Related papers: Hadwiger's conjecture holds for strongly monotypic…

A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the…

We present new families of weighted homogeneous and Newton non-degenerate line singularities that satisfy the Zariski multiplicity conjecture.

We introduce the notion of combinatorial positivity of translation-invariant valuations on convex polytopes that extends the nonnegativity of Ehrhart h*-vectors. We give a surprisingly simple characterization of combinatorially positive…

We prove a conjecture of R. Schwartz about the type of some complex hyperbolic triangle groups.

We show that the Jacobian conjecture of the two dimensional case is true.

In this article, we give a proof of the strong openness conjecture for plurisubharmonic functions posed by Demailly.

In this note, we present two arguments showing that the classical \textit{linear adjoint cone restriction conjecture} holds for the class of functions supported on the cone and invariant under the spatial rotation in all dimensions. The…

The article provides a counterexample to a conjecture by Blocki-Zwonek.

Hadwiger's conjecture asserts that if a simple graph $G$ has no $K_{t+1}$ minor, then its vertex set $V(G)$ can be partitioned into $t$ stable sets. This is still open, but we prove under the same hypotheses that $V(G)$ can be partitioned…

In this paper, we prove the Khavinson conjecture for hyperbolic harmonic functions on the unit ball. This conjecture was partially solved in \cite{JKM2020}.

We prove some general results on syzygies of smooth projective varieties with numerically trivial canonical line bundle. This allows to confirm several cases of Mukai's syzygies conjecture for finite quotients of abelian varieties in any…

In this very short note, we give a counterexample to a recent conjecture of Gilmer which would have implied the union-closed conjecture.

The monodromy conjecture is a mysterious open problem in singularity theory. Its original version relates arithmetic and topological/geometric properties of a multivariate polynomial $f$ over the integers, more precisely, poles of the…

We present a new proof of the Joints Theorem without taking derivatives. Then we generalize the proof to prove the Multijoints Conjecture and Carbery's generalization. All results are in any dimension over an arbitrary field.

In this note we prove the Weinstein conjecture for a class of symplectic manifolds including the uniruled manifolds based on Liu-Tian's result.

We introduce a notion of essential hyperbolic Coxeter polytope as a polytope which fits some minimality conditions. The problem of classification of hyperbolic reflection groups can be easily reduced to classification of essential Coxeter…

Given a finite collection P of convex n-polytopes in RP^n (n>1), we consider a real projective manifold M which is obtained by gluing together the polytopes in P along their facets in such a way that the union of any two adjacent polytopes…

We give a proof of the so-called generalized Waldhausen conjecture, which says that an orientable irreducible atoroidal 3-manifold has only finitely many Heegaard splittings in each genus, up to isotopy. Jaco and Rubinstein have announced a…

We investigate a novel setting for polytope rigidity, where a flex must preserve edge lengths and the planarity of faces, but is allowed to change the shapes of faces. For instance, the regular cube is flexible in this notion. We present…

A well-known theorem of Blind and Mani says that every simple polytope is uniquely determined by its graph. Kalai gave a very short and elegant proof of this result using the concept of acyclic orientations. As it turns out, Kalai's proof…