Dmitriy Bilyk

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Dmitriy Bilyk

Professor

School of Mathematics
University of Minnesota
Vincent Hall, office 328
206 Church St. SE
Minneapolis, MN, 55408 USA

E-mail: dbilyk@math.umn.edu

Curriculum Vitae

Publications

Talks

RESEARCH INTERESTS: Harmonic analysis, functional analysis, discrepancy theory.
The research is supported by the NSF grants DMS 0801036 (2008-2011), DMS 1101519 (2011-2015), DMS 1665007 (2017-2020), and the Simons Foundation Collaboration Grant (2016-2021).

PUBLICATIONS (published, accepted, submitted):

Energy on spheres and discreteness of minimizing measures, 2019, submitted. (with A. Glazyrin, R. Matzke, J. Park, O. Vlasiuk)

Optimal measures for p-frame energies on the sphere, 2019, submitted. (with A. Glazyrin, R. Matzke, J. Park, O. Vlasiuk)
General and refined Montgomery lemmata, Mathematische Annalen, 2019, 373 (3-4), 1283-1297. (with F. Dai and S. Steinerberger)

On the Fejes Tóth problem on the sum of acute angles, Proceedings of AMS, 2019, 147 (1), 51-59. (with R. Matzke)
Geodesic distance Riesz energy on the sphere, Transactions of AMS, 2019, 372 (5), 3141-3166. (with F. Dai)

Stolarsky principle and energy optimization on the sphere, Constructive Approximation, 2018, 48 (1), 31-60. (with F. Dai and R. Matzke)
BMO and exponential Orlicz space estimates of the discrepancy function in arbitrary dimension, Journal d'Analyse Math., 2018, 135 (1), 249-269, published online. (with L. Markhasin)
Tusnády's problem, the transference principle and non-uniform QMC sampling, Monte Carlo and Quasi-Monte Carlo Methods 2016, Springer Proceedings in Math. and Stat., 2018, 169-180, published online. (with C. Aistleitner and A. Nikolov)
The two-dimensional small ball inequality and binary nets, Journal of Fourier Analysis and Appl., 2017, 23 (4), 817-833. (with N. Feldheim)
Random tessellations, restricted isometric embeddings, and one bit sensing, 2017, submitted. (with M. Lacey)
One-bit sensing, discrepancy, and the Stolarsky principle, Sbornik Math., 2017, 208 (5-6), 744-763. Russian language version: Mat. Sb., 2017, 208 (6), 4-25. (with M. Lacey)

Estimates of the discrepancy function in exponential Orlicz spaces, 2015, submitted (with G. Amirkhanyan and M. Lacey)
Diophantine approximation and directional discrepancy of rotated lattices, Transactions of AMS, 368, 2016, pp. 3871-3897, published electronically Sep 2015. (with X. Ma, J.Pipher, and C.Spencer)
Chapter Roth's Orthogonal Function Method in Discrepancy Theory and Some New Connections in the book "Panorama of Discrepancy Theory", Lecture Notes in Math 2107 Springer Verlag, 2014. pp. 71-158.
Discrepancy theory and harmonic analysis in the book "Uniform Distribution and Quasi-Monte Carlo Methods", De Gruyter, 2014. pp. 45-62.
Dichotomy results for the L¹ norm of the discrepancy function, Journal of Math Analysis and Applications 410, 2013. (with G. Amirkhanyan and M. Lacey)
The L² discrepancy of irrational lattices, Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proceedings in Math. and Stat. 65, Springer, 2013.
The supremum norm of the discrepancy function: recent results and connections, Monte Carlo and Quasi-Monte Carlo Methods 2012, Springer Proceedings in Math. and Stat. 65 Springer, 2013. (with M. Lacey)
On the scientic work of Konstantin Ilyich Oskolkov, Recent Advances in Harmonic Analysis and Applications, Springer Proceedings in Math. and Stat. 25, Springer, 2013, pp. 3-22. (with L. De Carli, A. Petukhov, A. Stokolos, and B. Wick)
The L² discrepancy of two-dimensional lattices, Recent Advances in Harmonic Analysis and Applications, Springer Proceedings in Math. and Stat. 25, Springer, 2013, pp. 63-78. (with V.Temlyakov and R.Yu)
Fibonacci Sets and Symmetrization in Discrepancy Theory, Journal of Complexity 28, 2013, pp. 18-36. (with V.Temlyakov and R.Yu) Co-winner of 2012 Best Paper Award in the Journal of Complexity.
Directional Discrepancy in Two Dimensions, Bulletin of London Math. Soc. 43, 2011, pp. 1151-1166. (with X. Ma, J.Pipher, and C.Spencer)
Roth’s orthogonal function method in discrepancy theory, Uniform Distribution Theory 6 (2011), no. 1, 143–184.
A three-dimensional signed Small Ball Inequality, “Dependence in Probability, Analysis and Number Theory”, Walter Philipp memorial volume, 73–87, Kendrick Press, Heber City, UT, 2010. (with M. Lacey, I.Parissis, and A.Vagharshakyan)
Cyclic Shifts of the Van der Corput set, Proc. Amer. Math. Soc., 137 (2009), 2591-2600.
Exponential Squared Integrability of the Discrepancy Function in Two Dimensions, Mathematika, 55 (2009), no. 1-2, 1-27. (with M.Lacey, I.Parissis, and A.Vagharshakyan)
Composition of Haar Paraproducts: The Random Case, Analysis Math, 35:1 (2009). (with M.Lacey, X.Li, and B.Wick)
On the Small Ball Inequality in Three Dimensions, Duke Math. J. 143 (2008), no. 1, 81-115. (with M.Lacey)

On the Small Ball Inequality in All Dimensions, J. Funct. Anal. 254 (2008), no. 9, 2470-2502. (with M.Lacey and A.Vagharshakyan)

On the Signed Small Ball Inequality, Online Journal of Analytic Combinatorics, 3 (2008). (with M.Lacey and A.Vagharshakyan)

The interplay between distributional estimates and boundedness in harmonic analysis, The Bulletin of London Math. Soc., 37 (2005) 427-434. (with L. Grafakos)

The distributional estimates for the bilinear Hilbert transform, Journal of Geometric Analysis 16 (2006) 563-583. (with L. Grafakos)

A new way of looking at distributional estimates; applications for the bilinear Hilbert transform, Proceedings of the 7th International Conference on Harmonic Analysis and Partial Differential Equations [El Escorial, 2004], Collectanea Mathematica (2006), 141-169. (with L. Grafakos)

Narrow operators on vector-valued sup-normed spaces, Illinois J. Math., 45 (2002) 120-143. (with V. Kadets, R. Shvidkoy, G. Sirotkin, D. Werner)

Narrow operators the Daugavet property for ultraproducts, Positivity, 9 (2005) 45-62. (with V. Kadets, R. Shvidkoy, D. Werner)

SOME OF MY RECENT TALKS:

Discreteness of minimizing measures, JMM 2020, Dencer, CO.
Discrete minimizers of energies on the sphere, 10th Discrete Geometry and Algebraic Combinatorics Conference, South Padre Island, TX, Sep 2019.
Discrepancy bounds and related questions, Program "Equidistribution: Arithmetic, Computational and Probabilistic Aspects'', IMS, Singapore, May 2019.
Distributing points on a sphere, Conference ``Optimal Point Configurations and Potential Theory'', CIEM, Castro Urdiales, Spain, April 2019.
Tutorial talk Discrepancy Theory', RICAM, Linz, Austria, Nov 2018.
Point distributions and harmonic analysis, 4-lecture series, TU Graz, Austria, October 2018.
Discrepancy and energy optimization on the sphere, 6th International Conference on Uniform Distribution Theory (UDT2018), CIRM, Luminy, France, Sep 2018.
Energy minimization on the sphere, ICM 2018 Satellite Conference Harmonic Analysis, Federal University of Rio Grande do Sul, Porto Alegre, Brazil, July 24-29, 2018.
Plenary talk Point distributions on the sphere: energy minimization, discrepancy, and more, 34th Southeastern Analysis Meeting (SEAM) conference, Georgia Tech, Atlanta, GA, Mar 2018.
Point distributions on the sphere: energy minimization, discrepancy, and more, workshop "Optimal and Random Point Configurations", ICERM, Providence, RI, Feb 2018.
On some problems of uniform distribution and energy minimization on the sphere, , workshop "Discrepancy theory and quasi-Monte Carlo methods", ESI, Vienna, Sep 2017.
Uniform distribution problems on the sphere, Vilnius Conference in Combinatorics and Number Theory, July 2017.
Various uniform distribution problems on the sphere, workshop "Approximation Theory and Function Spaces", CRM, Barcelona, Spain, June 2017.
Discrepancy and energy optimization on the sphere, conference ``Optimal point configurations and orthogonal polynomials 2017", CIEM, Castro Urdiales, Spain, April 2017.
Lacunary Fourier series: from Weierstrass to our days, "HalloWeierstrass" (a special Halloween/Weierstrass birthday talk), University of Minnesota, Oct 31, 2016.
Stolarsky-type identities, energy optimization, uniform tessellations, and one-bit sensing, workshop "Optimal and Random Point Configurations: From Statistical Physics to Approximation Theory", Institut Henri Poincare, Paris, France, Jun-Jul 2016.
Points on a sphere: Stolarsky-type identities, energy optimization, uniform tessellations, and one-bit sensing, workshop on Function Spaces and High-dimensional Approximation, CRM, Barcelona, Catalunya, May 2016.
Tessellations of the sphere, one-bit sensing, restricted isometries, and discrepancy, AMS sectional meeting, Athens, GA, March 2016.
Geometric discrepancy, AIM workshop "Hereditary discrepancy and factorization norms", San Jose, CA, Feb 2016.
Uniform distribution: approximating continuous objects by discrete ones, "Introduction to Research" seminar (an introductory talk for graduate students), University of Minnesota, February 2016.
Methods of harmonic analysis in discrepancy theory (four 1.5-hr lectures), Winter School on Complexity and Discrepancy, Traunkirchen, Austria, Nov-Dec 2015.
Small ball inequality and low discrepancy constructions, "Diophantine approximation and related topics" conference, Aarhus, Denmark, July 2015.
Small ball inequality and low discrepancy constructions, Johannes-Kepler Universität, Linz, Austria, June 2015.
Sphere tessellations and discrepancy, Information-based Complexity program, Banach Center, Będlewo, Poland, April 2015.
Discrepancy theory and analysis, colloquium, Duke University, Durham, NC, February 2015.
The L2 discrepancy of Fibonacci (and other) lattices, workshop on "Optimal Point Configurations and Applications", program on "Minimal Energy Point Sets, Lattices, and Designs", Erwin Schrödinger Institute, Vienna, Austria, October 2014.
Discrepancy function estimates in various function spaces, Analysis Seminar, Brown University, Providence, RI, November 2014.
Tutorial talk Relationship of discrepancy with harmonic analysis, approximation theory and probability, ICERM, Brown University, Providence RI, October 2014.
Uniform distribution, lacunary Fourier series, and Riesz products, Hausdorff Institute, Bonn, Germany, May 2014.
The L1 dichotomy for the discrepancy function estimates, 11th International Conference on Monte-Carlo and Quasi-Monte Carlo Methods in Scientic Computing, MCQMC 2014, Leuven, Belgium, April 2014.
On some distributions with low L2 discrepancy, International conference "Nonlinear approximations and applications" dedicated to 60th birthday of V. Temlyakov, Steklov Institute, Moscow, Russia, October 2013.
Discrepancy theory and harmonic analysis, Johann Radon Institute for Computational and Applied Mathematics (RICAM), Linz, Austria, October 2013.
Small ball inequalities and discrepancy, Oberwolfach workshop "Uniform Distribution Theory and Applications", Oberwolfach, Germany, September 2013.
Discrepancy estimates, small ball inequalities, and hyperbolic cross approximations, workshop "Discrepancy, Numerical Integration and Hyperbolic Cross Approximation", Hausdorff Center for Mathematics, University of Bonn, September 2013.
Discrepancy theory and harmonic analysis, Palanga Conference in Combinatorics and Number Theory, Palanga, Lithuania September 2013.
Plenary lecture Discrepancy theory and analysis, 2nd International Conference "Mathematics in Armenia. Advances and Perspectives", Armenia, August 2013.
Simultaneous Diophantine approximation and rotated lattices, Georgia Analysis Symposium, University of Georgia, Athens, GA, March 2013.
Discrepancy theory and the small ball inequality, Analysis seminar, Universite Paris-Sud XI, Orsay, France, November 2012.

Here are some pictures of my native city, Kharkov