Preprints and publications

Recent

• Analytic mappings of the unit disk which almost preserve hyperbolic area
  with Artur Nicolau
  
• Inner functions and laminations
  with Mariusz Urbański
  
• Shapes of infinite conformally balanced trees
  with Peter Lin, Steffen Rohde, Emanuel Sygal
  updated Oct. 31, 2023.
• Inner functions, composition operators, symbolic dynamics and thermodynamic formalism
  with Mariusz Urbański
  updated Aug. 27, 2023.

Positive curvature

• On meromorphic functions whose image has finite spherical area
  Studia Math. 272 (2023), no. 2, 121-137.

Random geometry

• Homogenization of random quasiconformal mappings and random Delauney triangulations
  with Vladimir Marković
  J. Differential Geom. 124 (2023), no. 3, 523-551.

Function theory in the unit disk

• Simultaneous zero-free approximation and universal optimal polynomial approximants
  with Catherine Bénéteau, Myrto Manolaki and Daniel Seco
  J. Approx. Theory 256 (2020), 105389.
• Beurling-Carleson sets, inner functions and a semi-linear equation
  with Artur Nicolau
  To appear in Analysis and PDE.

Critical sets of inner functions

• Critical values of inner functions
  with U. Kreitner
  updated Nov. 12, 2023.
• Critical structures of inner functions
  J. Funct. Anal. 281 (2021), 109138.
• Stable convergence of inner functions
  J. London Math. Soc. 102 (2020), 257-286.
• Prescribing inner parts of derivatives of inner functions
  J. d'Analyse Math. 139 (2019), 495-519.

Makarov's principle and related topics

• Sparse Beltrami coefficients, integral means of conformal mappings and the Feynman-Kac formula
  Potential Analysis 48 (2018), no. 4, 437-457.
• 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
  Int. Math. Res. Not. 2019 (2019), no. 5, 1543-1567
• 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

• Critical values of inner functions
  Harmonic and Complex Analysis: Modern and Classical, Ramat Gan, Israel, June 20, 2023
• Entropy of universal covering maps
  Constructive Aspects of Complex Analysis, Honolulu, Hawaii, March 22, 2019
• Describing Blaschke Products by their critical points
  Knoxville, Tennessee, January 22, 2019
• 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