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We find and describe unexpected isomorphisms between two very different objects associated to hypersurface singularities. One object is the Milnor algebra of a function, while the other object associated to a singularity is the local ring…
This is a survey written in an expositional style on the topic of symplectic singularities and symplectic resolutions, which could also serve as an introduction to this subject.
In the current relativistic literature there are misleading considerations about some singular surfaces. An accurate geometric analysis allows to settle the question. No physical meaning is attributable to the spatial regions surrounded by…
Supersymmetric nonlinear sigma models have target spaces that carry interesting geometry. The geometry is richer the more supersymmetries the model has. The study of models with two dimensional world sheets is particularly rewarding since…
This is a survey on Zariski equisingularity. We recall its definition, main properties, and a variety of applications in Algebraic Geometry and Singularity Theory. In the first part of this survey, we consider Zariski equisingular families…
We give a full exceptional collection in the triangulated category of singularities in the sense of Orlov for a hypersurface singularity of Fermat type, and discuss its relation with homological mirror symmetry for simple elliptic…
In this note we provide a formula for the Tjurina number of a join of isolated hypersurface singularities in separated variables. From this we are able to provide a characterization of isolated hypersurface singularities whose difference…
We consider biharmonic submanifolds in both generalized complex and Sasakian space forms. After giving the biharmonicity conditions for submanifolds in these spaces, we study different particular cases for which we obtain curvature…
Non-degenerate real hypersurfaces of almost Hermite-like manifolds are examined. Tangential real hypersurfaces are introduced and the main identities of such hypersurfaces are obtained. With the help of these identities, contact metric…
This is a review article on the combinatorial aspects of the mixed Hodge structure of a Milnor fibre of the isolated hypersurface singularity. We give a purely combinatorial method to compute spectral pairs of the singularity under the…
We consider open symplectic manifolds which admit dilations (in the sense previously introduced by Solomon and the author). We obtain restrictions on collections of Lagrangian submanifolds which are pairwise disjoint (or pairwise…
We establish birational superrigidity for a large class of singular projective Fano hypersurfaces of index one. In the special case of isolated singularities, our result applies for instance to: (1) hypersurfaces with semi-homogeneous…
Motivated by a problem in local differential geometry of Cauchy--Riemann (CR) structures of hypersurface type, we find a canonical form for pairs consisting of a nondegenerate Hermitian form and a self-adjoint antilinear operator, or,…
In this paper, we deal with hypernormal forms of non-resonant double Hopf singularities. We investigate the infinite level normal form classification of such singularities with nonzero radial cubic part. We provide a normal form…
We devise a method that reduces the problem of classifying systems of forms and linear mappings to the problem of classifying systems of linear mappings. Canonical matrices of (i) bilinear or sesquilinear forms, (ii) pairs of symmetric,…
We study a generalization of the classical correspondence between homogeneous quadratic polynomials, quadratic forms, and symmetric/alternating bilinear forms to forms in $n$ variables. The main tool is combinatorial polarization, and the…
We give a survey on some aspects of deformations of isolated singularities. In addition to the presentation of the general theory, we report on the question of the smoothability of a singularity and on relations between different…
We study singular hypersurfaces in tensor multi-scalar theories of gravity. We derive in a distributional and then in an intrinsic way, the general equations of junction valid for all types of hypersurfaces, in particular for lightlike…
Modular and quasimodular solutions of specific second order differential equation in the upper-half plane which originates from a study of supersingular j-invariants are given explicitly. A characterization of the differential equation is…
We investigate the geometry of holomorphic curves and complex surfaces from the perspective of singularity theory. We show that, with a suitable choice of a complex bilinear symmetric form, the families of functions and mappings that…