Related papers: Finiteness of Disjoint Covering Systems with Preci…

We give a review of results on the minimum convex cover and maximum hidden set problems. In addition, we give some new results. First we show that it is NP-hard to determine whether a polygon has the same convex cover number as its hidden…

Suppose $M$ is a complete, embedded minimal surface in $\mathbb{R}^3$ with an infinite number of ends, finite genus and compact boundary. We prove that the simple limit ends of $M$ have properly embedded representatives with compact…

In this paper we apply a geometric covering method to study the number of ends on shrinkers. On one hand, we prove that the number of ends on any complete non-compact shrinker is at most polynomial growth with fixed degree. On the other…

The concept of (a,b)-module comes from the study the Gauss-Manin lattices of an isolated singularity of a germ of an holomorphic function. It is a very simple ''abstract algebraic structure'', but very rich, whose prototype is the formal…

We show that for any integer $s \geq 2$, there are only finitely many $s$-Cullen numbers that are repunits. More precisely, for fixed $s \ge 2$, there are only finitely many integers $n$, $b$, and $q$ with $n \geq 2$, $b \geq 2$ and $q \geq…

We prove the following three closely related results. The first is that every finite simple group has a profinite presentation with 2 generators and at most 18 relations. The second is that if G is a finite simple group, F a field and M an…

The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In…

We prove:(1) the existence, for every integer n > 3, of a noncompact smooth n-dimensional topological manifold whose diffeomorphism group contains an isomorphic copy of every finitely presented group; (2) a finiteness theorem on finite…

We study the problem of covering a given point set in the plane by unit disks so that each point is covered exactly once. We prove that 17 points can always be exactly covered. On the other hand, we construct a set of 657 points where an…

We study the maximum set coverage problem in the massively parallel model. In this setting, $m$ sets that are subsets of a universe of $n$ elements are distributed among $m$ machines. In each round, these machines can communicate with each…

Let P be a d-dimensional lattice polytope. We show that there exists a natural number c_d, only depending on d, such that the multiples cP have a unimodular cover for every natural number c >= c_d. Actually, a subexponential upper bound for…

Let $D(n)$ be the number of pairwise disjoint Steiner quadruple systems. A simple counting argument shows that $D(n) \leq n-3$ and a set of $n-3$ such systems is called a large set. No nontrivial large set was constructed yet, although it…

Understanding the structure of indecomposable $n$-dimensional persistence modules is a difficult problem, yet is foundational for studying multipersistence. To this end, Buchet and Escolar showed that any finitely presented rectangular…

For each odd $m \geq 3$ we completely solve the problem of when an $m$-cycle system of order $u$ can be embedded in an $m$-cycle system of order $v$, barring a finite number of possible exceptions. In cases where $u$ is large compared to…

We prove that there exist only finitely many families of Calabi-Yau quasismooth weighted complete intersections with every fixed dimension $m$. This generalizes a result of Johnson and Koll\'{a}r to higher codimensions.

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

We show that for every finite set of prime numbers S, there are at most finitely many singular moduli that are S-units. The key new ingredient is that for every prime number p, singular moduli are p-adically disperse. We prove analogous…

We prove that in a semi-bounded o-minimal expansion of an ordered group every non-empty open definable set is a finite union of open cells.

When branching for binary mixed integer linear programs with disjunctions of sparsity level $2$, we observe that there exists a finite list of $2$-sparse disjunctions, such that any other $2$-sparse disjunction is dominated by one…

We define the compact universal cover of a compact, metrizable connected space (i.e. a continuum) X to be the inverse limit of all continua that regularly cover X. We show that such covers do indeed form an inverse system with bonding maps…