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We will give a pure combinatorial proof of the Eisenbud-Goto conjecture for arbitrary monomial curves. Moreover, we will show that the conjecture holds for certain simplicial affine semigroup rings.

In this paper we correct the errors of Yu.~N.~Kuznetsova's paper on the continuous duality for Moore groups.

We exhibit an example of a finitely presented monoid that is congruence-free and simple but not bisimple.

In this paper, we give a simple counter example to the famous Hodge conjecture.

In this paper we use JSJ-decompositions to formalise a folk conjecture recorded by Pride on the structure of one-relator groups with torsion. We prove a slightly weaker version of the conjecture, which implies that the structure of…

In this note we provide some counterexamples for the conjectures of finite simple groups, one of the conjectures said "all finite simple groups $G$ can be determined using their orders $|G|$ and the number of elements of order $p$, where…

For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups.…

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

We show that, in certain circumstances, a Boolean ample monoid may be fully embedded into a Boolean inverse monoid in a way that generalizes how right reversible cancellative monoids may be embedded into groups. We use groupoids of…

Recently, F. Brenti put a preprint on the arXiv with several interesting open problems on Coxeter groups and unimodality. In this note, we refute one of these conjectures with a counterexample and provide supporting data related to it. This…

In this paper we characterize the monoid congruences of commutative semigroups by the help of the notion of the separator of subsets of semigroups. We show that every monoid congruence of a commutative semigroup S can be constructed by the…

Recently, G. Mason has produced a counterexample of order 128 to a conjecture in conformal field theory and tensor category theory in [Ma]. Here we easily produce an infinite family of counterexamples, the smallest of which has order 72.

We give a generalization of Quillen's $S^{-1}S$ construction for arbitrary $E_n$-monoids as an $E_{n-1}$-monoidal $\infty$-category and show that its realization models the group completion provided that $n \geq 2$. We will also show how…

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

We discuss a construction that gives counterexamples to various questions of unique determination of convex bodies.

We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an…

We prove Union-Closed sets conjecture.

We verify a finiteness conjecture of Feit on sources of simple modules over group algebras for various classes of finite groups related to the symmetric groups.

In this short note we present a family of counterexamples to the King's conjecture.

We obtain examples of smooth projective varieties over $\mathbb{C}$ that violate the integral Hodge conjecture and for which the total Chow group is of finite rank. Moreover, we show that there exist such examples defined over number…