Harmonic maps into loop spaces of compact Lie groups

We study harmonic maps from Riemann surfaces M to the loop spaces ΩG of compact Lie groups G, using the twistor approach. We conjecture that harmonic maps of the Riemann sphere ℂℙ¹ into ΩG are related to Yang-Mills G-fields on ℝ⁴. This work was partly supported by the RFBR (Grant Nos. 04-01-00236, 06-02-04012), by the program of Support of Scientific Schools (Grant No. 1542.2003.1), and by the Scientific Program of RAS “Nonlinear Dynamics”     

Tautological rings of stable map spaces

We find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero, admitting a simple geometrical interpretation. In particular, when the target is Pn, these give a complete presentation for the cohomology and Chow rings in the cases with/without marked points.     

Static Theory for Planar Ferromagnets and Antiferromagnets

Here we generalize the “BBH”-asymptotic analysis to a simplified mathematical model for the planar ferromagnets and antiferromagnets. To develop such a static theory is a necessary step for a rigorous mathematical justification of dynamical laws for the magnetic vortices formally derived in [1] and [2]. Received March 15, 2001, Accepted May 16, 2001     

Geostatistical computing of acoustic maps in the presence of barriers

Acoustic maps are the main diagnostic tools used by authorities for addressing the growing problem of urban acoustic contamination. Geostatistics models phenomena with spatial variation, but restricted to homogeneous prediction regions. The presence of barriers such as buildings introduces discontinuities in prediction areas. In this paper we investigate how to incorporate information of a geographical nature into the process of geostatistical prediction. In addition, we study the use of a Cost-Based distance to quantify the correlation between locations.     

Characterization of standard embeddings between complex Grassmannians by means of varieties of minimal rational tangents

In 1993,Tsal proved that a proper holomorphic mapping f:Ω→Ω' from an irreducible bounded symmetric domainΩof rank≥2 into a bounded symmetric domainΩ' is necessarily totally geodesic provided that r':=rank(Ω')≤rank(Ω):= r,proving a conjecture of the author's motivated by Hermitian metric rigidity.As a first step in the proof,Tsai showed that df preserves almost everywhere the set of tangent vectors of rank 1.Identifying bounded symmetric domains as open subsets of their compact duals by means of the Borel embedding,this means that the germ of f at a general point preserves the varieties of minimal rational tangents(VMRTs). In another completely different direction Hwang-Mok established with very few exceptions the Cartan- Fubini extension priniciple for germs of local biholomorphisms between Fano manifolds of Picard num- ber 1,showing that the germ of map extends to a global biholomorphism provided that it preserves VMRTs.We propose to isolate the problem of characterization of special holomorphic embeddings between Fano manifolds of Picard number 1,especially in the case of classical manifolds such as ratio- nal homogeneous spaces of Picard number 1,by a non-equidimensional analogue of the Cartan-Fubini extension principle.As an illustration we show along this line that standard embeddings between com- plex Grassmann manifolds of rank≤2 can be characterized by the VMRT-preserving property and a non-degeneracy condition,giving a new proof of a result of Neretin's which on the one hand paves the way for far-reaching generalizations to the context of rational homogeneous spaces and more generally Fano manifolds of Picard number 1,on the other hand should be applicable to the study of proper holomorphic mappings between bounded domains carrying some form of geometric structures.     

Isometries for Ky Fan norms between matrix spaces

We characterize linear maps between different rectangular matrix spaces preserving Ky Fan norms.

     

SOME REMARKS ON HOLOMORPHIC FUNCTIONS AND TAYLOR SERIES IN Cn

Some previous results on convergence of Taylor series in Cn [3] are improved by indicating outside the domain of convergence the points where the series diverges and simplifying some proofs. These results contain the Cauchy-Hadamard theorem in C. Some Cauchy integral formulas of a holomorphic function on a closed ball in Cn are constructed and the Taylor series expansion is deduced.     

On Representations Associated with Completely n-Positive Linear Maps on Pro-C^＊-Algebras

It is shown that an n × n matrix of continuous linear maps from a pro-C^＊-algebra A to L（H）, which verifies the condition of complete positivity, is of the form [V^＊TijФ（·）V]^n i,where Ф is a representation of A on a Hilbert space K, V is a bounded linear operator from H to K, and j=1,[Tij]^n i,j=1 is a positive element in the C^＊-algebra of all n×n matrices over the commutant of Ф（A） in L（K）. This generalizes a result of C. Y.Suen in Proc. Amer. Math. Soc., 112（3）, 1991, 709-712. Also, a covariant version of this construction is given.     

A continuity theorem for Stinespring's dilation

We show a continuity theorem for Stinespring's dilation: two completely positive maps between arbitrary C^∗-algebras are close in cb-norm if and only if we can find corresponding dilations that are close in operator norm. The proof establishes the equivalence of the cb-norm distance and the Bures distance for completely positive maps. We briefly discuss applications to quantum information theory.