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.

Schedule

10:00 AM - Opening Remarks - Doron Levy (UMD)
10:05 AM - Recent advances on the problem: "Can one hear the shape of a drum" - Steven Zelditch (Northwestern University)
11:00 AM - Mirzakhani's counting theorem - Viveka Erlandsson (University of Bristol)
1:00 PM - Singular Analytic Vector Fields: Analysis, Geometry & Dynamics (Alvaro Alvarez-Parrilla - Grupo Alxima SA de CV)
2:00 PM - Hot Spots on a Triangle Will Migrate Towards the Boundary - Chris Judge (Indiana University)
3:00 PM - The Spectral Theory of Random Surfaces - Alex Wright (University of Michigan)
4:00 PM - Reception - MC Joel Cohen (UMD)

Speakers

• Alvaro Alvarez-Parrilla
  (Grupo Alixmia SA de CV)
• Viveka Erlandsson
  (University of Bristol)
• Chris Judge
  (Indiana University)
• Alex Wright
  (University of Michigan)
• Steven Zelditch
  (Northwestern University)

About the Conference

This is the first conference, specifically on these two subjects, held at University of Maryland. The speakers were chosen to highlight connections between the two subjects, especially in areas related to Wolpert's contributions.

About Scott Wolpert

After receiving his Ph.D. from Stanford University in 1976, Scott Wolpert joined the Department of Mathematics at the University of Maryland, where he has remained ever since. Last year he achieved Emeritus status following his retirement.

Scott's research career included fundamental contributions to geometry and analysis of Riemann surfaces. His work on the Weil-Petersson geometry of Teichmüller space earned him an invitation to the International Congress of Mathematicians in 1986. His work in spectral geometry of Riemann surfaces included the resolution (with Gordon and Webb)  of Marc Kac's famous question: "Can you hear the shape of a drum?" 

His service record is notable, having served as Associate Chair for Undergraduate Studies, Associate Dean in the College of Science, Associate Dean in the Office of Undergraduate Affairs, and, most recently, Chair of the Department of Mathemaics. His work on Calculus Reform led to an award as University of Maryland Distinguished Scholar-Teacher, and his work as Associate Dean earned him a Kirwan Award at the University of Maryland.