Related papers: A note on Lefschetz spheres and their relatives

We prove that the three-sphere recognition problem lies in the complexity class NP. Our work relies on Thompson's original proof that the problem is decidable [Math. Res. Let., 1994], Casson's version of her algorithm, and recent results of…

We prove the $l^2$ Decoupling Conjecture for compact hypersurfaces with positive definite second fundamental form and also for the cone. This has a wide range of important consequences. One of them is the validity of the Discrete…

We classify isoparametric hypersurfaces in spheres with $(g,m)=(6,1)$ and thereby reprove a result of Dorfmeister and Neher.

The existence of certain monomial hyperovals $D(x^k)$ in the finite Desarguesian projective plane $PG(2,q)$, $q$ even, is related to the existence of points on certain projective plane curves $g_k(x,y,z)$. Segre showed that some values of…

In this article, we introduce and investigate polynomial curvelets on spheres, which form a class of Parseval frames for $L^2(\mathbb{S}^{d-1})$, $d \geq 3$. The proposed construction offers a directionally sensitive multiscale…

A very fundamental geometric problem on finite systems of spheres was independently phrased by Kneser (1955) and Poulsen (1954). According to their well-known conjecture if a finite set of balls in Euclidean space is repositioned so that…

Let $Z$ be a closed subscheme of a smooth complex projective variety $Y\subseteq \Ps^N$, with $\dim\,Y=2r+1\geq 3$. We describe the intermediate N\'eron-Severi group (i.e. the image of the cycle map $A_r(X)\to H_{2r}(X;\mathbb{Z})$) of a…

We give a new geometric proof of the classification of $T$-polygons, a theorem originally due to Kasprzyk, Nill and Prince, using ideas from mirror symmetry. In particular, this gives a completely geometric proof that any two toric…

We investigate the group of points of the $3$-sphere modulo a prime, point out connections to other known groups and the Chebyshev polynomials, and show that there is an infinite series which converges if and only if there are finitely many…

I prove "Lefschetz principle"-type theorems for semistable and curve semistable Higgs sheaves on smooth projective varieties defined over an algebraically closed field of characteristic $0$. These theorems are applied to reduce a…

In a paper of Bosio and Meersseman (Real quadrics in Cn, complex manifolds and convex polytopes) the following is conjectured: If P is dual neighborly, then Zp is diffeomorphic to the connected sum of products of spheres. In this paper a…

Remarkable parallelism between the theory of integrable systems of first-order quasilinear PDE and some old results in projective and affine differential geometry of conjugate nets, Laplace equations, their Bianchi-Baecklund transformations…

The result of Padrol asserts that for every $d\geq 4$, there exist $2^{\Omega(n\log n)}$ distinct combinatorial types of $\lfloor d/2\rfloor$-neighborly simplicial $(d-1)$-spheres with $n$ vertices. We present a construction showing that…

We solve the isoperimetric problem in the Lens spaces with large fundamental group. Namely, we prove that the isoperimetric surfaces are geodesic spheres or tori of revolution about geodesics. We also show that the isoperimetric problem in…

We prove Bertini type theorems and give some applications of them. The applications are in the context of Lefschetz theorem for Nori fundamental group for normal varieties as well as for geometric formal orbifolds. In another application,…

In this paper, we give a survey of various sphere theorems in geometry. These include the topological sphere theorem of Berger and Klingenberg as well as the differentiable version obtained by the authors. These theorems employ a variety of…

While hard-sphere models form the foundation of theoretical condensed matter physics, real systems often exhibit some degree of softness. We present a theoretical and numerical study of a class of nearly hard-sphere systems, generalized…

In this paper we introduce a polynomial frame on the unit sphere $\sph$ of $\mathbb{R}^d$, for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere $\sph$, such as…

In [7], Higashitani, Kummer, and Micha{\l}ek pose a conjecture about the symmetric edge polytopes of complete multipartite graphs and confirm it for a number of families in the bipartite case. We confirm that conjecture for a number of new…

For each g > 2 and h > 1, we explicitly construct (1) fiber sum indecomposable relatively minimal genus g Lefschetz fibrations over genus h surfaces whose monodromies lie in the Torelli group, (2) fiber sum indecomposable genus g surface…