Related papers: Unexpected hypersurfaces of type $(d+k,d)$
C.T.C. Wall and the first author discovered an extension of Arnold's strange duality embracing on one hand series of bimodal hypersurface singularities and on the other, isolated complete intersection singularities. In this paper, we derive…
We introduce a $N=1$ supergravity model with a very simple hidden sector coupled to the electroweak gauge and Higgs sectors of the MSSM. At the classical level, supersymmetry and $SU(2)\times U(1)$ are both spontaneously broken, with…
We argue that extra dimensions with a properly chosen compactification scheme could be a natural source for emergent gauge symmetries. Actually, some proposed vector field potential terms or polynomial vector field constraints introduced in…
Let $M$ be a compact hypersurface with boundary $\partial M=\partial D_1 \cup \partial D_2$, $\partial D_1 \subset \Pi _1$, $\partial D_2 \subset \Pi _2$, $\Pi_1$ and $\Pi _2$ two parallel hyperplanes in $\mathbb{R}^{n+1}$ ($n \geq 2$).…
A hypersurface is said to be quasihomogeneous if in suitable coordinates with assigned weights, its equation becomes weighted homogeneous in its variables. For an irreducible quasihomogeneous plane curve, the equation necessarily becomes a…
In the present survey paper, we present several new classes of Hochster's spectral spaces "occurring in nature", actually in multiplicative ideal theory, and not linked to or realized in an explicit way by prime spectra of rings. The…
For an isolated hypersurface singularity which is neither simple nor simple elliptic, it is shown that there exists a distinguished basis of vanishing cycles which contains two basis elements with an arbitrary intersection number. This…
We study supergravity models in four dimensions where the hidden sector is superconformal and strongly-coupled over several decades of energy below the Planck scale, before undergoing spontaneous breakdown of scale invariance and…
We investigate how exotic differential structures may reveal themselves in particle physics. The analysis is based on the A. Connes' construction of the standard model. It is shown that, if one of the copies of the spacetime manifold is…
This article concerns a natural generalization of the classical asymptotic Plateau problem in hyperbolic space. We prove the existence of a smooth complete hypersurface of constant scalar curvature with a prescribed asymptotic boundary at…
In this paper we introduce a new geometric flow --- the hyperbolic gradient flow for graphs in the $(n+1)$-dimensional Euclidean space $\mathbb{R}^{n+1}$. This kind of flow is new and very natural to understand the geometry of manifolds. We…
The Su-Schrieffer-Heeger model is extended to the three and higher dimensional systems. Nearly or absolutely flat midgap surface and hypersurface bands are predicted based on the topological analysis, which do not require fine tuning of…
An ordinary hypersphere of a set of points in real $d$-space, where no $d+1$ points lie on a $(d-2)$-sphere or a $(d-2)$-flat, is a hypersphere (including the degenerate case of a hyperplane) that contains exactly $d+1$ points of the set.…
Let $X\subset\mathbb{P}^{n+e}$ be any $n$-dimensional closed subscheme. We are mainly interested in two notions related to syzygies: one is the property $\mathbf{N}_{d,p}~(d\ge 2, ~p\geq 1)$, which means that $X$ is $d$-regular up to $p$-th…
In this talk, we present three examples of new non-trivial vacuum structures that can occur in supersymmetric field theories, along with explicit models in which they arise. The first vacuum structure is one in which supersymmetry is broken…
A classical open problem in combinatorial geometry is to obtain tight asymptotic bounds on the maximum number of k-level vertices in an arrangement of n hyperplanes in d dimensions (vertices with exactly k of the hyperplanes passing below…
In this paper we study Hardy-Sobolev inequalities on hypersurfaces of $\mathbb{R}^{n+1}$, all of them involving a mean curvature term and having universal constants independent of the hypersurface. We first consider the celebrated Sobolev…
We prove new results on projective normality, normal presentation and higher syzygies for a surface of general type $X$ embedded by adjoint line bundles $L_r = K + rB$, where $B$ is a base point free, ample line bundle. Our main results…
A new realization of extended quantum-mechanical supersymmetry (QM SUSY) with central extension is investigated. We first show that two different sets of $d+2$ ($d+1$) supercharges for $d=$ even (odd), each of which satisfies an…
The quotient singularities of dimensions two and three obtained from polyhedral groups and the corresponding binary polyhedral groups admit natural resolutions of singularities as Hilbert schemes of regular orbits whose exceptional fibres…