Topological Designs

We give an exponential upper and a quadratic lower bound on the number of pairwise non-isotopic simple closed curves can be placed on a closed surface of genus g such that any two of the curves intersects at most once. Although the gap is large, both bounds are the best known for large genus. In genus one and two, we solve the problem exactly. Our methods generalize to variants in which the allowed number of pairwise intersections is odd, even, or bounded, and to surfaces with boundary components.Comment: 14 p., 4 Figures. To appear in Geometriae Dedicat

Almost-Fisher families

A classic theorem in combinatorial design theory is Fisher's inequality, which states that a family $\mathcal F$ of subsets of $[n]$ with all pairwise intersections of size $\lambda$ can have at most $n$ non-empty sets. One may weaken the condition by requiring that for every set in $\mathcal F$, all but at most $k$ of its pairwise intersections have size $\lambda$. We call such families $k$-almost $\lambda$-Fisher. Vu was the first to study the maximum size of such families, proving that for $k=1$ the largest family has $2n-2$ sets, and characterising when equality is attained. We substantially refine his result, showing how the size of the maximum family depends on $\lambda$. In particular we prove that for small $\lambda$ one essentially recovers Fisher's bound. We also solve the next open case of $k=2$ and obtain the first non-trivial upper bound for general $k$.Comment: 27 pages (incluiding one appendix

New Complexity Bounds for Certain Real Fewnomial Zero Sets

Consider real bivariate polynomials f and g, respectively having 3 and m monomial terms. We prove that for all m>=3, there are systems of the form (f,g) having exactly 2m-1 roots in the positive quadrant. Even examples with m=4 having 7 positive roots were unknown before this paper, so we detail an explicit example of this form. We also present an O(n^{11}) upper bound for the number of diffeotopy types of the real zero set of an n-variate polynomial with n+4 monomial terms.Comment: 8 pages, no figures. Extended abstract accepted and presented at MEGA (Effective Methods in Algebraic Geometry) 200

The number of k-intersections of an intersecting family of r-sets

The Erdos-Ko-Rado theorem tells us how large an intersecting family of r-sets from an n-set can be, while results due to Lovasz and Tuza give bounds on the number of singletons that can occur as pairwise intersections of sets from such a family. We consider a natural generalization of these problems. Given an intersecting family of r-sets from an n-set and 1\leq k \leq r, how many k-sets can occur as pairwise intersections of sets from the family? For k=r and k=1 this reduces to the problems described above. We answer this question exactly for all values of k and r, when n is sufficiently large. We also characterize the extremal families.Comment: 10 pages, 1 figur

Limits in PMF of Teichmuller geodesics

We consider the limit set in Thurston's compactification PMF of Teichmueller space of some Teichmueller geodesics defined by quadratic differentials with minimal but not uniquely ergodic vertical foliations. We show that a) there are quadratic differentials so that the limit set of the geodesic is a unique point, b) there are quadratic differentials so that the limit set is a line segment, c) there are quadratic differentials so that the vertical foliation is ergodic and there is a line segment as limit set, and d) there are quadratic differentials so that the vertical foliation is ergodic and there is a unique point as its limit set. These give examples of divergent Teichmueller geodesics whose limit sets overlap and Teichmueller geodesics that stay a bounded distance apart but whose limit sets are not equal. A byproduct of our methods is a construction of a Teichmueller geodesic and a simple closed curve $\gamma$ so that the hyperbolic length of the geodesic in the homotopy class of gamma varies between increasing and decreasing on an unbounded sequence of time intervals along the geodesic.Comment: 39 pages, 4 figure

Bipolar Lawson Tau-Surfaces and Generalized Lawson Tau-Surfaces

Recently Penskoi [J. Geom. Anal. 25 (2015), 2645-2666, arXiv:1308.1628] generalized the well known two-parametric family of Lawson tau-surfaces $\tau_{r,m}$ minimally immersed in spheres to a three-parametric family $T_{a,b,c}$ of tori and Klein bottles minimally immersed in spheres. It was remarked that this family includes surfaces carrying all extremal metrics for the first non-trivial eigenvalue of the Laplace-Beltrami operator on the torus and on the Klein bottle: the Clifford torus, the equilateral torus and surprisingly the bipolar Lawson Klein bottle $\tilde{\tau}_{3,1}$. In the present paper we show in Theorem 1 that this three-parametric family $T_{a,b,c}$ includes in fact all bipolar Lawson tau-surfaces $\tilde{\tau}_{r,m}$. In Theorem 3 we show that no metric on generalized Lawson surfaces is maximal except for $\tilde{\tau}_{3,1}$ and the equilateral torus.Comment: arXiv admin note: text overlap with arXiv:1308.1628 by other author