Compact connected components in relative character varieties of punctured spheres

We prove that some relative character varieties of the fundamental group of a punctured sphere into the Hermitian Lie groups $\mathrm{SU}(p,q)$ admit compact connected components. The representations in these components have several counter-intuitive properties. For instance, the image of any simple closed curve is an elliptic element. These results extend a recent work of Deroin and the first author, which treated the case of $\textrm{PU}(1,1) = \mathrm{PSL}(2,\mathbb{R})$. Our proof relies on the non-Abelian Hodge correspondance between relative character varieties and parabolic Higgs bundles. The examples we construct admit a rather explicit description as projective varieties obtained via Geometric Invariant Theory.

Fukaya categories of surfaces, spherical objects and mapping class groups

We prove that every spherical object in the derived Fukaya category of a closed surface of genus at least $2$ whose Chern character represents a nonzero Hochschild homology class is quasi-isomorphic to a simple closed curve equipped with a rank $1$ local system. (The homological hypothesis is necessary.) This largely answers a question of Haiden, Katzarkov and Kontsevich. It follows that there is a natural surjection from the autoequivalence group of the Fukaya category to the mapping class group. The proofs appeal to and illustrate numerous recent developments: quiver algebra models for wrapped categories, sheafifying the Fukaya category, equivariant Floer theory for finite and continuous group actions and homological mirror symmetry. An application to high-dimensional symplectic mapping class groups is included.

Shortest paths in arbitrary plane domains

Let $\Omega $ be a connected open set in the plane and $\gamma : [0,1] \to \overline {\Omega }$ a path such that $\gamma ((0,1)) \subset \Omega $ . We show that the path $\gamma $ can be “pulled tight” to a unique shortest path which is homotopic to $\gamma $ , via a homotopy h with endpoints fixed whose intermediate paths $h_t$ , for $t \in [0,1)$ , satisfy $h_t((0,1)) \subset \Omega $ . We prove this result even in the case when there is no path of finite Euclidean length homotopic to $\gamma $ under such a homotopy. For this purpose, we offer three other natural, equivalent notions of a “shortest” path. This work generalizes previous results for simply connected domains with simple closed curve boundaries.

Existence and Hardness of Conveyor Belts

An open problem of Manuel Abellanas asks whether every set of disjoint closed unit disks in the plane can be connected by a conveyor belt, which means a tight simple closed curve that touches the boundary of each disk, possibly multiple times. We prove three main results: For unit disks whose centers are both $x$-monotone and $y$-monotone, or whose centers have $x$-coordinates that differ by at least two units, a conveyor belt always exists and can be found efficiently. It is NP-complete to determine whether disks of arbitrary radii have a conveyor belt, and it remains NP-complete when we constrain the belt to touch disks exactly once. Any disjoint set of $n$ disks of arbitrary radii can be augmented by $O(n)$ "guide" disks so that the augmented system has a conveyor belt touching each disk exactly once, answering a conjecture of Demaine, Demaine, and Palop.

Geometric Multicut: Shortest Fences for Separating Groups of Objects in the Plane

We study the following separation problem: Given a collection of pairwise disjoint coloured objects in the plane with k different colours, compute a shortest “fence” F, i.e., a union of curves of minimum total length, that separates every pair of objects of different colours. Two objects are separated if F contains a simple closed curve that has one object in the interior and the other in the exterior. We refer to the problem as geometrick-cut, as it is a geometric analog to the well-studied multicut problem on graphs. We first give an $$O(n^4\log ^3\!n)$$ O ( n 4 log 3 n ) -time algorithm that computes an optimal fence for the case where the input consists of polygons of two colours with n corners in total. We then show that the problem is NP-hard for the case of three colours. Finally, we give a randomised $$4/3\cdot 1.2965$$ 4 / 3 · 1.2965 -approximation algorithm for polygons and any number of colours.

ITERASI FUNGSI KUADRAT KOMPLEKS DAN KONSTRUKSI HIMPUNAN JULIA

This research aims to: (1) study and explain the definition of the Julia set, (2) study and explain the characteristics of the Julia set, (3) construct and visualize the Julia set in computer. This research is a qualitative descriptive research in the form of literature study. The method used in this research is examine various scientific literatures such as books and scientific journals about the definition, characteristics, and the method of constructing and visualizing Julia's set. The main reference of this research is the book Fractal Geometry Mathematical Foundations and Applications edition by Kenneth Falconer (2003). The results of this research are as follows: (1) the Julia set is the boundary of filled- in Julia sets, which is the set of points where in every neighbourhood of there are points and with and ; (2) there are two characteristics of Julia set based on parameters , namely Julia set in a totally disconnected curve and Julia set in a simple closed curve ; (3) the steps in constructing the Julia set are to take the function of the Julia set, choose the value, determine the number of iterations to be performed, do the calculations according to the iteration that has been determined, then visualize. In visualizing the Julia set, it is needed computer application assistance, one of which is Matlab, so that the results for the Julia set can be modeled in an interesting form. Keywords: Complex Quadratic Functions; Julia Set; Iteration

Consider the less traveled path, because of the Jordan Curve Theorem

“Consider the less traveled path, because of the Jordan Curve Theorem” offers a basic introduction to simple, closed curves and explains why the theorem asserting that every simple closed curve in the plane separates the plane into an “inside” and an “outside” is best appreciated when considering pathological curves. A pathological curve, such as a space-filling curve or the Koch snowflake, is one that lacks features of so-called well-behaved curves. The discussion is enhanced by numerous hand-drawn sketches and a reference to “The Road Not Taken” by poet Robert Frost. Mathematics students and enthusiasts are encouraged to consider the less traveled path in mathematical and life pursuits. At the chapter’s end, readers may check their understanding by working on a problem. A solution is provided.

Arrangements of Pseudocircles: Triangles and Drawings

A pseudocircle is a simple closed curve on the sphere or in the plane. The study of arrangements of pseudocircles was initiated by Grünbaum, who defined them as collections of simple closed curves that pairwise intersect in exactly two crossings. Grünbaum conjectured that the number of triangular cells $$p_3$$ p 3 in digon-free arrangements of n pairwise intersecting pseudocircles is at least $$2n-4$$ 2 n - 4 . We present examples to disprove this conjecture. With a recursive construction based on an example with 12 pseudocircles and 16 triangles we obtain a family of intersecting digon-free arrangements with $$p_3({\mathscr {A}})/n \rightarrow 16/11 = 1.\overline{45}$$ p 3 ( A ) / n → 16 / 11 = 1 . 45 ¯ . We expect that the lower bound $$p_3({\mathscr {A}}) \ge 4n/3$$ p 3 ( A ) ≥ 4 n / 3 is tight for infinitely many simple arrangements. It may however be true that all digon-free arrangements of n pairwise intersecting circles have at least $$2n-4$$ 2 n - 4 triangles. For pairwise intersecting arrangements with digons we have a lower bound of $$p_3 \ge 2n/3$$ p 3 ≥ 2 n / 3 , and conjecture that $$p_3 \ge n-1$$ p 3 ≥ n - 1 . Concerning the maximum number of triangles in pairwise intersecting arrangements of pseudocircles, we show that $$p_3 \le \frac{4}{3}\left( {\begin{array}{c}n\\ 2\end{array}}\right) +O(n)$$ p 3 ≤ 4 3 n 2 + O ( n ) . This is essentially best possible because there are families of pairwise intersecting arrangements of n pseudocircles with $$p_3 = \frac{4}{3}\left( {\begin{array}{c}n\\ 2\end{array}}\right) $$ p 3 = 4 3 n 2 . The paper contains many drawings of arrangements of pseudocircles and a good fraction of these drawings was produced automatically from the combinatorial data produced by our generation algorithm. In the final section we describe some aspects of the drawing algorithm.

ENUMERATION OF MEANDERS AND MASUR–VEECH VOLUMES

A meander is a topological configuration of a line and a simple closed curve in the plane (or a pair of simple closed curves on the 2-sphere) intersecting transversally. Meanders can be traced back to H. Poincaré and naturally appear in various areas of mathematics, theoretical physics and computational biology (in particular, they provide a model of polymer folding). Enumeration of meanders is an important open problem. The number of meanders with $2N$ crossings grows exponentially when $N$ grows, but the long-standing problem on the precise asymptotics is still out of reach. We show that the situation becomes more tractable if one additionally fixes the topological type (or the total number of minimal arcs) of a meander. Then we are able to derive simple asymptotic formulas for the numbers of meanders as $N$ tends to infinity. We also compute the asymptotic probability of getting a simple closed curve on a sphere by identifying the endpoints of two arc systems (one on each of the two hemispheres) along the common equator. The new tools we bring to bear are based on interpretation of meanders as square-tiled surfaces with one horizontal and one vertical cylinder. The proofs combine recent results on Masur–Veech volumes of moduli spaces of meromorphic quadratic differentials in genus zero with our new observation that horizontal and vertical separatrix diagrams of integer quadratic differentials are asymptotically uncorrelated. The additional combinatorial constraints we impose in this article yield explicit polynomial asymptotics.