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We study singularities obtained by the contraction of the maximal divisor in compact (non kaehlerian) surfaces which contain global spherical shells. These singularities are of genus 1 or 2, may be Q-Gorenstein, numerically Gorenstein or…

We give a complete complexity classification for the problem of finding a solution to a given system of equations over a fixed finite monoid, given that a solution over a more restricted monoid exists. As a corollary, we obtain a complexity…

We find a class of solutions to cosmological Einstein equations that generalizes the four dimensional cylindrically symmetric spacetimes to higher dimensions. The AdS soliton is a special member of this class with a unique singularity…

We study the dimensional properties of Moran sets and Moran measures in doubling metric spaces. In particular, we consider local dimensions and $L^q$-dimensions. We generalize and extend several existing results in this area.

Several notions of multiplicativity are introduced for forms of degree $d\geq 3$ over a field of characteristic 0 or greater than d. Examples of multiplicative and strongly multiplicative forms of higher degree are given. Conditions…

General criteria are given for when an embedding of a Mori dream space into another satisfies certain nice combinatorial conditions on some of their associated cones. An explicit example of such an embedding is studied.

This is a survey on recent results regarding singularities that occur on higher dimensional stable varieties.

We find the explicit local models of isolated singularities of conformal hyperbolic metrics by Complex Analysis, which is interesting in its own and could potentially be extended to high-dimensional case.

We study the Besov-Morrey spaces $ \mathcal{N}^{s}_{u,p,q}(\mathbb{R}^{d}) $ and show that under certain conditions on the parameters these spaces can be characterized in terms of higher-order differences. Furthermore we prove that some of…

We enumerate all isotopy classes of degree three Morse polynomials ${\mathbb R}^3 \to {\mathbb R}^1$ with nonsingular principal homogeneous parts, proving that there are exactly 37 of them. We also count all 2258 isotopy classes of {\em…

We give new, explicit and asymptotically sharp, lower bounds for dimensions of irreducible modular representations of finite symmetric groups.

We refer to the classic definition of a singularity in Einstein's general relativity (based on geodesic incompletness) as well as to some other criteria to evaluate the nature of singularities in cosmology. We review what different…

All components of complements of discriminant varieties of simple real function singularities are explicitly listed. New invariants of such components (for not necessarily simple singularities) are introduced. A combinatorial algorithm…

The study of noncommutative solitons is greatly facilitated if the field equations are integrable, i.e. result from a linear system. For the example of a modified but integrable U(n) sigma model in 2+1 dimensions we employ the dressing…

In this paper we will give the calculus, the criterion, and the existence of the arithmetic Galois covers of higher relative dimensions.

Let $A$ be an Artin algebra. We investigate subalgebras of $A$ with certain conditions and obtain some classes of algebras whose finitistic dimensions are finite.

In this article, we show multiple inequalities for the singular values of the difference of matrix means. The obtained results refine and complement some well established results in the literature. Although we target singular values…

We discuss a general procedure to generate a class of (everywhere regular) solutions of Einstein equations that can have an (a-priori fixed) arbitrary number of horizons. We then report on work currently in progress i) to find a suitable…

We prove a formula for the polar degree of projective hypersurfaces in terms of the Milnor data of the singularities, extending to 1-dimensional singularities the Dimca-Papadima result for isolated singularities. We discuss the…

We study the singularities of the projective dual variety.