## Ilia Binder: Research

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

- [1] (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. - (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. - (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,*Volume 37, Issue 2, pp. 369-388 - (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. - (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. - (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. - (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. - (with C. Rojas and M. Yampolsky) “Computable Caratheodory Theory”,
*Advances in Mathematics,*Volume 265, 10 November 2014, Pages 280-312. - (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*. - (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. - (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* - (with M. Braverman, C. Rojas, M. Yampolsky) “Computability of Brolin-Lyubich Measure”,
*Communications in Mathematical Physics, 2011,**Volume 308, Number 3**, Pages 743-771*. - (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, 359–390*.* - (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, 391–408*.* - “Phase transition for the universal bounds on the integral means spectrum”,
*Nonlinearity*22 (2009), 1857-1867 - (with B. Duplantier)“Harmonic Measure and Winding of Random Conformal Paths: A Coulomb Gas Perspective”,
*Nuclear Physics B*, Volume**802**, Issue 3, p. 494-513 - (with M. Braverman and M.Yampolsky)“On computational complexity of Riemann mapping”,
*Arkìv för matematik,***45**(2007), no. 2, 221 - 239 - (with M. Braverman and M.Yampolsky) “Filled Julia sets with empty interior are computable”,
*Foundations of Computational Mathematics*,**7**(2007), no. 4, 405 - 416. - (with M. Braverman and M.Yampolsky) “On computational complexity of Siegel Julia sets”,
*Communications in Mathematical Physics,***264**(2006), no. 2, 317 - 334 - (with N. Makarov and S. Smirnov)“Harmonic measure and polynomial Julia sets”,
*Duke Mathematical Journal***117**(2003), no. 2, 343 - 365 - (with L. DeMarco) “Dimension of pluriharmonic measure and polynomial endomorphisms of
**C**^{n}”,*International Math Research Notes***2003:11**(2003) 613 - 625 - (with B. Duplantier) “Harmonic Measure and Winding of Conformally Invariant Curves”,
*Physical Review Letters***89**, 264101 (2002)*.* - “A theorem on correction up to gradients of harmonic functions.” St.-Petersburg Math. Journal, 5 (1994), no. 2, 91--107.
- “On weak convergence of charges”, Vestnik Leningrad Univ. Math.
**24**(1991), no. 1, 13--18. - (with C. Rojas and M. Yampolsky) “Caratheodory convergence and harmonic measure”, To appear in
*Potential Analysis*. - (with D. Damanik, M. Goldstein, and M. Lukic) “ Almost periodicity in time of solutions of the KdV equation ”. To appear in
*Duke Mathematical Journal*. - (with M. Braverman) “The complexity of simulating Brownian Motion”.
*Proceedings of ACM-SIAM Symposium on Discrete Algorithms*(2009), 58 - 67. - (with M. Braverman) “Derandomization of Euclidean Random Walks”, Lecture Notes in Computer Science, Springer Berlin / Heidelberg, Volume
**4627**/2007, Pages 353-365. - “Rotational spectrum of planar domains”, PhD Thesis, 1997.
- (with D. Kreimer, editor) “Universality and Renormalization: From Stochastic Evolution to Renormalization of Quantum Fields”, 404 pages, Fields Institute Communications (Book 50), AMS, 2007
- “Asymptotic for the integral mixed spectrum of the basin of attraction of infinity for the polynomials
*z(z+ δ)*”. Preprint. - “Harmonic measure and rotation of planar domains”. Preprint.