Computational Geometric and Algebraic Topology

Computational topology is a young, emerging field of mathematics that seeks out practical algorithmic methods for solving complex and fundamental problems in geometry and topology. It draws on a wide variety of techniques from across pure mathematics (including topology, differential geometry, combinatorics, algebra, and discrete geometry), as well as applied mathematics and theoretical computer science. In turn, solutions to these problems have a wide-ranging impact: already they have enabled significant progress in the core area of geometric topology, introduced new methods in applied mathematics, and yielded new insights into the role that topology has to play in fundamental problems surrounding computational complexity. At least three significant branches have emerged in computational topology: algorithmic 3-manifold and knot theory, persistent homology and surfaces and graph embeddings. These branches have emerged largely independently. However, it is clear that they have much to offer each other. The goal of this workshop was to be the first significant step to bring these three areas together, to share ideas in depth, and to pool our expertise in approaching some of the major open problems in the field

Doctor of Philosophy

dissertationComputer programs have complex interactions with their underlying hardware, exhibiting complex behaviors as a result. It is critical to understand these programs, as they serve an importantrole: researchers use them to express new ideas in computer science, while many others derive production value from them. In both cases, program understanding leads to mastery over these functions, adding value to human endeavors. Memory behavior is one of the hallmarks of general program behavior: it represents the critical function of retrieving data for the program to work on; it often reflects the overall actions taken by the program, providing a signature of program behavior; and it is often an important performance bottleneck, as the the memory subsystem is typically much slower than the processor. These reasons justify an investigation into the memory behavior of programs. A memory reference trace is a list of memory transactions performed by a program at runtime, a rich data source capturing the whole of a program's interaction with the memory subsystem, and a clear starting point for investigating program memory behavior. However, such a trace is extremely difficult to interpret by mere inspection, as it consists solely of many, many addresses and operation codes, without any more structure or context. This dissertation proposes to use visualization to construct images and animations of the data within a reference trace, thereby visually transmitting structures and events as encoded in the trace. These visualization approaches are designed with different focuses, meant to expose various aspects of the trace. For instance, the time dimension of the reference traces can be handled either with animation, showing events as they occur, or by laying time out in a spatial dimension, giving a view of the entire history of the trace at once. The approaches also vary in their level of abstraction from the hardware: some are concretely connected to representations of the memory itself, while others are more free-form, using more abstract metaphors to highlight general behaviors and patterns, which in turn characterize the program behavior. Each approach delivers its own set of insights, as demonstrated in this dissertation

Multiscale Topological Trajectory Classification with Persistent Homology

Abstract—Topological approaches to studying equivalence classes of trajectories in a configuration space have recently received attention in robotics since they allow a robot to reason about trajectories at a high level of abstraction. While recent work has approached the problem of topological motion planning under the assumption that the configuration space and obsta-cles within it are explicitly described in a noise-free manner, we focus on trajectory classification and present a sampling-based approach which can handle noise, which is applicable to general configuration spaces and which relies only on the availability of collision free samples. Unlike previous sampling-based approaches in robotics which use graphs to capture information about the path-connectedness of a configuration space, we construct a multiscale approximation of neighbor-hoods of the collision free configurations based on filtrations of simplicial complexes. Our approach thereby extracts additional homological information which is essential for a topological trajectory classification. By computing a basis for the first persistent homology groups, we obtain a multiscale classification algorithm for trajectories in configuration spaces of arbitrary dimension. We furthermore show how an augmented filtration of simplicial complexes based on a cost function can be defined to incorporate additional constraints. We present an evaluation of our approach in 2, 3, 4 and 6 dimensional configuration spaces in simulation and using a Baxter robot. I

Aspects of Invariant Manifold Theory and Applications

Recent years have seen a surge of interest in "data-driven" approaches to determine the equations governing complex systems. Yet in spite of modern computing advances, the high dimensionality of many systems --- such as those occurring in biology and robotics --- renders direct machine learning approaches infeasible. This dissertation develops tools for the experimental study of complex systems, based on mathematical concepts from dynamical systems theory. Our approach uses the fact that parsimonious assumptions often lead to strong insights from dynamical systems theory; such insights can be leveraged in learning algorithms to mitigate the “curse of dimensionality” and make these algorithms practical. Our first contribution concerns nonlinear oscillators. Oscillators are ubiquitous in nature, and usually associated with the existence of an "asymptotic phase" which governs the long-term dynamics of the oscillator. We show that asymptotic phase can be expressed as a line integral with respect to a uniquely defined closed differential 1-form, and provide an algorithm for estimating this "ToF" from observational data. Unlike all previously available data-driven phase estimation methods, our algorithm can: (i) use observations that are much shorter than a cycle; (ii) recover phase within the entire region for which data convergent to the limit cycle is available; (iii) recover the phase response curves (PRC-s) that govern weak oscillator coupling; (iv) show isochron curvature, and recover nonlinear features of isochron geometry. Our method may find application wherever models of oscillator dynamics need to be constructed from measured or simulated time-series. Our next contribution concerns locomotion systems which are dominated by viscous friction in the sense that without power expenditure they quickly come to a standstill. From geometric mechanics, it is known that in the ``Stokesian'' (viscous; zero Reynolds number) limit, the motion is governed by a reduced order "connection'' model that describes how body shape change produces motion for the body frame with respect to the world. In the "perturbed Stokes regime'' where inertial forces are still dominated by viscosity, but are not negligible (low Reynolds number), we show that motion is still governed by a functional relationship between shape velocity and body velocity, but this function is no longer connection-like. We derive this model using results from noncompact NHIM theory in a singular perturbation framework. Using a normal form derived from theoretical properties of this reduced-order model, we develop an algorithm that estimates an approximation to the dynamics near a cyclic body shape change (a "gait") directly from observational data of shape and body motion. Our algorithm has applications to the study of optimality of animal gaits, and to hardware-in-the-loop optimization to produce gaits for robots. Finally, we make fundamental contributions to NHIM theory: we prove that the global stable foliation of a NHIM is a $C^0$ disk bundle, and we prove that the dynamics restricted to the stable manifold of a compact inflowing NHIM are globally topologically conjugate to the linearized transverse dynamics at the NHIM restricted to the stable vector bundle. We also give conditions ensuring $C^k$ versions of our results, and we illustrate the theory by giving applications to geometric singular perturbation theory in the case of an attracting critical manifold: we show that the domain of the Fenichel Normal Form can be extended to the entire global stable manifold, and under additional nonresonance assumptions we derive a smooth global linear normal form.PHDElectrical and Computer EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/147642/1/kvalheim_1.pd