Alexey Garber

Research

I am working on different problems of discrete and computational geometry and combinatorics in general. My main interests are polytopes and related tilings and patterns, and their combinatorial and geometric properties.

Currently my research is partially supported by Simons Foundation.

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

Teaching

Currently (Spring 2025) I am teaching the following courses.

• MATH 4390-90 Math Project

Last updated in 2025.

Preprints

1. H. Edelsbrunner, A. Garber, M. Saghafian, On Spheres with $k$ points inside, preprint, arXiv:2410.21204, 2024, accepted to SoCG 2025.
2. A. Garber, Ž. Virk, N. Zava, On the metric spaces of lattices and periodic point sets, preprint, arXiv:2310.07594, 2023.
3. H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, Flips in Two-dimensional Hypertriangulations, preprint, arXiv:2212.11380, 2022.

Eternal Preprints

• 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, Voronoi conjecture for five-dimensional parallelohedra, Inventiones Mathematicae, (2025), DOI:10.1007/s00222-025-01325-0. Initial versions of this paper were written together with Alexander Magazinov. Unfortunately, he decided to step down as an author of the paper due to personal circumstances.
2. H. Edelsbrunner, A. Garber, M. Saghafian, Order-2 Delaunay Triangulations Optimize Angles, Advances in Mathematics, 461 (2025), 110055, DOI:10.1016/j.aim.2024.110055.
3. D. Frettlöh, A. Garber, N. Mañibo, Substitution tilings with transcendental inflation factor, arXiv:2208.01327, Discrete Analysis, 2024:11, DOI:10.19086/da.125449.
4. A. Garber, On Helly number for crystals and cut-and-project sets, Studia Scientiarum Mathematicarum Hungarica, 61:3 (2024), 203--214, DOI:10.1556/012.2024.04314.
5. N. Dolbilin, A. Garber, E. Schulte, M. Senechal, Bounds for the Regularity Radius of Delone Sets, preprint, arXiv:2306.11127, 2023, to appear in Discrete & Computational Geometry, DOI:10.1007/s00454-024-00666-6.
6. E. Bajo, R. Davis, J.A. De Loera, A. Garber, S. Garzón Mora, K. Jochemko, J. Yu, Weighted Ehrhart Theory: Extending Stanley's nonnegativity theorem, Advances in Mathematics, 444 (2024), 109627, DOI:10.1016/j.aim.2024.109627.
7. H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, M. Wintraecken, Brillouin Zones of Integer Lattices and Their Perturbations, SIAM Journal on Discrete Math, 38:2 (2024), 1784--1807, DOI:10.1137/22M1489071.
8. H. Edelsbrunner, A. Garber, M. Ghafari, T. Heiss, M. Saghafian, On Angles in Higher Order Brillouin Tessellations and Related Tilings in the Plane, Discrete & Computational Geometry, 72 (2024), 29--48, DOI:10.1007/s00454-023-00566-1.
9. D. Frettlöh, A. Garber, N. Mañibo, Catalan numbers as discrepancies for a family of substitutions on infinite alphabets, Indagationes Mathematicae, 35:5 (2024), 890--913, DOI:10.1016/j.indag.2023.06.010.
10. 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.
11. A. Garber, On combinatorics of Voronoi polytopes for perturbations of the dual root lattices, Experimental Mathematics, 33:1 (2024), 86--99, DOI:10.1080/10586458.2021.1994488.
12. 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--1414, DOI:10.3934/dcds.2021157.
13. 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.
14. A. Garber, I. Pak, Concrete polytopes may not tile the space, Mathematika, 66:4 (2020), 920--926, DOI:10.1112/mtk.12052.
15. 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.
16. M. Dutour-Sikirić, A. Garber, Periodic triangulations of Zⁿ, Electronic Journal of Combinatorics, 27:2 (2020), P2.36, DOI:10.37236/8298.
17. A. Garber, On triangular paperfolding patterns, European Journal of Combinatorics, 89 (2020), DOI:10.1016/j.ejc.2020.103167.
18. 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.
19. 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.
20. 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.
21. 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.
22. A. Garber, On π-surfaces of four-dimensional parallelohedra, Annals of Combinatorics, 21:4 (2017), 551--572, DOI:10.1007/s00026-017-0366-9
23. 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.
24. 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 .
25. 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.
26. 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.
27. A. Garber, Belt diameter of Π-zonotopes, European Journal of Combinatorics, 34:5 (2013), 923--933. DOI:10.1016/j.ejc.2013.01.005.
28. A.I. Garber, Belt distance between facets of space-filling zonotopes, Mathematical Notes, 92:3-4 (2012), 345--355. DOI:10.1134/S0001434612090064.
29. 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.
30. 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.
31. A.I. Garber, Graphs of linear operators, Proceedings of the Steklov Institute of Mathematics, 263 (2008), 57--64. DOI:10.1134/S0081543808040056.
32. 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

• 2024-2029. Simons Foundation travel support for mathematicians.
• 2020-2023. Alexander von Humboldt Foundation Fellowship for experienced researchers.
• 2019-2024. 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).

• 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.
• May 22 - August 24, 2023, Alexander von Humboldt Fellow at Bielefeld Univesity, Germany.

• “On spheres with k points inside” (joint work with Herbert Edelsbrunner and Morteza Saghafian from IST Austria), Conference “Combinatorics and Geometry in Ioannina”, September 5, 2024, Ioannina, Greece.
• “On the metric spaces of lattices and periodic point sets” (based on a joint work with Žiga Virk from University of Ljubljana and Nicolò from IST Austria), MIF++ seminar, December 12, 2023, Liverpool (online), UK.
• “Counting tiles in substitution tilings” (joint work with Dirk Frettlöh and Neil Mañibo from Bielefeld University), 36th Summer Topology Conference, July 19, 2022, University of Vienna, Vienna, Austria.
• “Concrete polytopes may not tile the space” (joint work with Igor Pak from UCLA), Discrete Geometry Seminar, June 24, 2022, Freie Universität, Berlin, Germany.
• “The Voronoi conjecture”, Discrete Geometry Seminar, July 15, 2021, Freie Universität, Berlin, Germany.
• “Helly numbers for crystals and cut-and-project sets”, “Helly and Tverberg type Theorems” workshop, October 7, 2019, Casa Matematica Oaxaca, Oaxaca, Mexico.
• “Voronoi conjecture for parallelohedra” (based on joint works with Mathieu Dutour-Sikirić from Rudjer Bosković Institute, Andrey Gavrilyuk from Higher School of Economics, Alexander Magazinov from Yandex, Achill Schürmann from Rostock University, and Clara Waldmann from TU Munich), “Soft Packings, Nested Clusters, and Condensed Matter” workshop, October 1, 2019, Casa Matematica Oaxaca, Oaxaca, Mexico.
• “Measuring weighted cut-and-project sets” (joint work with Dirk Frettlöh from Bielefeld University), AMS Eastern Sectional Meeting, April 21, 2018, Northeastern University, Boston, MA, USA.
• “Enumeration of five-dimensional Dirichlet-Voronoi parallelohedra” (joint work with Mathieu Dutour-Sikirić from Rudjer Bosković Institute, Achill Schürmann from Rostock University, and Clara Waldmann from TU Munich), AMS Eastern Sectional Meeting, November 14, 2015, Rutgers University, New Brunswick, NJ, USA.
• “Parallelohedra and the Voronoi Conjecture” (partially joint work with Andrey Gavrilyuk and Alexander Magazinov from Steklov Mathematical Institute), November 28, 2013, Discrete Geometry Seminar, Free University Berlin, Berlin, Germany.
• “Another ham sandwich in the plane” (joint work with Alexey Balitskiy and Roman Karasev from Moscow Insitute of Physics and Technology), November 9, 2013, Kolloquium über Kombinatorik, TU Ilmenau, Ilmenau, Germany.
• “Tilings with unique vertex corona” (joint work with Dirk Frettlöh from Bielefeld Univesity), April 18, 2013, Third Texas Southmost Geometry and Topology Conference, University of Texas at Brownsville, Brownsville, TX, USA.
• “Belt diameter of some class of space-filling zonotopes”, May 22, 2012, International Topological Conference “Alexandroff Readings”, Moscow State University, Moscow, Russia.
• “Quasicrystals, bi-Lipschitz equivalence and bounded movement” (joint work with Dirk Frettlöh from Bielefeld Univesity), April 17, 2010, Third Texas Southmost Geometry and Topology Conference, University of Texas at Brownsville, Brownsville, TX, USA.

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.