Ilia Binder: Research

Research Interests:

Geometric Function Theory
Complex Analysis and Potential Theory
Complexity and Computability in Analysis and Dynamical Systems
Complex Dynamics
Stochastic Löwner Evolution
Multifractal Analysis
Quasi-periodic and random Shroedinger and Jacobi operators

Papers:

  1. (with L. Richards) “ Power rate of convergence of discrete curves: framework and applications.” Communications in Mathematical Physics, to appear.
  2. (with D. Damanik, M. Goldstein, and M. Lukic) “ Comb domains of Schrödinger operators with small quasi-periodic potentials.” Journal of Spectral Theory, 16(3) (2026), 1143-1196.
  3. (with H. Hakobyan and W. Li) “ Conformal Dimension of the Brownian Graph. ” Duke Mathematical Journal, 174(7) (2025), 1341-1405.
  4. (with Q. He, Z. Li, and  Y. Zhang )“ On computability of equilibrium states.” International Mathematics Research Notices, 2025(6), 1-35.
  5. (With A. Glucksam, C. Rojas, and  M. Yampolsky)“ Computability in Harmonic Analysis.” Foundations of Computational Mathematics, 22(3), pages 849-873 (2022).
  6. (with C. Rojas and M. Yampolsky) “Caratheodory convergence and harmonic measure”, Potential Analysis, 2019, 51(4), 499-509.
  7. (with D. Damanik, M. Goldstein, and M. Lukic) “ Almost periodicity in time of solutions of the KdV equation ”. Duke Mathematical Journal, 2018, Volume 167, Issue 14, pp. 2633 - 2678.
  8. (with D. Damanik, M. Lukic, and T. VandenBoom) “Almost Periodicity in Time of Solutions of the Toda Lattice”. C. R. Math. Rep. Acad. Sci. , 2018, Volume 40(1), pp. 1-28.
  9. (with D. Kinzebulatov and M. Voda) “Non-Perturbative Localization with Quasiperiodic Potential in Continuous Time”. Communications in Mathematical Physics, 2017, Volume 351, Issue 3, pp. 1149-1175.
  10. (with M. Goldstein and M. Voda) “On the Sum of the Non-Negative Lyapunov Exponents for Some Cocycles Related to the Anderson Model”, Ergodic Theory and Dynamical Systems, 2017, Volume 37, Issue 2, pp. 369-388
  11. (with T. Alberts and F. Viklund) “ A Dimension Spectrum for SLE Boundary Collisions”, Communications in Mathematical Physics, 2016, Volume 343, Issue 1, pp 273-298.
  12. (with M. Goldstein and M. Voda)“On fluctuations and localization length for the Anderson model on a strip ”, Journal of Spectral Theory, Volume 5, Issue 1, 2015, pp. 193-225.
  13. (with L. Chayes and  H. K. Lei) “On the Rate of Convergence for Critical Crossing Probabilities”, Annales de l’Institut Henri Poincaré Probabilités et Statistiques, 2015, Vol. 51, No. 2, 672-715.
  14. (with C. Rojas and M. Yampolsky) “Non-Computable impressions of computable external rays of quadratic polynomials”, Communications in Mathematical Physics, Volume 335, Issue 2 (2015), Page 739-757.
  15. (with C. Rojas and M. Yampolsky) “Computable Caratheodory Theory”, Advances in Mathematics, Volume 265, 10 November 2014, Pages 280-312.
  16. (with M. Voda) “On Optimal Separation of Eigenvalues for a Quasiperiodic Jacobi Matrix”, Communications in Mathematical Physics, 2014, Volume 325, Number 3, Pages 1063-1106.
  17. (with M. Voda) “An Estimate on the Number of Eigenvalues of a Quasiperiodic Jacobi Matrix of Size n Contained in an Interval of Size n−C”,  Journal of Spectral Theory, pp. 1-45, Volume 3, Issue 1, 2013.
  18. (with M. Braverman) “The rate of convergence of the Walk on Spheres Algorithm”, Geometric and Functional Analysis: Volume 22, Issue 3 (2012), Page 558-587
  19. (with M. Braverman, C. Rojas, M. Yampolsky) “Computability of Brolin-Lyubich Measure”, Communications in Mathematical Physics, 2011, Volume 308, Number 3, Pages 743-771.
  20. (with L. Chayes and  H. K. Lei) “ On Convergence to SLE6 I: Conformal Invariance for Certain Models of the Bond--Triangular Type”, Journal of Statistical Physics, 141 (2010), no. 2, 359–390.
  21. (with L. Chayes and  H. K. Lei) “ On Convergence to SLE6 II: Discrete Approximations and Extraction of Cardy's Formula for General Domains”, Journal of Statistical Physics, 141 (2010), no. 2, 391–408.
  22. “Phase transition for the universal bounds on the integral means spectrum”, Nonlinearity 22 (2009), no. 8, 1857-1867
  23. (with B. Duplantier)“Harmonic Measure and Winding of Random Conformal Paths: A Coulomb Gas Perspective”, Nuclear Physics B, Volume 802, Issue 3 (2008), p. 494-513
  24. (with M. Braverman and M.Yampolsky)“On the computational complexity of the Riemann mapping”, Arkiv för Matematik, 45 (2007), no. 2, 221 - 239
  25. (with M. Braverman and M.Yampolsky) “Filled Julia sets with empty interior are computable”, Foundations of Computational Mathematics, 7 (2007), no. 4, 405 - 416.
  26. (with M. Braverman and M.Yampolsky) “On computational complexity of Siegel Julia sets”, Communications in Mathematical Physics,  264  (2006), no. 2, 317 - 334
  27. (with N. Makarov and S. Smirnov)“Harmonic measure and polynomial Julia sets”, Duke Mathematical Journal 117 (2003), no. 2, 343 - 365
  28. (with L. DeMarco) “Dimension of pluriharmonic measure and polynomial endomorphisms of Cn”, International Math Research Notes 2003:11 (2003) 613 - 625
  29. (with B. Duplantier) “Harmonic Measure and Winding of Conformally Invariant Curves”, Physical Review Letters 89(26), 264101 (2002).
  30. “A theorem on correction up to gradients of harmonic functions.” St. Petersburg Math. J., 5 (1994), no. 2, 301-315 (translated from Algebra i Analiz 5 (1993), no. 2, 91-107).
  31. “On weak convergence of charges”, Vestnik Leningrad Univ. Math. 24 (1991), no. 1, 13--18.
  32. (with M. Braverman) “The complexity of simulating Brownian Motion”. Proceedings of ACM-SIAM Symposium on Discrete Algorithms (2009), 58 - 67.
  33. (with M. Braverman) “Derandomization of Euclidean Random Walks”, Lecture Notes in Computer Science, Springer Berlin / Heidelberg, Volume 4627/2007, Pages 353-365.
  34. “Rotational spectrum of planar domains”, PhD Thesis, 1997.

Non-refereed papers:

  1. “After Leiden.” CMS Notes, 58(4) (2026), 1-5.
  2. (with D. Damanik, D. Goldstein, M. Lukic, and W. Schlag) “Preface: In Memory of Michael Goldstein.” Journal of Spectral Theory, 16(3) (2026), 983-993.
  3. “My Coxeter.” CMS Notes, 56(4) (2024), 1-2.
  4. “Rate of convergence of critical interfaces to SLE curves.” In Extended Abstracts Fall 2019, Trends in Mathematics: Research Perspectives CRM Barcelona 12, Birkhäuser (2021), 43-50.

Books edited:

  1. (with D. Kinzebulatov and J. Mashreghi, editors) “Function Spaces, Theory and Applications”, 485 pages, Fields Institute Communications (Volume 87), Springer Nature, 2023.
  2. (with D. Kreimer, editor) “Universality and Renormalization: From Stochastic Evolution to Renormalization of Quantum Fields”, 404 pages, Fields Institute Communications (Book 50), AMS, 2007

Preprints:

  1. “Asymptotic for the integral mixed spectrum of the basin of attraction of infinity for the polynomials z(z+ δ)”. Preprint.
  2. “Harmonic measure and rotation of planar domains”. Preprint.