Related papers: A note on the abundance conjecture
The Union Closed Sets Conjecture states that in every finite, nontrivial set family closed under taking unions there is an element contained in at least half of all the sets of the family. We investigate two new directions with respect to…
In the present paper a criteria for a rectangular diagram to admit a simplification is given in terms of Legendrian knots. It is shown that there are two types of simplifications which are mutually independent in a sense. A new proof of the…
We establish isomorphism ranges for the comparison maps between algebraic and topological K-groups, extending classical Quillen-Lichtenbaum conjecture to separated complex schemes of finite type after refinement. Additionally, we…
In this article, we use the cone of nef curves to study minimal log discrepancies. The first result is an improvement of the nef cone theorem in the case of log Calabi-Yau dlt pairs. Then, we prove that the ascending chain condition for…
In 2009 Lurie published an expository article outlining a proof for a higher version of the cobordism hypothesis conjectured by Baez and Dolan in 1995. In this note we give a proof for the 1-dimensional case of this conjecture. The proof…
The union-closed sets conjecture (sometimes referred to as Frankl's conjecture) states that every finite, nontrivial union-closed family of sets has an element that is in at least half of its members. Although the conjecture is known to be…
We show that if the minimum entropy for a polynomial with roots on the unit circle is attained by polynomials with equally spaced roots, then, under a generic hypothesis about the nature of the extremum, the Krzyz conjecture on the maximum…
We discuss some aspects of swampland constraints - especially the swampland distance conjecture - in a large number of space-time dimensions $D$. We analyze Kaluza-Klein (KK) states at large $D$ and find that some KK spectra possess an…
The study of minimal complements in a group or a semigroup was initiated by Nathanson. The notion of minimal complements and being a minimal complement leads to the notion of co-minimal pairs which was considered in a prior work of the…
Dual canonical bases are expected to satisfy a certain (double) triangularity property by Leclerc's conjecture. We propose an analogous conjecture for common triangular bases of quantum cluster algebras. We show that a weaker form of the…
We prove a subadjunction theorem which relates the multi-adjoint linear system of the ambient space and the linear system of the restricted bundle on a subvariety.
We prove an analogue for holonomic DQ-modules of the codimension-three conjecture for microdifferential modules recently proved by Kashiwara and Vilonen. Our result states that any holonomic DQ-module having a lattice extends uniquely…
In this paper, we continue the study of Serrano's conjecture in low dimensions. We focus on two special cases of the log version of Serrano's conjecture: the ampleness conjecture and the log version of Campana--Peternell's conjecture. In…
The Knot Entropy Conjecture states that the exponential growth rate of the number of $n$-edge lattice polygons with knot-type $K$ is the same as that for unknot polygons. Moreover, the next order growth follows a power law in $n$ with an…
We present a proof for a conjecture previously formulated by Dzhafarov, Kujala, and Larsson (Foundations of Physics, in press, arXiv:1411.2244). The conjecture specifies a measure for the degree of contextuality and a criterion (necessary…
String theory suggests modifications of our spacetime such as extra dimensions and the existence of a mininal length scale. In models with addidional dimensions, the Planck scale can be lowered to values accessible by future colliders.…
We show that there is a strong minimal pair in the computably enumerable Turing degrees.
The well-known middle levels conjecture asserts that for every integer $n\geq 1$, all binary strings of length $2(n+1)$ with exactly $n+1$ many 0s and 1s can be ordered cyclically so that any two consecutive strings differ in swapping the…
An MDS matrix is a matrix whose minors all have full rank. A question arising in coding theory is what zero patterns can MDS matrices have. There is a natural combinatorial characterization (called the MDS condition) which is necessary over…
The union-closed sets conjecture (Frankl's conjecture) says that for any finite union-closed family of finite sets, other than the family consisting only of the empty set, there exists an element that belongs to at least half of the sets in…