History and Perspectives of Quantum Groups

The advent of Quantum Groups in the course of the working out the quantum analogue of the Inverse Scattering Method from the soliton theory gives an instructive example of interinfluence of different domains of mathematics. Here I give a rather personal account of this development. Leonardo da Vinci Lecture held on November 7, 2005 Received: June 2006     

Bäcklund Algebra and the KdV Hierarchy

This article presents the Korteweg-de Vries hierarchy in the framework of the Lie algebra of B?cklund transformations and points to the problems raised by its vanishing residues characterization. This paper is in final form and no version of it will be submitted for publication elsewhere. Leonardo da Vinci Lecture held on April 26, 2004 Received: February 2005     

A Survey of Total Positivity

This paper contains an exposition of the classical work of Schoenberg and Gantmacher, Krein on the theory of totally positive matrices. It also contains a generalization of this theory to the Lie theoretic context and an application to the recent results of Fock and Goncharov on Teichmüller theory. Leonardo da Vinci lecture held on May 28, 2007 Received: June 2007     

Fourier Transform Over Local Fields

The Fourier transform in the space of distributions on finite-dimensional vector spaces over local fields is defined, and a strong similarity between the case of archimedian and non-archimedian fields is discussed. Furthermore, the definition of the Fourier transform for the case of “functions” on finite-dimensional vector spaces over 2-dimensional local fields is introduced. Leonardo da Vinci lecture held on June 29, 2005 Received: October 2005     

Details on the mathematical interplay between Leonardo da Vinci and Luca Pacioli

Leonardo da Vinci (1452–1519) maintained a strong relationship with mathematics, but the benefits of this connection unfortunately had little impact in his time. His meeting with Luca Pacioli (1445–1517) in Milan in 1496 was crucial: the two men would become friends and a deep mutual respect would preside over Leonardo da Vinci's improving his mathematical knowledge while contributing to the drawings of Pacioli's De divina proportione (Milano, 1496–98) (Pacioli 1509a Pacioli, L, De divina proportione […], Venezia: Paganino de’ Paganini, 1509a. [Google Scholar]). Based on my recent works on the relationship between mathematics and physics during the Renaissance, here I resume some results on the conceptual mathematical frameworks between two Italian scholars.     

Filtered Ends, Proper Holomorphic Mappings of Kähler Manifolds to Riemann Surfaces, and Kähler Groups

The main result of this paper is that a connected bounded geometry complete K?hler manifold which has at least 3 filtered ends admits a proper holomorphic mapping onto a Riemann surface. As an application, it is also proved that any properly ascending HNN extension with finitely generated base group, as well as Thompson’s groups V, T, and F, are not K?hler. The results and techniques also yield a different proof of the theorem of Gromov and Schoen that, for a connected compact K?hler manifold whose fundamental group admits a proper amalgamated product decomposition, some finite unramified cover admits a surjective holomorphic mapping onto a curve of genus at least 2. Received: January 2006, Revision: November 2006, Accepted: March 2007     

On the dispersion of nonstationary surface waves in anisotropic elastic media

The whispering gallery modes propagating along the surface of an anisotropic elastic body are investigated with the use of space-time caustic expansions and space-time ray series. Each surface mode modulated in amplitude and frequency, is interpreted as a wave packet, with its amplitude’s maximum moving at a group velocity. On the boundary surface, asymptotic expressions for the group velocity (as a function of time and coordinates) are derived, which are in agreement with analogous formulas for Rayleigh waves of SV type in the isotropic case. Bibliography: 6 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 332, 2006, pp. 299–312.     

Quantum statistical theory of radiation friction of a relativistic electron

We comprehensively investigate the effect of quantum space-time nonlocality that accounts for retardation of the electron interaction with both the electron’s own radiation field and the fluctuation field of the electromagnetic vacuum. We rigorously show that the quantum nonlocality effect eliminates the self-acceleration and the causality violation paradoxes that are inherent in the classical theory of radiation friction. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 158, No. 3, pp. 478–496, March, 2009.     

On closed sets of relational constraints and classes of functions closed under variable substitutions

Pippenger’s Galois theory of finite functions and relational constraints is extended to the infinite case. The functions involved are functions of several variables on a set A and taking values in a possibly different set B, where any or both of A and B may be finite or infinite. Received April 30, 2004; accepted in final form February 8, 2005.     

The minimal cost maximum matching of a graph

Summary This paper deals with the problem of finding the maximum matching of a weighted graph, having the minimum cost. This problem is solved via a branch and bound algorithm, derived directly fromLand andDoig's technique. The linear programming problem associated with each step of the procedure is solved through a primal-dual algorithm.

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Zusammenfassung Diese Arbeit beschäftigt sich mit demmatching problem. In einem gewichteten Graphen ist dasmaximal matching mit minimalen Kosten gesucht. Zur Lösung wird ein vonLand undDoig beschriebener Branch-and-Bound-Algorithmus verwendet. Das sich je Iterationsschritt ergebende LP-Problem wird mit einem Primal-Dual-Algorithmus gelöst.

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The authors are with the Laboratory of Automatic Control, Istituto di Elettrotecnica ed Elettronica, Politecnico di Milano, I-20133 Milano, Piazza Leonardo da Vinci, 32.     

Ortho-u-monoids

In this paper, we study the class of ortho-u-monoids which are generalized orthogroups within the class of E(S)-semiabundant semigroups. After introducing the concept of (∼)-Green’s relations, and obtaining some important properties of (∼)-Green’s relations and super E(S)-semiabundant semigroups, we have given the semilattice decomposition of ortho-u-monoids and a structure theorem for regular ortho-u-monoids. The main techniques that we used in the study are the (∼)-Green’s relations, and the semi-spined product of semigroups.     

The Halting Probability Omega: Irreducible Complexity in Pure Mathematics

Some G?del centenary reflections on whether incompleteness is really serious, and whether mathematics should be done somewhat differently, based on using algorithmic complexity measured in bits of information. Enriques Lecture held on October 30, 2006 Received: November 2006     

Statistical Causality,Extremal Measures and Weak Solutions of Stochastic Differential Equations with Driving Semimartingales

In this paper we consider different concepts of causality in filtered probability spaces. Especially, we consider a generalization of a causality relationship “G is a cause of J within H ” which was first given by Mykland (1986) and which is based on Granger’s definition of causality (Granger, Econometrica 37:424–438, 1969). Then we apply this concept on weak solutions of stochastic differential equations with driving semimartingales. We also show that the given causality concept is closely connected to the concept of extremality of measures and links Granger’s causality with the concept of adapted distribution. Finally, the concept of causality is applied on solution of martingale problem.     

Terms which are Mal’cev modulo some functions

We derive consequences from the existence of a term which satisfies Mal’cev identities (characterizing permutability) modulo two functions F and G from admissible relations to admissible relations. We also provide characterizations of varieties having a Mal’cev term modulo F and G. Received September 21, 2006; accepted in final form March 20, 2007.     

Twists of quantum groups and noncommutative field theory

We discuss the role of quantum universal enveloping algebras of symmetries in constructing the noncommutative geometry of space-time and the corresponding field theory. It is shown that in the framework of the twist theory of quantum groups, the noncommutative (super)space-time defined by coordinates with Heisenberg commutation relations is (super)Poincaré invariant, as well as the corresponding field theory. Noncommutative parameters of global transformations are introduced. Bibliography: 30 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 335, 2006, pp. 188–204.     

Shohat-Favard and Chebyshev’s methods in d-orthogonality

This paper is concerned with the Shohat-Favard, Chebyshev and Modified Chebyshev methods for d-orthogonal polynomial sequences d∈ℕ. Shohat-Favard’s method is presented from the concept of dual sequence of a sequence of polynomials. We deduce the recurrence relations for the Chebyshev and the Modified Chebyshev methods for every d∈ℕ. The three methods are implemented in the Mathematica programming language. We show several formal and numerical tests realized with the software developed. This revised version was published online in June 2006 with corrections to the Cover Date.     

Formalization of the “Twin Paradox” for Non-uniformly Accelerated Motions

The use of hyperbolic numbers for studying space-time geometry and trigonometry is extended for demonstrating the Frenet’s formulas in space-time. By means of this introduction the twin paradox for non-uniformly accelerated motions is formalized in a straightforward way.     

Completions of GBL-algebras: negative results

We provide a simple sufficient criterion to show that a given variety of GBL-algebras does not admit (local) completions. As corollaries, we obtain that no variety of GBL-algebras containing Chang’s chain, no nontrivial variety of ℓ-groups, nor the variety of product algebras admit completions. The first result strengthens a result of Gehrke and Priestley. Received August 10, 2006; accepted in final form March 8, 2007.     

Bounds on Supremum Norms for Hecke Eigenfunctions of Quantized Cat Maps

We study extreme values of desymmetrized eigenfunctions (so called Hecke eigenfunctions) for the quantized cat map, a quantization of a hyperbolic linear map of the torus. In a previous paper it was shown that for prime values of the inverse Planck’s constant N = 1/h, such that the map is diagonalizable (but not upper triangular) modulo N, the Hecke eigenfunctions are uniformly bounded. The purpose of this paper is to show that the same holds for any prime N provided that the map is not upper triangular modulo N. We also find that the supremum norms of Hecke eigenfunctions are ≪_ε N^ε for all ε > 0 in the case of N square free. Submitted: March 6, 2006; Accepted: April 30, 2006     

Flow-Invariant Hypersurfaces in Semi-Dispersing Billiards

This work results from our attempts to solve Boltzmann–Sinai’s hypothesis about the ergodicity of hard ball gases. A crucial element in the studies of the dynamics of hard balls is the analysis of special hypersurfaces in the phase space consisting of degenerate trajectories (which lack complete hyperbolicity). We prove that if a flow-invariant hypersurface J in the phase space of a semi-dispersing billiard has a negative Lyapunov function, then the volume of the forward image of J grows at least linearly in time. Our proof is independent of the solution of the Boltzmann–Sinai hypothesis, and we provide a complete and self-contained argument here. Submitted: March 14, 2006. Accepted: August 2, 2006.