Representations of polygons of finite groups

We construct discrete and faithful representations into the isometry group of a hyperbolic space of the fundamental groups of acute negatively curved even-sided polygons of finite groups.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol9/paper43.abs.htm

Flexible Object Manipulation

Flexible objects are a challenge to manipulate. Their motions are hard to predict, and the high number of degrees of freedom makes sensing, control, and planning difficult. Additionally, they have more complex friction and contact issues than rigid bodies, and they may stretch and compress. In this thesis, I explore two major types of flexible materials: cloth and string. For rigid bodies, one of the most basic problems in manipulation is the development of immobilizing grasps. The same problem exists for flexible objects. I have shown that a simple polygonal piece of cloth can be fully immobilized by grasping all convex vertices and no more than one third of the concave vertices. I also explored simple manipulation methods that make use of gravity to reduce the number of fingers necessary for grasping. I have built a system for folding a T-shirt using a 4 DOF arm and a fixed-length iron bar which simulates two fingers. The main goal with string manipulation has been to tie knots without the use of any sensing. I have developed single-piece fixtures capable of tying knots in fishing line, solder, and wire, along with a more complex track-based system for autonomously tying a knot in steel wire. I have also developed a series of different fixtures that use compressed air to tie knots in string. Additionally, I have designed four-piece fixtures, which demonstrate a way to fully enclose a knot during the insertion process, while guaranteeing that extraction will always succeed

Rigidity and Fluidity in Living and Nonliving Matter

Many of the standard equilibrium statistical mechanics techniques do not readily apply to non-equilibrium phase transitions such as the fluid-to-disordered solid transition found in repulsive particulate systems. Examples of repulsive particulate systems are sand grains and colloids. The first part of this thesis contributes to methods beyond equilibrium statistical mechanics to ultimately understand the nature of the fluid-to-disordered solid transition, or jamming, from a microscopic basis. In Chapter 2 we revisit the concept of minimal rigidity as applied to frictionless, repulsive soft sphere packings in two dimensions with the introduction of the jamming graph. Minimal rigidity is a purely combinatorial property encoded via Laman\u27s theorem in two dimensions. It constrains the global, average coordination number of the graph, for instance. Minimal rigidity, however, does not address the geometry of local mechanical stability. The jamming graph contains both properties of global mechanical stability at the onset of jamming and local mechanical stability. We demonstrate how jamming graphs can be constructed using local rules via the Henneberg construction such that these graphs are of the constraint percolation type, where percolation is the study of connected structures in disordered networks. We then probe how jamming graphs destabilize, or become fluid-like, by deleting an edge/contact in the graph and computing the resulting rigid cluster distribution. We also uncover a new potentially diverging lengthscale associated with the random deletion of contacts. In Chapter 3 we study several constraint percolation models, such as k-core percolation and counter-balance percolation, on hyperbolic lattices to better understand the role of loops in such models. The constraints in these percolation models incorporate aspects of local mechanical rigidity found in jammed systems. The expectation is that since these models are indeed easier to analyze than the more complicated problem of jamming, we will gain insight into which constraints affect the nature of the jamming transition and which do not. We find that k = 3-core percolation on the hyperbolic lattice remains a continuous phase transition despite the fact that the loop structure of hyperbolic lattices is different from Euclidean lattices. We also contribute towards numerical techniques for analyzing percolation on hyperbolic lattices. In Chapters 4 and 5 we turn to living matter, which is also nonequilibrium in a very local way in that each constituent has its own internal energy supply. In Chapter 4 we study the fluidity of a cell moving through a confluent tissue, i.e. a group of cells with no gaps between them, via T1 transitions. A T1 transition allows for an edge swap so that a cell can come into contact with new neighbors. Cell migration is then generated by a sequence of such swaps. In a simple four cell system we compute the energy barriers associated with this transition. We then find that the energy barriers in a larger system are rather similar to the four cell case. The many cell case, however, more easily allows for the collection of statistics of these energy barriers given the disordered packings of cell observed in experiments. We find that the energy barriers are exponentially distributed. Such a finding implies that glassy dynamics is possible in a confluent tissue. Finally, in chapter 5 we turn to single cell migration in the extracellular matrix, another native environment of a cell. Experiments suggest that the migration of some cells in the three-dimensional ext ra cellular matrix bears strong resemblance to one-dimensional cell migration. Motivated by this observation, we construct and study a minimal one-dimensional model cell made of two beads and an active spring moving along a rigid track. The active spring models the stress fibers with their myosin-driven contractility and alpha-actinin-driven extendability, while the friction coefficients of the two beads describe the catch/slip bond behavior of the integrins in focal adhesions. Net motion arises from an interplay between active contractility (and passive extendability) of the stress fibers and an asymmetry between the front and back of the cell due to catch bond behavior of integrins at the front of the cell and slip bond behavior of integrins at the back. We obtain reasonable cell speeds with independently estimated parameters. Our model highlights the role of alpha-actinin in three-dimensional cell motility and does not require Arp2/3 actin filament nucleation for net motion

Lagrangian Floer potential of orbifold spheres

For each sphere with three orbifold points, we construct an algorithm to compute the open Gromov–Witten potential, which serves as the quantum-corrected Landau–Ginzburg mirror and is an infinite series in general. This gives the first class of general-type geometries whose full potentials can be computed. As a consequence we obtain an enumerative meaning of mirror maps for elliptic curve quotients. Furthermore, we prove that the open Gromov–Witten potential is convergent, even in the general-type cases, and has an isolated singularity at the origin, which is an important ingredient of proving homological mirror symmetry.National Research Foundation of Korea; 2010-0019516; 2012R1A1A2003117; 2013R1A1A1058646 - National Research Foundation of Kore

Concurrent geometrico-topological tuning of nanoengineered auxetic lattices fabricated by material extrusion for enhancing multifunctionality: multiscale experiments, finite element modeling and data-driven prediction

This study demonstrates the multifunctional performance of innovative 2D auxetic lattices through a combination of multiscale experiments, finite element modeling and data-driven prediction. A geometric modeling approach utilizing Voronoi partitioning and a unique branch-stem-branch (BSB) structure, patterned according to 2D wallpaper symmetries, enables precise concurrent geometric and topological tuning of lattices across a continuous parameter space. Selected architectures are physically realized via material extrusion of polylactic acid (PLA) infused with carbon black (CB). Experimental characterizations, supported by Finite Element modeling, reveal the significant influence of BSB structure's design parameters on mechanical and piezoresistive performance under tensile loading, with a remarkable Poisson’s ratio of -0.74, accompanied by a 15-fold increase in elastic stiffness and a 34-fold increase in strain sensitivity. Additionally, architecturally, and topologically tailored lattice structures exhibit tunable damage sensitivity, reflecting the rate of conductive network destruction within the lattice. This offers insights into the rapidity of cell wall failure, with a steeper slope of the piezoresistance curve in the inelastic regime indicating a faster breakdown and quicker onset of mechanical failure. Integration of Gaussian Process Regression enables accurate exploration of the design space beyond realized structures, highlighting the potential of these intelligent lattice structures for applications such as sensors and in situ health monitoring, marking a significant advancement in multifunctional materials

Discrete Geometry (hybrid meeting)

A number of important recent developments in various branches of discrete geometry were presented at the workshop, which took place in hybrid format due to a pandemic situation. The presentations illustrated both the diversity of the area and its strong connections to other fields of mathematics such as topology, combinatorics, algebraic geometry or functional analysis. The open questions abound and many of the results presented were obtained by young researchers, confirming the great vitality of discrete geometry

Spatial Random Processes and Statistical Mechanics

The workshop focused on the broad area of spatial random processes and their connection to statistical mechanics. The subjects of interest included random walk in random environment, interacting random walks, polymer models, random fields and spin systems, dynamical problems, metastability as well as problems involving two-dimensional conformal geometry. The workshop brought together many leading researchers in these fields who reported to each other on their recent achievements and exchanged ideas for new problems and potential solutions