Nikolay Petrovich Dolbilin
Steklov Mathematical Institute of the Russian Academy of Sciences
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Talk title
Delone Sets: From Local to Global
Abstract
In the talk, we present the local theory of regular systems, motivated by a fundamental question concerning the origin of crystalline order. This theory was initiated half a century ago by Delone’s geometric school at the Steklov Institute.
The atomic structure of an ideal crystal is modeled by a Delone set with a crystallographic symmetry group. A remarkable feature is the emergence of this order during crystallization, when atoms are rearranged from a disordered state of a liquid into a highly organized crystal structure.
Physicists explain this phenomenon as follows. During crystallization, atoms tend to surround themselves with clusters that minimize the internal energy. Clearly, the energy-minimizing clusters around identical atoms should be the same. And finally, the decisive argument (R. Feynman): the repetition of clusters throughout the structure leads to its global symmetry. This argument seemed to be obvious and did not need proof. However, the aperiodic Penrose tilings (1974) and the discovery of quasicrystals by D. Shechtman (1982) showed that the recurrence of clusters in a Delone set does not always lead to a crystallographic symmetry group.
Any Delone set with crystallographic symmetry is a finite union of regular systems. A regular system is defined as an orbit of a point under a crystallographic group. One of the main goals of the local theory of regular systems is to find local geometric conditions guaranteeing that a Delone set is a regular system.
Given a Delone set X ⊂ ℝᵈ, x ∈ X, and ρ > 0, a ρ-cluster Cₓ(ρ) at x is the set of all points of X whose distance from x does not exceed ρ. Two clusters Cₓ(ρ) and Cₓ′(ρ) are equivalent if there is an isometry g such that g(x) = x′ and g(Cₓ(ρ)) = Cₓ′(ρ). The regularity radius is the smallest number ρ̂d > 0 such that every Delone set whose ρ̂d-clusters are pairwise equivalent is a regular system.
We plan to discuss the local criterion for regular systems, which provides an upper bound for ρ̂d; in the plane, ρ̂₂ = 4R, where R is the covering radius; and in 3-space, 6R ≤ ρ̂₃ ≤ 10R.
Biography
Nikolay Dolbilin (b. 1943), Doctor of Sciences and Professor, graduated from Moscow State University in 1965. He joined the Steklov Mathematical Institute in 1966 for postgraduate study under B. Delaunay. Since 1969, he has worked in its Department of Geometry, and since 1984 in the Geometry and Topology department, where he is a Leading Research Fellow.
His interests are discrete geometry, combinatorial topology, and applications to crystallography and statistical physics. Principal results include the theory of lattice coverings of space by equal balls, the local criteria for regular point systems and isohedral tilings, the local theory of regular systems, and, with coauthors, the first rigorous proof of the celebrated Kac–Ward determinant formula for the partition function of the two-dimensional Ising model.
In mathematics education and popularization, he served on the Executive Committee of the International Commission on Mathematical Instruction (ICMI, 2003–2006) and the International Programme Committee of ICME-10 (2004). He was an invited speaker at ICME-9 (2000) in Japan and shared a Russian Academy of Sciences gold medal for the popularization of science.