Related papers: Counterexamples to Siegel's Conjecture
The purpose of this paper is to initiate Arakelov theory in a noncommutative setting. More precisely, we are concerned with noncommutative arithmetic surfaces. We introduce a version of arithmetic intersection theory on noncommutative…
New examples of noncommutative 4-spheres are introduced.
There exists a surface of a convex polyhedron P and a partition L of P into geodesic convex polygons such that there are no connected "edge" unfoldings of P without self-intersections (whose spanning tree is a subset of the edge skeleton of…
We prove a noncommutative version of Bishop's peak interpolation-set theorem.
In this paper we give an example to show Clemens' conjecture is not a first order deformation problem.
We show that there is a set which is not a set of multiple recurrence despite being a set of recurrence for nil-Bohr sets. This answers Huang, Shao, and Ye's \enquote{higher-order} version of Katznelson's Question on Bohr recurrence and…
We disprove a conjecture of Simon for higher-order Szego theorems for orthogonal polynomials on the unit circle and propose a modified version of the conjecture.
We exhibit an example of a finitely presented monoid that is congruence-free and simple but not bisimple.
We prove that seminormality of cut polytopes is equivalent to normality. This settles two conjectures regarding seminormality of cut polytopes.
Statements analogous to the Hard Lefschetz Theorem (HLT) and the Hodge-Riemann bilinear relations (HRR) hold in a variety of contexts: they impose restrictions on the cohomology algebra of a smooth compact K\"ahler manifold or on the…
We discuss several counterexamples to a rigidity conjecture of K. Khanin, which states that under some quantitative condition on non-existence of periodic orbits, $C^0$ conjugacy implies $C^1$ (even $C^\infty$) conjugacy. We construct…
There is empirical evidence supporting the claim that almost all cubic non-Hamiltonian graphs are bridge graphs. In this paper, we pose a related conjecture and prove that the original claim holds for non-3-connected graphs if the…
Beginning from the resolution of the Dirichlet L function, using the inner product formula between two infinite-dimensional vectors in the complex space, the author proved the baffling problem--Hecke conjecture.
We find general geometric conditions on a convex body of revolution K, in dimensions four and six, so that its intersection body IK is not a polar zonoid. We exhibit several examples of intersection bodies which are are not polar zonoids.
In a previous article, we proved tight lower bounds for the coefficients of the generalized $h$-vector of a centrally symmetric rational polytope using intersection cohomology of the associated projective toric variety. Here we present a…
Weighted enumeration of reduced pipe dreams (or rc-graphs) results in a combinatorial expression for Schubert polynomials. The duality between the set of reduced pipe dreams and certain antidiagonals has important geometric implications [A.…
As a discrete analog to Minkowski's theorem on convex bodies, Wills conjectured that the Ehrhart coefficients of a centrally symmetric lattice polytope with exactly one interior lattice point are maximized by those of the cube of side…
We prove two conjectures on weighted complete intersections and give the complete classification of threefold weighted complete intersections in weighted projective space that are canonically or anticanonically embedded.
Let $G$ be a semiabelian variety and $C$ a curve in $G$ that is not contained in a proper algebraic subgroup of $G$. In this situation, conjectures of Pink and Zilber imply that there are at most finitely many points contained in the…
We construct a compact nonpositively curved squared 2-complex whose universal cover contains a flat plane that is not the limit of periodic flat planes.