest over the years. We will discuss effective versions of these theorems for hyperbolic 3-manifolds and for rational maps.  Both Kleinian groups and rational maps define dynamical systems on the Riemann sphere and they are expected to behave analogously in view of Sullivan’s dictionary. We will  explain how our theorems fit in this dictionary.