Semi-algebraic colorings of complete graphs

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of $m$ is relevant, e.g., to problems concerning the number of distinct distances determined by a point set. For $p\ge 3$ and $m\ge 2$, the classical Ramsey number $R(p;m)$ is the smallest positive integer $n$ such that any $m$-coloring of the edges of $K_n$, the complete graph on $n$ vertices, contains a monochromatic $K_p$. It is a longstanding open problem that goes back to Schur (1916) to decide whether $R(p;m)=2^{O(m)}$, for a fixed $p$. We prove that this is true if each color class is defined semi-algebraically with bounded complexity. The order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle {\em et al.}, and on a Szemer\'edi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erd\H{o}s and Shelah

Hamilton-Jacobi theory in multisymplectic classical field theories

The geometric framework for the Hamilton-Jacobi theory developed in previous works is extended for multisymplectic first-order classical field theories. The Hamilton-Jacobi problem is stated for the Lagrangian and the Hamiltonian formalisms of these theories as a particular case of a more general problem, and the classical Hamilton-Jacobi equation for field theories is recovered from this geometrical setting. Particular and complete solutions to these problems are defined and characterized in several equivalent ways in both formalisms, and the equivalence between them is proved. The use of distributions in jet bundles that represent the solutions to the field equations is the fundamental tool in this formulation. Some examples are analyzed and, in particular, the Hamilton-Jacobi equation for non-autonomous mechanical systems is obtained as a special case of our results.Comment: 44 p

Geometric evolution equations and p-harmonic theory with applications in differential geometry.

In this dissertation, we consider parabolic (e.g. Ricci flow) and elliptic (e.g. p-harmonic equations) partial differential equations on Riemannian manifolds and use them to study geometric and topological problems. More specifically, to classify a special class of Ricci flow equations, we constructed a family of new entropy functionals in the sense of Perelman. We study the monotonicity of these functionals and use this property to prove that a compact steady gradient Ricci breather is necessarily Ricci-flat. We introduce a new approach to prove the monotonicity formula of Perelman's W -entropy functional and we construct similar entropy functionals on expanders from this new viewpoint. We prove that a large family of complete non-compact Riemannian manifolds cannot be stably minimally immersed into Euclidean space as a hypersurface which serves as a non-existence theorem considering the Generalized Bernstein Conjecture. We give another yet simpler proof for a theorem of do Carmo and Peng, concerning stable minimal hypersurfaces in Euclidean space with certain integral curvature condition. In the study of p-harmonic geometry, we develop a classification theory of Riemannian manifolds by using p-superharmonic functions in the weak sense. We gave sharp estimates as sufficient conditions for a p-parabolic manifold. By developing a Generalized Uniformization Theorem, a Generalized Bochner's Method, and an iterative method, we approach various geometric and variational problems in complete noncompact manifolds of general dimensions

Packing 1-plane Hamiltonian cycles in complete geometric graphs

Counting the number of Hamiltonian cycles that are contained in a geometric graph is #P-complete even if the graph is known to be planar [15]. A relaxation for problems in plane geometric graphs is to allow the geometric graphs to be 1-plane, that is, each of its edges is crossed at most once. We consider the following question: For any set P of n points in the plane, how many 1-plane Hamiltonian cycles can be packed into a complete geometric graph Kn? We investigate the problem by taking two different situations of P, namely, when P is in convex position, wheel configurations position. For points in general position we prove the lower bound of k − 1 where n = 2k + h and 0 ≤ h < 2k. In all of the situations, we investigate the constructions of the graphs obtained

Circumscribing Polygons and Polygonizations for Disjoint Line Segments

Given a planar straight-line graph G=(V,E) in R^2, a circumscribing polygon of G is a simple polygon P whose vertex set is V, and every edge in E is either an edge or an internal diagonal of P. A circumscribing polygon is a polygonization for G if every edge in E is an edge of P. We prove that every arrangement of n disjoint line segments in the plane has a subset of size Omega(sqrt{n}) that admits a circumscribing polygon, which is the first improvement on this bound in 20 years. We explore relations between circumscribing polygons and other problems in combinatorial geometry, and generalizations to R^3. We show that it is NP-complete to decide whether a given graph G admits a circumscribing polygon, even if G is 2-regular. Settling a 30-year old conjecture by Rappaport, we also show that it is NP-complete to determine whether a geometric matching admits a polygonization

Relocating Units in Robot Swarms with Uniform Control Signals is PSPACE-Complete

This paper investigates a restricted version of robot motion planning, in which particles on a board uniformly respond to global signals that cause them to move one unit distance in a particular direction on a 2D grid board with geometric obstacles. We show that the problem of deciding if a particular particle can be relocated to a specified location on the board is PSPACE-complete when only allowing 1x1 particles. This shows a separation between this problem, called the relocation problem, and the occupancy problem in which we ask whether a particular location can be occupied by any particle on the board, which is known to be in P with only 1x1 particles. We then consider both the occupancy and relocation problems for the case of extremely simple rectangular geometry, but slightly more complicated pieces consisting of 1x2 and 2x1 domino particles, and show that in both cases the problems are PSPACE-complete