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Regular version of the site

REU (Research Experience for Undergraduates) in Mathematics

This program is a part of the HSE Summer University and is designed for undergraduate students majoring in Mathematics or related areas. Participants will work on research projects under supervision of distinguished mathematicians. Guaranteed are at least 8 hours of consultations by assigned advisors and their assistants. A written report is expected as a result of the project; it will be reviewed and evaluated by the advisor. The program takes three weeks during the period from June 20 till August 20, 2017. The exact time span of the program is to be chosen by the participant and confirmed by the advisor.

Some projects offered by the Faculty of Mathematics (more to be announced):

Chris Brav, Linear programming and fundamental domains of complex hyperbolic groups
Sergey Lando: Invariants of Graphs and Knots
Sergey Natanzon: Formal Solutions of Integrable Systems
Alexandra Skripchenko: Hausdorff Dimension in Dynamics and Probability
Alexey Gorodentsev: Diophantine Aproximations and Semiorthogonal Geometry
Satoshi Kondo: p-adic Numbers

Alexey Zykin: Number theoretic constructions of dense sphere packing

Sphere packings, lattices, codes, and number theory (2 ECTS credits, 16 academic hours, HSE Moscow)
Dates: July, 3 - 16

Alexey Zykin

Research Fellow: Laboratory of Algebraic geometry and its Applications,
Professor: University of French Polynesia

 

The problem of finding maximally dense sphere packings in the n dimensional Euclidian space has been attracting the efforts of many mathematicians since the ancient times. The question is surprisingly difficult. For instance, a complete answer in the case of dimension 3 was obtained only in 1998 by Thomas Hales who succeeded in proving the conjecture on the optimality of the cannon ball packing, formulated by Johannes Kepler as early as in 1611. The corresponding problem in dimension 4 still remains unsolved.

In 2016 Maryna Viazovska made a spectacular breakthrough by proving the optimality of the  lattice packing in, as well as of the Leech lattice packing  in, the last result being a joint work with H. Cohn, A. Kumar, S. D. Miller, and D. Radchenko. The proofs, in which methods from the theory of modular forms are used, are remarkably short and accessible, especially compared to the 300 pages proof of the Kepler’s conjecture.

The so called error correcting codes can be regarded as a finite analogue of sphere packings. Apart from their applications to the information transmission problems, the mathematical methods used to study them turn out to be remarkably rich and beautiful, ranging from combinatorics and analysis to number theory and algebraic geometry. Moreover, methods and results of the theory of error correcting codes turn out to have many applications to the sphere packing problem.

The main goal of the course is to provide an introduction to the theory of error correcting codes, lattices, and sphere packings by showing the wide variety of mathematical techniques used therein, with a particular accent on the modern number theoretical tools that seem to give the strongest results.



Karine Kuyumzhiyan: Toric geometry

Note that some advisors may have time restrictions. Applicants may also request a specific topic or a specific advisor (chosen among the HSE Math faculty members); we will do our best to find a match.

Please check “Research Experience for Undergraduates in Mathematics”.
Please address questions to vtimorin@hse.ru (Vladlen Timorin, Dean of the Faculty of Mathematics).