Related papers: The Hirsch conjecture holds for normal flag comple…
Flag manifolds are in general not symmetric spaces. But they are provided with a structure of $\mathbb{Z}_2^k$-symmetric space. We describe the Riemannian metrics adapted to this structure and some properties of reducibility. We detail for…
The Chen-Ng\^o Conjecture predicts that the Hitchin morphism from the moduli stack of $G$-Higgs bundles on a smooth projective variety surjects onto the space of spectral data. The conjecture is known to hold for the group $GL_n$ and any…
Why withdrawn: Main theorem can only be proved, if the flag manifold F with b2=1 is additionally hermitian symmetric.Mistake made at the following place: If F is not hermitian symmetric, certain vector fields are not left invariant
A maniplex of rank n s an n-valent properly edge-coloured graph that generalises, simultaneously, maps on surfaces and abstract polytopes. The problem of stability in maniplexes is a natural variant of the problem of stability in graphs. A…
We disprove Hitchin's conjecture to the effect that for a generic complex structure on a simply connected spin complex surface the square root of the canonical bundle has no more cohomology then is predicted by the Riemann--Roch theorem.…
We prove the equivariant Leray-Hirsch theorem combinatorially for sufficiently good torus equivariant fiber bundles consisting of homogeneous spaces of Lie groups. We apply this theorem to determining the equivariant integral cohomology…
The doubling conjecture predicts that a manifold admits positive scalar curvature with mean convex boundary if and only if its double admits positive scalar curvature. We show that it holds true for manifolds where the inclusion of the…
We study complete minimal graphs in HxR, which take asymptotic boundary values plus and minus infinity on alternating sides of an ideal inscribed polygon Γ in H. We give necessary and sufficient conditions on the "lenghts" of the sides…
These notes focus on the Lipschitz geometry of sets that are definable in o-minimal structures (expanding the real field). We show that every set which is definable in a polynomially bounded o-minimal structure admits a stratification which…
A prismatoid is a polytope with all its vertices contained in two parallel facets, called its bases. Its width is the number of steps needed to go from one base to the other in the dual graph. The author recently showed in arXiv:1006.2814…
We prove the Nitsche--Hopf conjecture for non-parametric minimal graphs over disks. If \(S\) is a minimal graph over a disk of radius \(R\), and if \(\xi\) is the point above the center, then \[ W(\xi)^2 |K(\xi)|<\frac{\pi^2}{2R^2}. \] Here…
The celebrated Erdos-Hajnal conjecture states that for every $n$-vertex undirected graph $H$ there exists $\eps(H)>0$ such that every graph $G$ that does not contain $H$ as an induced subgraph contains a clique or an independent set of size…
The disproved Nash Williams conjecture states that every 4-regular 4-connected graph has a hamiltonian cycle. We show that a modification of this conjecture is equivalent to the Dominating Cycle Conjecture.
We prove a splitting theorem for a smooth noncompact manifold with (possibly noncompact) boundary. We show that if a noncompact manifold of dimension $n\geq 2$ has $\lambda_1(-\alpha\Delta+\operatorname{Ric})\geq 0$ for some…
We obtain a geometric construction of a ``standard monomial basis'' for the homogeneous coordinate ring associated with any ample line bundle on any flag variety. This basis is compatible with Schubert varieties, opposite Schubert…
A flag complex can be defined as a simplicial complex whose simplices correspond to complete subgraphs of its 1-skeleton taken as a graph. In this article, by introducing the notion of s-dismantlability, we shall define the s-homotopy type…
We prove that if the number of edges does not exceed 7 then the asymptotics of eigenvalues of the Dirichlet problem uniquely determine the shape of the graph.
Let $n$ and $r$ be integers with $n-2\ge r\ge 3$. We prove that any $r$-uniform hypergraph $\mathcal{H}$ on $n$ vertices with spectral radius $\lambda(\mathcal{H}) > \binom{n-2}{r-1}$ must contain a Hamiltonian Berge cycle unless…
The graph reconstruction conjecture states that all graphs on at least three vertices are determined up to isomorphism by their deck. In this paper, a general framework for this problem is proposed to simply explain the reconstruction of…
Combinatorially and stochastically defined simplicial complexes often have the homotopy type of a wedge of spheres. A prominent conjecture of Kahle quantifies this precisely for the case of random flag complexes. We explore whether such…