István Prause

Professor, Åbo Akademi University, Finland

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Gallery

A gallery of 3D limit shapes (to be updated time to time)

These were produced by the "tangent plane method" introduced in paper [3], and further developed in [1,2] below. Limit shape formation is a general phenomenon which says that random discrete surfaces converge (on a macroscopic scale) to a continuous deterministic limit surface.

See also Alexei & Matvey Borodin's page for 3D models for discrete surfaces sampled randomly.

Selected talks

Limit shapes

• Random Young tableaux and the tangent plane method, Galileo Galilei Institute, Firenze, Italy, May 2022 (video) & Probability and Mathematical Physics, ICM Satellite conference, June 2022 (video)

• Integrability of limit shapes, Solvable lattice models seminar, November 2021 (video)

• Probabilistic limit shapes and harmonic functions, CAvid seminar, October 2021 (video)

• Arctic curves beyond the arctic circle, 6th Workshop on Fourier Analysis and Related Fields, Pécs, Hungary, August 2017

 Holomorphic interpolation

• Schwarz lemma and quasiconformal mappings, Analysis aspects of dynamics, Imperial College, London, UK, May 2018

• Holomorphic interpolation: from dynamics to analysis, Topics in Analysis and Holomorphic Dynamics, Simons Semester in Banach Center, Warsaw, Poland, September 2015

• Burkholder integrals, Morrey's problem and quasiconformal mappings, Ahlfors-Bers Colloquium, Houston, USA, March 2011

Multifractal spectra

• Quasidisks and twisting of the Riemann map, Analysis and Dynamics, BIRS-CMO Workshop, Oaxaca, Mexico, September 2016 (video)

• Asymptotic variance of the Beurling transform, Workshop on Conformal Methods in Analysis, Random Structures and Dynamics, Chales, France, April 2015

• Area distortion, quasicircles and harmonic measure, Quasiconformal Geometry and Elliptic PDEs, IPAM, Los Angeles, USA, May 2013 (video)

• Bilipschitz rotation versus quasiconformal stretching, Harmonic and Complex Analysis and its Applications (HCAA), Puerto de la Cruz, Tenerife, Spain, March 2012

Publications and preprints

Submitted

• Dimer models and conformal structures (with Kari Astala, Erik Duse and Xiao Zhong), arXiv:2004.02599, preprint 2023

• Asymptotics of Bounded Lecture-Hall Tableaux (with David Keating and Zhongyang Li), arXiv:2309.15235, preprint 2023

Published papers (incl. accepted)

1. Limit shapes from harmonicity: dominos and the five vertex model (with Rick Kenyon), J. Phys. A: Math. Theor. 57 035001, 2024, arXiv:2310.06429

2. The genus-zero five-vertex model (with Rick Kenyon), Probability and Mathematical Physics, Vol. 3 (2022), No. 4, 707–729, arXiv:2101.04195

3. Gradient variational problems in R2 (with Rick Kenyon), Duke Math. J. 171 (14), 2022, 3003 - 3022, arXiv:2006.01219

4. On mappings of finite distortion that are quasiconformal in the unit disk (with Olli Hirviniemi, Lauri Hitruhin and Eero Saksman), J. Anal. Math., 2022,  arXiv:2104.10961

5. Localized regularity of planar maps of finite distortion (with Olli Hirviniemi and Eero Saksman), Rev. Mat. Iberoam. 38 (2), 2022, 615--634, arXiv:2001.04173

6. Stretching and rotation of planar quasiconformal mappings on a line (with Olli Hirviniemi and Eero Saksman), Potential Anal., arXiv:2007.07735, 2021

7. Quasidisks and twisting of the Riemann map, Int. Math. Res. Not. IMRN (7), 2019, 1936--1954, arXiv:1611.06734

8. Quasiconformal extensions, Loewner chains, and the lambda-Lemma (with Pavel Gumenyuk), Anal. Math. Phys. (8), 2018, 621--635, A. Vasiliev Memorial Volume, arXiv:1712.08632

9. Asymptotic variance of the Beurling transform (with Kari Astala, Oleg Ivrii and Antti Perälä), GAFA 25 (6), 2015, 1647--1687, arXiv:1502.00459

10. A hunt for sharp L^p-estimates and rank-one convex variational integrals (with Kari Astala, Tadeusz Iwaniec and Eero Saksman), Filomat 29 (2), 2015, 245--261, arXiv:1403.1095

11. On a Hölder constant in the theory of quasiconformal mappings, Comput. Methods. Funct. Theory 14, 2014, 483--486, F.W. Gehring Memorial Volume, arXiv:1309.6748

12. Bilipschitz and quasiconformal rotation, stretching and multifractal spectra (with Kari Astala, Tadeusz Iwaniec and Eero Saksman), Publ. Math. Inst. Hautes Études Sci. 121 (1), 2015, 113--154, arXiv:1303.2325

13. Burkholder integrals, Morrey's problem and quasiconformal mappings (with Kari Astala, Tadeusz Iwaniec and Eero Saksman), J. Amer. Math. Soc. 25 (2), 2012, 507--531, arXiv:1012.0504

14. Hausdorff measure of quasicircles (with Xavier Tolsa and Ignacio Uriarte-Tuero), Adv. Math. 229 (2), 2012, 1313--1328, arXiv:0912.3365

15. Quasisymmetric distortion spectrum (with Stas Smirnov), Bull. London Math. Soc. 43 (2), 2011, 267--277, arXiv:0910.4723

16. Flatness properties of quasispheres, Comput. Methods Funct. Theory 7, 2007, 527–541.

17. On distortion of Hausdorff measures under quasiconformal mappings, Conform. Geom. Dyn. 11, 2007, 219--223.

18. A remark on quasiconformal dimension distortion on the line, Ann. Acad. Sci. Fenn. Math. 32, 2007 no. 2, 341--352.