Singular analytic vector fields: analysis, geometry & dynamics.

Alvaro Alvarez–Parrilla
Grupo Alximia SA de CV
Ensenada, Baja California, México
Singular analytic objects over a Riemann surface M are holomorphic objects with singular sets composed of zeros, poles, essential singularities and their accumulation points.
There is a canonical correspondence between different singular objects on Riemann surfaces M:

• vector fields X,
• 1–forms ωX,
• functions (distinguished parameters) ΨX : M → Cb that provide a uniformization of the vector field X,
• flat Riemann surfaces RX associated to these functions,
• certain maximal domains ΩX of the global flow,
• and certain differential equations associated to the Schwarzian of ΨX.

The above relationship is exploited in the study of certain families of vector fields, from the analytical, geometric and dynamical perspective.

 Mirzakhani’s curve counting theorem 

Viveka Erlandsson
University of Bristol 
Bristol, UK 

In her thesis, Mirzakhani established the asymptotic behavior of the number of simple closed geodesics of a given type in a hyperbolic surface. Here we say that two geodesics are of the same type if they differ up to homotopy by a homeomorphism. In this talk I will discuss this theorem, the extension to geodesics which are not simple, and some applications.

Hot Spots On A Triangle Will Mitigate Towards the Boundary

Chris Judge
Indiana University
Bloomington, IN, USA

A temperature distribution on a perfectly insulated, perfectly homogeneous body will tend to a constant temperature distribution as time tends to infinity. Jeff Rauch conjectured in the 1974 that, moreover, every local extremum of the temperature distribution will migrate towards the boundary as time tends to infinity. This is now believed to be true for convex domains and is known as the ‘Hot spots conjecture’. In joint work with Sugata Mondal we verify Rauch’s conjecture for triangles. This answers the question posed in Polymath 7

The Spectral Theory of Random Surfaces

Alex Wright
University of Michigan
Ann Arbor, MI, USA

We will discuss how ideas of Mirzakhani can be combined with the Selberg Trace Formula and new bounds on non-simple closed geodesics to give results on the first eigenvalue of the Laplacian of Weil-Petersson random high genus surfaces. This is joint work in progress with Michael Lipnowski.

Recent Advances on the Problem, 'Can You Hear the Shape of a Drum?'

Steven Zelditch
Northwestern University
Evanston, IL, USA

This problem was posed by M. Kac in 1965. He proved that a disk is determined by its Dirichlet or Neumann eigenvalues. Until last year, it remained the only domain which was known to be determined by its eigenvalues among all smooth domains. Now it is proved (by Hamid Hezari and S. Z.) that ellipses of small eccentricity are also determined by their Dirichlet (or, Neumann) eigenvalues. Basic ingredients include recent purely dynamical inverse results of Avila, Kaloshin and de Simoi on the Birkhoff conjecture that ellipses are the only “integrable billiards”. My talk will review the dynamical results, the link to the spectrum, and the new arguments involving the wave equation.