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Alexey Garber

Associate Professor

Ph.D. (2009, Steklov Mathematical Institute, Russian Academy of Sciences)

Contact information:

LHSB 2.518
School of Mathematical & Statistical Sciences
1 West University blvd.
The University of Texas Rio Grande Valley
Brownsville, TX, 78520

tel: +1 956 882 6672

e-mail: alexey.garber@utrgv.edu

General information

Research

I am working on different problems of discrete geometry and combinatorics. My current research interests are parallelohedra theory (in particular, the Voronoi conjecture on parallelohedra), geometrical properties of quasiperiodic point sets (bounded distance and bilipschitz equivalence to a lattice), and measures equipartitions.

You can find some more structured information below, or in my CV.

Teaching

Currently (Fall 2022) I am teaching the following courses.

  • MATH 3352-31R Modern Geometry I;
  • MATH 4390-09 Math Project;
  • MATH 6330-90L Linear Algebra.
Publications and preprints

Last updated in 2022.

Preprints

  1. D. Frettlöh, A. Garber, N. Mañibo, Catalan numbers as discrepancies for a family of substitutions on infinite alphabets, preprint, arXiv:2211.02548, 2022.
  2. D. Frettlöh, A. Garber, N. Mañibo, Substitution tilings with transcendental inflation factor, preprint, arXiv:2208.01327, 2022.
  3. H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane, preprint, arXiv:2204.01076, 2022.
  4. H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, M. Wintraecken, Brillouin Zones of Integer Lattices and Their Perturbations, preprint, arXiv:2204.01077, 2022.
  5. A. Garber, A. Magazinov, Voronoi conjecture for five-dimensional parallelohedra, preprint, arXiv:1906.05193, 2019.
  6. A. Garber, On Helly number for crystals and cut-and-project sets, preprint, arXiv:1605.07881, 2016, accepted to Contemporary Mathematics.

Eternal Preprints

  1. D. Frettlöh, A. Garber, Bounded distance and bilipschitz equivalence of Delone sets, preprint, available at the web-page of Dirk Frettlöh.

Refereed journal publications

  1. A. Garber, A. Magazinov, On the Voronoi conjecture for four- and five-dimensional parallohedra (in Russian), Uspekhi Mat. Nauk, 77:1 (2022), 185--186, DOI:10.4213/rm10020.
  2. A. Garber, On combinatorics of Voronoi polytopes for perturbations of the dual root lattices, preprint, arXiv:2104.07895, to appear in Experimental Mathematics, DOI:10.1080/10586458.2021.1994488, 2021.
  3. D. Frettlöh,, A. Garber, L. Sadun, Number of bounded distance equivalence classes in hulls of repetitive Delone sets, Discrete & Continuous Dynamical Systems, 42:3 (2022), 1403--144, DOI:10.3934/dcds.2021157.
  4. N. Dolbilin, A. Garber, U. Leopold, E. Schulte, M. Senechal, On the regularity radius of Delone sets in R3, Discrete & Computational Geometry, 66 (2021), 996--1024, DOI:10.1007/s00454-021-00292-6.
  5. A. Garber, I. Pak, Concrete polytopes may not tile the space, Mathematika, 66:4 (2020), 920--926, DOI:10.1112/mtk.12052.
  6. M. Dutour-Sikirić, A. Garber, A. Magazinov, On the Voronoi Conjecture for combinatorially Voronoi parallelohedra in dimension 5, SIAM Journal on Discrete Mathematics, 34:5 (2020), 2481--2501, DOI:10.1137/18M1235004.
  7. M. Dutour-Sikirić, A. Garber, Periodic triangulations of Zn, Electronic Journal of Combinatorics, 27:2 (2020), P2.36, DOI:10.37236/8298.
  8. A. Garber, On triangular paperfolding patterns, European Journal of Combinatorics, 89 (2020), DOI:10.1016/j.ejc.2020.103167.
  9. A. Garber, E. Roldán-Pensado, On a Helly-type question for central symmetry, Periodica Mathematica Hungarica, 79:1 (2019), 78--85, DOI:10.1007/s10998-018-0263-y.
  10. D. Frettlöh, A. Garber, Weighted 1×1 cut-and-project sets in bounded distance to a lattice, Discrete & Computational Geometry, 64:3 (2019), 649--661, DOI:10.1007/s00454-018-0005-1.
  11. I. A. Baburin, M. Bouniaev, N. Dolbilin, N. Yu. Erokhovets, A. Garber, S. V. Krivovichev, E. Schulte, On the Origin of Crystallinity: a Lower Bound for the Regularity Radius of Delone Sets, Acta Crystallographica Section A, A74:6 (2018), 616--629, DOI:10.1107/S2053273318012135.
  12. D. Frettlöh, A. Garber, Pisot substitution sequences, one dimensional cut-and-project sets and bounded remainder sets with fractal boundary, Indagationes Mathematicae, 29:4 (2018), 1114--1130, DOI:10.1016/j.indag.2018.05.012.
  13. A. Garber, On π-surfaces of four-dimensional parallelohedra, Annals of Combinatorics, 21:4 (2017), 551--572, DOI:10.1007/s00026-017-0366-9
  14. M. Dutour-Sikirić, A. Garber, A. Schürmann, C. Waldmann, The complete classification of five-dimensional Dirichlet-Voronoi polyhedra of translational lattices, Acta Crystallographica A72 (2016), 673--683, DOI:10.1107/S2053273316011682.
  15. D. Frettlöh, A. Garber, Symmetries of Monocoronal Tilings, Discrete Mathematics & Theoretical Computer Science, 17:2 (2015), 203--234, the paper is available at the journal's web-site .
  16. A. Balitskiy, A. Garber, R. Karasev, Another ham sandwich in the plane, Annals of Combinatorics, 19:2 (2015), DOI:10.1007/s00026-015-0270-0.
  17. A. Garber, A. Gavrilyuk, A. Magazinov, The Voronoi conjecture for parallelohedra with simply connected δ-surface, Discrete & Computational Geometry, 53:2 (2015), DOI:10.1007/s00454-014-9660-z.
  18. A. Garber, Belt diameter of Π-zonotopes, European Journal of Combinatorics, 34:5 (2013), 923--933. DOI:10.1016/j.ejc.2013.01.005.
  19. A.I. Garber, Belt distance between facets of space-filling zonotopes, Mathematical Notes, 92:3-4 (2012), 345--355. DOI:10.1134/S0001434612090064.
  20. A. Garber, The second Voronoi conjecture on parallelohedra for zonotopes, Moscow Journal of Combinatorics and Number Theory, 1:2 (2011), 113--119. The paper on MJCNT web-site.
  21. A.I. Garber, On equivalence classes of separated nets (in Russian), Modeling and Analysis of Information Systems, 16:2 (2009), 109--118. The paper on MathNet web-site.
  22. A.I. Garber, Graphs of linear operators, Proceedings of the Steklov Institute of Mathematics, 263 (2008), 57--64. DOI:10.1134/S0081543808040056.
  23. A.I. Garber, Graph of difference operator for p-ary sequences, Functional Analysis and Other Mathematics, 1:2 (2006), 159--173. DOI:10.1007/s11853-007-0011-y.

Non-refereed publications and proceedings

  1. A.I. Garber, Complicated sequences due to V.I. Arnold (in Russian), Materials of the 9th International Seminar “Discrete Mathematics and its Applications” devoted to 75th Anniversary of the academician O.B. Lupanov, 2007, 374--376.
  2. A.I. Garber, A.P. Poyarkov, On permutahedra (in Russian), Vestnik MGU, ser. 1, iss. 2 (2006), 3--8.
  3. A.I. Garber, A.A. Glazyrin, Rigidity of some classes of cubillages, Proceedings of Second COE Workshop on Sphere Packings, May 30 - June 5, 2005, pp. 86--89.
  4. A.I. Garber, A.P. Poyarkov, On permutahedra, Voronoi's Impact in Modern Science. Book 3: Proceedings of Voronoi Conference on analytic number theory and spatial tesselations, Kyiv, Institute of Mathematics, 2005, pp. 137--145.
Grants and awards

Grants and awards

  • 2020-present. Alexander von Humboldt Foundation Fellowship for experienced researchers.
  • 2019-present. NSF Conference grant DMS-1904635, PI.
  • 2018-present. SQuaREs program “Delone Sets: Local Rules in Crystalline Structures” at American Institute of Mathematics, San Jose, CA.
  • 2016-2018. NSF Conference grant DMS-1623600, co-PI.
  • 2014. “Dynasty” Foundation grant for young mathematicians.
  • October-December, 2013. Winner of joint DAAD (German Academic Exchange Service) and Moscow State University Scholarships Program “Vladimir Vernadskii” for joint research in Germany.
  • 2011-2013. Russian Federal Program “Kadry”. Co-PI of the project 16.740.11.0568 “Functional Analysis and Discrete Geometry”.
  • 2011-2013. Participation in Russian Government “Megagrant” program in the project 11.G34.31.0053 as research fellow of Delone Laboratory of Discrete and Computational Geometry.
  • 2010-2014. Russian Government Program for the Support of Leading Scientific Schools (grants ??-5413.2010.1 and ??-4995-2012.1, member of research team).
  • 2008-2014. Participation in several research projects funded by the Russian Foundation for Basic Research. RFBR grants 08-01-00565 (project in 2008-2010, member of research team), 08-01-91202 (joint Russian-Japanese RFBR-JSPS project in 2008-2009, member of research team), 11-01-00633 (project in 2011-2013, member of research team), 11-01-00735 (project in 2011-2013, member of research team).
Visiting researcher
  • February 23 - March 9, 2006, Institute of Statistical Mathematics, Tokyo, Japan.
  • March 18 - April 1, 2009, Institute of Statistical Mathematics, Tokyo, Japan.
  • November 16 - December 12, 2009, Collaborative Research Center 701 of Bielefeld University, Bielefeld, Germany.
  • September 26 - October 31, 2010, Centre Interfacultaire Bernoulli, Ecole Polytechnique Federal de Lausanne, Switzerland.
  • September 29 - October 26, 2011, Queen's University, Kingston, ON, Canada.
  • September 25 - October 10, 2012, Queen's University, Kingston, ON, Canada.
  • October 30 - December 1, 2012, Collaborative Research Center 701 of Bielefeld University, Bielefeld, Germany.
  • October 15 - December 14, 2013, Research Center for Mathematical Modelling of Bielefeld University, Bielefeld, Germany.
  • January 6 - February 1, 2014, Collaborative Research Center 701 of Bielefeld University, Bielefeld, Germany.
  • May 5 - May 14, 2014, Rostock University, Rostock, Germany.
  • June 19 - July 2, 2016, Collaborative Research Center 701 of Bielefeld University, Bielefeld, Germany.
  • February 1 - April 30, 2021, Visiting Professor at IST Austria, Klosterneuburg, Austria.
  • May 16 - August 20, 2021, Alexander von Humboldt Fellow at Bielefeld Univesity, Germany.
  • May 23 - August 25, 2022, Alexander von Humboldt Fellow at Bielefeld Univesity, Germany.
Selected presentations
Software, etc.
If any of the links is not working, please contact me by email alexey.garber@utrgv.edu.
  • scc 2.0 - Secondary Cone Cruiser. SCC is an implementation of Voronoi's reduction theory. It allows to cruise through the secondary cones of Delone triangulation (also known as L-type domains). More information at SCC guide link.
  • Supporting computations for the paper establishing 10R upper bound for the regularity radius in the three-dimensional space are available at the following link.
  • Supporting computations for the paper verifying GGM condition for all five-dimensional Voronoi parallelohedra are available at the following link.