Minimum vertex degree conditions for loose spanning trees in 3-graphs

In 1995, Koml\'os, S\'ark\"ozy and Szemer\'edi showed that every large $n$-vertex graph with minimum degree at least $(1/2 + \gamma)n$ contains all spanning trees of bounded degree. We consider a generalization of this result to loose spanning hypertrees in 3-graphs, that is, linear hypergraphs obtained by successively appending edges sharing a single vertex with a previous edge. We show that for all $\gamma$ and $\Delta$, and $n$ large, every $n$-vertex 3-uniform hypergraph of minimum vertex degree $(5/9 + \gamma)\binom{n}{2}$ contains every loose spanning tree $T$ with maximum vertex degree $\Delta$. This bound is asymptotically tight, since some loose trees contain perfect matchings.Comment: 18 pages, 1 figur

Simplicial spanning trees in random Steiner complexes

A spanning tree $T$ in a graph $G$ is a sub-graph of $G$ with the same vertex set as $G$ which is a tree. In 1981, McKay proved an asymptotic result regarding the number of spanning trees in random $k$-regular graphs. In this paper we prove a high-dimensional generalization of McKay's result for random $d$-dimensional, $k$-regular simplicial complexes on $n$ vertices, showing that the weighted number of simplicial spanning trees is of order $(\xi_{d,k}+o(1))^{\binom{n}{d}}$ as $n\to\infty$, where $\xi_{d,k}$ is an explicit constant, provided $k> 4d^2+d+2$. A key ingredient in our proof is the local convergence of such random complexes to the $d$-dimensional, $k$-regular arboreal complex, which allows us to generalize McKay's result regarding the Kesten-McKay distribution.Comment: 26 pages, 2 figure

Random Discrete Morse Theory and a New Library of Triangulations

1) We introduce random discrete Morse theory as a computational scheme to measure the complicatedness of a triangulation. The idea is to try to quantify the frequence of discrete Morse matchings with a certain number of critical cells. Our measure will depend on the topology of the space, but also on how nicely the space is triangulated. (2) The scheme we propose looks for optimal discrete Morse functions with an elementary random heuristic. Despite its na\"ivet\'e, this approach turns out to be very successful even in the case of huge inputs. (3) In our view the existing libraries of examples in computational topology are `too easy' for testing algorithms based on discrete Morse theory. We propose a new library containing more complicated (and thus more meaningful) test examples.Comment: 35 pages, 5 figures, 7 table

From Large to In nite Random Simplicial Complexes.

PhD ThesesRandom simplicial complexes are a natural higher dimensional generalisation to the models of random graphs from Erd}os and R enyi of the early 60s. Now any topological question one may like to ask raises a question in probability - i.e. what is the chance this topological property occurs? Several models of random simplicial complexes have been intensely studied since the early 00s. This thesis introduces and studies two general models of random simplicial complexes that includes many well-studied models as a special case. We study their connectivity and Betti numbers, prove a satisfying duality relation between the two models, and use this to get a range of results for free in the case where all probability parameters involved are uniformly bounded. We also investigate what happens when we move to in nite dimensional random complexes and obtain a simplicial generalisation of the Rado graph, that is we show the surprising result that (under a large range of parameters) every in nite random simplicial complexes is isomorphic to a given countable complex X with probability one. We show that this X is in fact homeomorphic to the countably in nite ball. Finally, we look at and construct nite approximations to this complex X, and study their topological properties

Planning in constraint space for multi-body manipulation tasks

Robots are inherently limited by physical constraints on their link lengths, motor torques, battery power and structural rigidity. To thrive in circumstances that push these limits, such as in search and rescue scenarios, intelligent agents can use the available objects in their environment as tools. Reasoning about arbitrary objects and how they can be placed together to create useful structures such as ramps, bridges or simple machines is critical to push beyond one's physical limitations. Unfortunately, the solution space is combinatorial in the number of available objects and the configuration space of the chosen objects and the robot that uses the structure is high dimensional. To address these challenges, we propose using constraint satisfaction as a means to test the feasibility of candidate structures and adopt search algorithms in the classical planning literature to find sufficient designs. The key idea is that the interactions between the components of a structure can be encoded as equality and inequality constraints on the configuration spaces of the respective objects. Furthermore, constraints that are induced by a broadly defined action, such as placing an object on another, can be grouped together using logical representations such as Planning Domain Definition Language (PDDL). Then, a classical planning search algorithm can reason about which set of constraints to impose on the available objects, iteratively creating a structure that satisfies the task goals and the robot constraints. To demonstrate the effectiveness of this framework, we present both simulation and real robot results with static structures such as ramps, bridges and stairs, and quasi-static structures such as lever-fulcrum simple machines.Ph.D