Volume 1 (2001), Number 1. Abstracts

M. Blank. Dynamics of Traffic Jams: Order and Chaos [PDF]

By means of a novel variational approach, we study the ergodic properties of a model of a multi lane traffic flow, considered as a (deterministic) wandering of interacting particles on an infinite lattice. For a class of initial configurations of particles (roughly speaking satisfying the Law of Large Numbers) the complete description of their limit behaviour (in time) is obtained, as well as estimates of the transient period. In this period the main object of interest is the dynamics of `traffic jams', which is rigorously defined and studied. It is shown that the dynamical system under consideration is chaotic in the sense that its topological entropy (calculated explicitly) is positive. Statistical quantities describing limit configurations are obtained as well.

Keywords. Traffic flow, dynamical system, variational principle, topological entropy

2000 Mathematics Subject Classification. Primary 37B99; Secondary 37B15, 37B40, 37A60, 60K

J. Guckenheimer and Yu. Ilyashenko. The Duck and the Devil: Canards on the Staircase [PDF]

Slow-fast systems on the two-torus $T^2$ provide new effects not observed for systems on the plane. Namely, there exist families without auxiliary parameters that have attracting canard cycles for arbitrary small values of the time scaling parameter $\epsilon$. In order to demonstrate the new effect, we have chosen a particularly simple family, namely $\dot x = a - \cos x -\cos y$, $\dot y = \epsilon$, $a \in (1,2)$ being fixed. There is no doubt that a similar effect may be observed in generic slow-fast systems on $T^2$. The proposed paper is the first step in the proof of this conjecture.

Keywords. Slow-fast systems on the torus, canard solution, devil's staircase, Poincaré map

2000 Mathematics Subject Classification. Primary 34A26, 34E15

A. Kirillov. Introduction to Family Algebras [PDF]

Classical and quantum family algebras, previously introduced by the author and playing an important role in the theory of semi-simple Lie algebras and their representations are studied. Basic properties, structure theorems and explicit fomulas are obtained for both types of family algebras in many significant cases. Exact formulas (based on experimental calculations) for quantum eigenvalues, their multiplicities, and the trace of the so-called matrix of special elements are conjectured.

Keywords. Semi-simple Lie algebra, symmetric algebra, universal enveloping algebra, irreducible representation, simple spectrum, classical and quantum family algebras

2000 Mathematics Subject Classification. Primary 15A30, 22E60

V. Ostrik. Dimensions of Quantized Tilting Modules [PDF]

Let $U$ be the quantum group with divided powers at $p$-th root of unity for prime $p$. To any two-sided cell $A$ in the corresponding affine Weyl group, one associates the tensor ideal in the category of tilting modules over $U$. In this note we show that for any cell $A$ there exists a tilting module $T$ from the corresponding tensor ideal such that the greatest power of $p$ which divides $\dim T$ is $p^{a(A)}$, where $a(A)$ is Lusztig's $a$-function. This result is motivated by a conjecture of J.Humphreys.

Keywords. Quantum groups at roots of unity, tilting modules, special representations of Weyl groups

2000 Mathematics Subject Classification. Primary 20G05; Secondary 17B37

S. Shlosman and M. Tsfasman. Random Lattices and Random Sphere Packings: Typical Properties [PDF]

We review results about the density of typical lattices in $\mathbb{R}^n$. This density is of the order of $2^{-n}$. We then obtain similar results for random (non-lattice) sphere packings in $\mathbb{R}^n$: after suitably taking a fraction $\nu$ of centers of spheres in a typical random packing $\sigma$, the resulting packing $\tau$ has density $C(\nu) 2^{-n}$ with a reasonable $C(\nu)$. We obtain estimates of $C(\nu)$.

Keywords. Geometric density, random field, vertex covering number, sphere packing

2000 Mathematics Subject Classification. 82B05

V. Vassiliev. Combinatorial Formulas for Cohomology of Knot Spaces [PDF]

We develop homological techniques for finding explicit combinatorial formulas for finite-type cohomology classes of spaces of knots in $\mathbb{R}^n$, $n \ge 3$, generalizing the Polyak-Viro formulas [PV] for invariants (i.e., 0-dimensional cohomology classes) of knots in $\mathbb{R}^3$. As the first applications, we give such formulas for the (reduced mod 2) {generalized Teiblum-Turchin cocycle} of order 3 (which is the simplest cohomology class of long knots $\mathbb{R}^1 \hookrightarrow \mathbb{R}^n$ not reducible to knot invariants or their natural stabilizations), and for all integral cohomology classes of orders 1 and 2 of spaces of {compact knots} $S^1 \hookrightarrow \mathbb{R}^n$. As a corollary, we prove the nontriviality of all these cohomology classes in spaces of knots in $\mathbb{R}^3$.

Keywords. Knot theory, discriminant, combinatorial formula, simplicial resolution, spectral sequence, chord diagram, finite-type cohomology class

2000 Mathematics Subject Classification. Primary 57M25, 55R80; Secondary 57Q45, 55T99, 54F05

S. Vlăduţ Isogeny Class and Frobenius Root Statistics for Abelian Varieties Over Finite Fields [PDF]

Let $I(g,q,N)$ be the number of isogeny classes of $g$-dimensional abelian varieties over a finite field $\mathbb{F}$ having a fixed number $N$ of $\mathbb{F}$-rational points. We describe the asymptotic (for $q\to\infty$) distribution of $I(g,q,N)$ over possible values of $N$. We also prove an analogue of the Sato-Tate conjecture for isogeny classes of $g$-dimensional abelian varieties.

Keywords. Abelian variety, isogeny class, Frobenius root, elliptic curve, Sato-Tate conjecture, probability measure

2000 Mathematics Subject Classification. Primary: 11G25, 14G15, 14K15; Secondary: 11G10, 14K02, 28A33