Research Statement         Gallery         Math 108C         Math 191

Preprints and publications

Critical sets of inner functions

• Stable convergence of inner functions
  updated March 17, 2018
  (new: Theorem 1.4 is now proved in full generality)
• Prescribing inner parts of derivatives of inner functions
  to appear in J. d'Analyse Math.

Makarov's principle and related topics

• Sparse Beltrami coefficients, integral means of conformal mappings and the Feynman-Kac formula
  to appear in Potential Analysis
• Makarov's principle for the Bloch unit ball
  with Ilgiz Kayumov
  Sbornik: Mathematics 208 (2017), no. 3, 399-412
• On Makarov's principle in conformal mapping
  to appear in Int. Math. Res. Not.
• Quasicircles of dimension 1+k^2 do not exist
  updated March 29, 2017
• Asymptotic variance of the Beurling transform
  with Kari Astala, Antti Perälä and István Prause
  Geom. Funct. Anal. 25 (2015), no. 6, 1647-1687

Weil-Petersson geometry

• Decorated rescaling limits of Blaschke products
  draft of July 2016, a little sketchy in some places and the last two sections are yet be written
• The geometry of the Weil-Petersson metric in complex dynamics
  to appear in the Trans. Amer. Math. Soc

Curious things

Slides

• Dimension of sparse quasicircles
  Analysis and dynamics conference, Oaxaca, Mexico, Sept. 20, 2016
• Makarov's principle
  Everything is complex, Saas Fee, Switzerland, Mar. 9, 2016
• Weil-Petersson geometry
  PhD Defense, Harvard University, Apr. 23, 2014

Some notes on various topics – not for publication

• Sigma^2 > 0.87914
• Approximating Jordan Curves by Julia Sets
• Approximation by Cantor Sets Julia Sets
• Asymptotic expansion of the Hausdorff dimension of the push-forward of Lebesgue measure

Expositions from my undergraduate studies

• Centaurs: Here, There, Everywhere!
  with Dimitri Dziabenko
  College Mathematical Journal 39 (September 2008), 267-272
• My notes from Prof. Edward Bierstone's class on Complex Analysis