Research

I am working on different problems of discrete and combinatorial geometry. My current research interests are parallelohedra theory (in particular, the Voronoi conjecture on parallelohedra), geometrical properties of quasicrystals (wobbling and bilipschitz equivalence to a lattice), and measures equipartitions.

List of publications and preprints in reverse chronological order

1. D. Frettlöh, A. Garber, Bilipschitzequivalence and wobbling equivalence of Delone sets, preprint.
2. D. Frettlöh, A. Garber, Symmetries of Monocoronal Tilings, prepint, arXiv:1402.4658, 2014.
3. A. Garber, On π-surfaces of four-dimensional parallelohedra, prepint, arXiv:1309.7661, 2013.
4. A. Balitskiy, A. Garber, R. Karasev, Another ham sandwich in the plane, preprint, arXiv:1307.7698, 2013.
5. A. Garber, A. Gavrilyuk, A. Magazinov, The Voronoi conjecture for parallelohedra with simply connected δ-surface, preprint, arXiv:1212.1019, 2012.
6. A. Garber, Belt diameter of Π-zonotopes, European Journal of Combinatorics, 34:5 (2013), 923--933. DOI:10.1016/j.ejc.2013.01.005.
7. A.I. Garber, Belt distance between facets of space-filling zonotopes, Mathematical Notes, 92:3-4 (2012), 345--355. DOI:10.1134/S0001434612090064.
8. 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.
9. 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.
10. A.I. Garber, Graphs of linear operators, Proceedings of the Steklov Institute of Mathematics, 263 (2008), 57--64. DOI:10.1134/S0081543808040056.
11. A.I. Garber, Complicated sequences due to V.I. Arnold (in Russian), Materials of the 9th International Seminar “Discrete Mathematics and its Application” devoted to 75th Anniversary of the academician O.B. Lupanov, 2007, 374--376.
12. 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.
13. A.I. Garber, A.P. Poyarkov, On permutahedra (in Russian), Vestnik MGU, ser. 1, iss. 2 (2006), 3--8.
14. 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.
15. 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

• 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-present. 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-present. Russian Government Program for the Support of Leading Scientific Schools (grants НШ-5413.2010.1 and НШ-4995-2012.1, member of research team).
• 2008-present. 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

• 18.03-01.04.2009, Institute of Statistical Mathematics, Tokyo, Japan.
• 16.11-12.12.2009, Collaborative Research Center 701 of Bielefeld University, Bielefeld, Germany.
• 26.09-31.10.2010, Centre Interfacultaire Bernoulli, Ecole Polytechnique Federal de Lausanne, Switzerland.
• 29.09-26.10.2011, Queen's University, Kingston, ON, Canada.
• 25.09-18.10.2012, Queen's University, Kingston, ON, Canada.
• 30.10-01.12.2012, Collaborative Research Center 701 of Bielefeld University, Bielefeld, Germany.
• 15.10-14.12.2013, Research Center for Mathematical Modelling of Bielefeld University, Bielefeld, Germany.

Selected presentations

• “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.
• “Parallelohedra and the Voronoi Conjecture” (partially joint work with Andrey Gavrilyuk and Alexander Magazinov from Steklov Mathematical Institute), November 28, 2012, Collaborative Research Center 701 “Spectral Structures and Topological Methods in Mathematics” Seminar, Bielefeld University, Bielefeld, Germany.
• “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.