Related papers: Counterexamples to Siegel's Conjecture

We construct a K3 surface whose transcendental lattice has a self-isomorphism which is not a linear combination of self-isomorphisms over $\mathbb{Q}$ which preserve cup products up to nonzero multiples. Products of it with itself give…

We prove that if nonlinear complex polynomials of the same degree have orbits with infinite intersection, then the polynomials have a common iterate. We also prove a special case of a conjectured dynamical analogue of the Mordell-Lang…

Suppose that $A$ and $B$ are closed subsets of a Euclidean space such that $A\cap B\neq\varnothing$, and we aim to find a point in this intersection with the help of the sequences $(a_n)_\nnn$ and $(b_n)_\nnn$ generated by the \emph{method…

The main theorem of the paper provides a way to produce examples such that the movable cone of an ample divisor does not coincide with the movable cone of its ambient variety.

We provide a counterexample to the Lagrangian Poincar\'e recurrence conjecture of Ginzburg and Viterbo in all dimensions $6$ and greater.

The Hard Lefschetz theorem is known to hold for the intersection cohomology of the toric variety associated to a rational convex polytope. One can construct the intersection cohomology combinatorially from the polytope, hence it is well…

We give a counterexample to a recently conjectured variant of the Penrose inequality.

Trinh and Xue have proposed a startling conjecture on intersections of blocks of cyclotomic Hecke algebras occurring in modular representation theory of finite reductive groups. We prove this conjecture for all exceptional type groups apart…

The Hirsch conjecture was posed in 1957 in a letter from Warren M. Hirsch to George Dantzig. It states that the graph of a d-dimensional polytope with n facets cannot have diameter greater than n - d. Despite being one of the most…

We give counterexamples to Okounkov's log-concavity conjecture for Littlewood-Richardson coefficients.

We construct new examples of cubic surfaces, for which the Hasse principle fails. Thereby, we show that, over every number field, the counterexamples to the Hasse principle are Zariski dense in the moduli scheme of non-singular cubic…

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

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

We construct symmetric convex bodies that are not intersection bodies, but all of their central hyperplane sections are intersection bodies. This result extends the studies by Weil in the case of zonoids and by Neyman in the case of…

The scissors congruence conjecture for the unimodular group is an analogue of Hilbert's third problem, for the equidecomposability of polytopes. Liu and Osserman studied the Ehrhart quasi-polynomials of polytopes naturally associated to…

We give in this paper some squares of abelian surfaces that are counterexamples to a conjecture formulated by Zhang about the intersection of subvarieties and preperiodic points.

In this article we develop counterexamples to the Hasse principle using only techniques from undergraduate number theory and algebra. By keeping the technical prerequisites to a minimum, we hope to provide a path for nonspecialists to this…

A conjecture of Woods from 1972 is disproved.

We prove that the billiard claimed to be a possible counterexample to the Birkhoff-Poritsky conjecture is actually not a counterexample. We also show that for a billiard in a table obtained by the string construction over any convex…

In this paper, we propose the study of a conjecture whose affirmative solution would provide an example of a non-convex Chebyshev set in an infinite-dimensional real Hilbert space.