Tjurina and Milnor Numbers of Matrix Singularities

To gain understanding of the deformations of determinants andPfaffians resulting from deformations of matrices, the deformationtheory of composites f F with isolated singularities is studied,where f : YC is a function with (possibly non-isolated) singularityand F : XY is a map into the domain of f, and F only is deformed.The corresponding T¹(F) is identified as (something like) thecohomology of a derived functor, and a canonical long exactsequence is constructed from which it follows that = µ(f F) – ß₀ + ß₁, where is the length of T¹(F) and ß_i is the lengthof Tor_i^O_Y(O_Y/J_f, O_X). This explains numerical coincidences observedin lists of simple matrix singularities due to Bruce, Tari,Goryunov, Zakalyukin and Haslinger. When f has Cohen–Macaulaysingular locus (for example when f is the determinant function),relations between and the rank of the vanishing homology ofthe zero locus of f F are obtained.     

On the Milnor and Tjurina Numbers of Zero-Dimensional Singularities

Functional Analysis and Its Applications - In this paper we study relationships between some topological and analytic invariants of zero-dimensional germs, or multiple points. Among other things,...     

The Relative Index for Corner Singularities

We study pseudo-differential operators on a cylinder where B has conical singularities. Configurations of that kind are the local model of corner singularities with cross section B. Operators in our calculus are assumed to have symbols a which are meromorphic in the complex covariable with values in the algebra of all cone operators on B. We show an explicit formula for solutions of the homogeneous equation if a is independent of the axial variable Each non-bijectivity point of the symbol in the complex plane corresponds to a finite-dimensional space of solutions. Moreover, we give a relative index formula.     

Families Index for Manifolds with Hyperbolic Cusp Singularities

Manifolds with fibered hyperbolic cusp metrics include hyperbolicmanifolds with cusps and locally symmetric spaces of -rank 1. We extend Vaillant's treatment of Dirac-typeoperators associated to these metrics by weakening the hypotheseson the boundary families through the use of Fredholm perturbationsas in the family index theorem of Melrose and Piazza, and bytreating the index of families of such operators. We also extendthe index theorem of Moroianu and Leichtnam–Mazzeo–Piazzato families of perturbed Dirac-type operators associated tofibered cusp metrics (sometimes known as fibered boundary metrics).     

Integrability,computation and applications

The study of integrable systems and the notion of integrability has been re-energized with the discovery that infinite-dimensional systems such as the Korteweg-de Vries equation are integrable. In this paper, the following novel aspects of integrability are described: (i) solutions of Darboux, Brioschi, Halphen-type systems and their relationships to monodromy problems and automorphic functions, (ii) computational chaos in integrable systems, (iii) we explain why we believe that homoclinic structures and homoclinic chaos associated with nonlinear integrable wave problems, will be observed in appropriate laboratory experiments.     

Invariant currents and dynamical Lelong numbers

Let ƒ be a polynomial automorphism of ℂ^k of degree λ, whose rational extension to ℙ^k maps the hyperplane at infinity to a single point. Given any positive closed current S on ℙ^k of bidegree (1,1), we show that the sequence λ⁻ⁿ(ƒⁿ)*S converges in the sense of currents on ℙ^k to a linear combination of the Green current T₊ of ƒ and the current of integration along the hyperplane at infinity. We give an interpretation of the coefficients in terms of generalized Lelong numbers with respect to an invariant dynamical current for ƒ⁻¹.     

Groupoid C *-Algebras and Index Theory on Manifolds with Singularities

The simplest case of a manifold with singularities is a manifold M with boundary, together with an identification M M × P, where P is a fixed manifold. The associated singular space is obtained by collapsing P to a point. When P = Z/k or S ¹, we show how to attach to such a space a noncommutative C ^*-algebra that captures the extra structure. We then use this C ^*-algebra to give a new proof of the Freed–Melrose Z/k-index theorem and a proof of an index theorem for manifolds with S ¹ singularities. Our proofs apply to the real as well as to the complex case. Applications are given to the study of metrics of positive scalar curvature.     

Lelong numbers,complex singularity exponents,and Siu's semicontinuity theorem

In this note, we describe a relation between Lelong numbers and complex singularity exponents. As an application, we obtain a new proof of Siu's semicontinuity theorem for Lelong numbers.     

On the Index of Differential Operators on Manifolds with Conical Singularities

The paper contains the proof of the index formula for manifolds with conical points. For operators subject to an additional condition of spectral symmetry, the index is expressed as the sum of multiplicities of spectral points of the conormal symbol (indicial family) and the integral from the Atiyah–Singer form over the smooth part of the manifold. The obtained formula is illustrated by the example of the Euler operator on a two-dimensional manifold with conical singular point.     

Integrability and beyond

An algebraic approach to solving nonlinear functional equations in the Riemann theta functions is stated. By the inverse scattering method and some general methods of the theory of partial differential equations, the solution of the initial boundary value problem for the nonlinear Schrödinger equation is presented. Bibliography:17 titles.     

On the Index of Vector Fields Tangent to Hypersurfaces with Non-Isolated Singularities

Let F be a germ of a holomorphic function at 0 in Cⁿ⁺¹, having0 as a critical point not necessarily isolated, and let be a germ of a holomorphic vectorfield at 0 in Cⁿ⁺¹ with an isolated zero at 0, and tangent toV := F^–1(0). Consider the O_V,0-complex obtained by contractingthe germs of Kähler differential forms of V at 0 (0.1) with the vector field X:=|_Von V: (0.2)     

Complex Hessian Operator and Lelong Number for Unbounded m-subharmonic Functions

m-subharmonic functions are the right class of admissible solutions to the complex Hessian equation. In this paper, we generalize the definition of the complex Hessian operator to some unbounded m-subharmonic functions, and we prove that the complex Hessian operator is continuous on the monotonically decreasing sequences of m-subharmonic functions. Moreover we establish the Lelong-Jensen type formula and introduce the Lelong number for m-subharmonic functions. A useful inequality for the mixed Hessian operator is showed.     

Noncommutative Integrability,Moment Map and Geodesic Flows

The purpose of this paper is to discuss the relationship betweencommutative and noncommutative integrability of Hamiltonian systemsand to construct new examples of integrable geodesic flows onRiemannian manifolds. In particular, we prove that the geodesic flowof the bi-invariant metric on any bi-quotient of a compact Lie group isintegrable in the noncommutative sense by means of polynomial integrals, andtherefore, in the classical commutative sense by means ofC -smooth integrals.     

Vector optimization: Singularities, regularizations

We discuss three scalarizations of the multiobjectie optimization from the point of view of the parametric optimization. We analyze three important aspects:

Singularities, Shocks, and Instabilities in Interface Growth

Two-dimensional interface motion is examined in the setting of geometric crystal growth. We focus on the relationships between local curvature and global shape evolution displaying the dual role of singularities and shocks depending on the parameterization of the curve—the crystal surface. Discontinuities in surface slope accompany regions of asymptotically decreasing curvature during transient growth, whereas an absence of discontinuities preempts such asymptotic curvature evolution. In one parameterization, these discontinuities manifest themselves as a finite-time continuous blowup of curvature, and in another, as a shock and hence a localized divergence of curvature. Previously, it has been conjectured, based on numerical evidence, that the minimum blowup time is preempted by shock formation. We prove this conjecture in the present paper. Additionally we prove that a class of local geometric models preserves the convexity of the surface. These results are connected to experiments on crystal growth.