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

Quantum error correction thresholds for non-Abelian Turaev-Viro codes

We consider a two-dimensional quantum memory of qubits on a torus which encode the extended Fibonacci string-net code, and devise strategies for error correction when those qubits are subjected to depolarizing noise. Building on the concept of tube algebras, we construct a set of measurements and of quantum gates which map arbitrary qubit errors to the string-net subspace and allow for the characterization of the resulting error syndrome in terms of doubled Fibonacci anyons. Tensor network techniques then allow to quantitatively study the action of Pauli noise on the string-net subspace. We perform Monte Carlo simulations of error correction in this Fibonacci code, and compare the performance of several decoders. For the case of a fixed-rate sampling depolarizing noise model, we find an error correction threshold of 4.7% using a clustering decoder. To the best of our knowledge, this is the first time that a threshold has been estimated for a two-dimensional error correcting code for which universal quantum computation can be performed within its code space

Clustering-Based Robot Navigation and Control

In robotics, it is essential to model and understand the topologies of configuration spaces in order to design provably correct motion planners. The common practice in motion planning for modelling configuration spaces requires either a global, explicit representation of a configuration space in terms of standard geometric and topological models, or an asymptotically dense collection of sample configurations connected by simple paths, capturing the connectivity of the underlying space. This dissertation introduces the use of clustering for closing the gap between these two complementary approaches. Traditionally an unsupervised learning method, clustering offers automated tools to discover hidden intrinsic structures in generally complex-shaped and high-dimensional configuration spaces of robotic systems. We demonstrate some potential applications of such clustering tools to the problem of feedback motion planning and control. The first part of the dissertation presents the use of hierarchical clustering for relaxed, deterministic coordination and control of multiple robots. We reinterpret this classical method for unsupervised learning as an abstract formalism for identifying and representing spatially cohesive and segregated robot groups at different resolutions, by relating the continuous space of configurations to the combinatorial space of trees. Based on this new abstraction and a careful topological characterization of the associated hierarchical structure, a provably correct, computationally efficient hierarchical navigation framework is proposed for collision-free coordinated motion design towards a designated multirobot configuration via a sequence of hierarchy-preserving local controllers. The second part of the dissertation introduces a new, robot-centric application of Voronoi diagrams to identify a collision-free neighborhood of a robot configuration that captures the local geometric structure of a configuration space around the robot’s instantaneous position. Based on robot-centric Voronoi diagrams, a provably correct, collision-free coverage and congestion control algorithm is proposed for distributed mobile sensing applications of heterogeneous disk-shaped robots; and a sensor-based reactive navigation algorithm is proposed for exact navigation of a disk-shaped robot in forest-like cluttered environments. These results strongly suggest that clustering is, indeed, an effective approach for automatically extracting intrinsic structures in configuration spaces and that it might play a key role in the design of computationally efficient, provably correct motion planners in complex, high-dimensional configuration spaces

LIPIcs, Volume 274, ESA 2023, Complete Volume

LIPIcs, Volume 274, ESA 2023, Complete Volum

Metrics for Materials Discovery

The vast corpus of experimental solid state data has enabled a variety of statistical methods to be applied in high throughput materials discovery. There are many techniques for representing a material into a numeric vector, and many investigations apply the Euclidean distance between these vectors to judge similarity. This thesis investigates applications of non-Euclidean metrics, in particular optimal transport measures, or the Earth Mover’s Distance (EMD), to quantify the similarity between two materials for use in computational workflows, with a focus on solid state electrolytes (SSEs). Chapter 1 introduces the field of lithium conducting SSEs for use in batteries, as well as an introductory precursor for some of the machine learning concepts, for those without exposure to this field. The EMD is a function which returns the minimal quantity of work that is required to transform one distribution into another, and a tutorial on how to compute the EMD using the simplest known technique is provided given its relevance to later chapters. In chapter 2 the discussion around the EMD is continued, and we introduce the workflow that has been developed for quantifying the chemical similarity of materials with the Element Movers Distance (ElMD). Given the affect that minor dopants can have on physical properties, it is imperative that we use techniques that capture nuanced differences in stoichiometry between materials. The relationships between the binary compounds of the ICSD are shown to be well captured using this metric. Larger scale maps of materials space are generated, and used to explore some of the known SSE chemistries. At the beginning of the PhD, there were no substantial datasets of lithium SSEs available, as such chapter 3 outlines the lengthy process of gathering this data. This resulted in the Liverpool ionics dataset, containing 820 entries, with 403 unique compositions having conductivities measured at room temperature. The performance of leading composition based property prediction models against this dataset is rigorously assessed. The resultant classification model gives a strong enough improvement over human guesswork that it may be used for screening in future studies. At present, materials datasets are disparate and scattered. Using the ElMD in chapter 4, we investigate how different metric indexing methods may be used to partition gathered datasets of compositions. This enables very fast nearest neighbour queries allowing the automated retrieval of similar compounds across millions of records in milliseconds. Chapter 5 introduces the technique Percifter for characterizing crystal structures, based on the principles of persistent homology (PH). This increasingly popular technique is used in materials science to describe the topology of a crystal. Percifter seeks to improve the stability of these representations for different choices of unit cells. These similarities may be observed directly, or compared through the EMD

Ahlfors circle maps and total reality: from Riemann to Rohlin

This is a prejudiced survey on the Ahlfors (extremal) function and the weaker {\it circle maps} (Garabedian-Schiffer's translation of "Kreisabbildung"), i.e. those (branched) maps effecting the conformal representation upon the disc of a {\it compact bordered Riemann surface}. The theory in question has some well-known intersection with real algebraic geometry, especially Klein's ortho-symmetric curves via the paradigm of {\it total reality}. This leads to a gallery of pictures quite pleasant to visit of which we have attempted to trace the simplest representatives. This drifted us toward some electrodynamic motions along real circuits of dividing curves perhaps reminiscent of Kepler's planetary motions along ellipses. The ultimate origin of circle maps is of course to be traced back to Riemann's Thesis 1851 as well as his 1857 Nachlass. Apart from an abrupt claim by Teichm\"uller 1941 that everything is to be found in Klein (what we failed to assess on printed evidence), the pivotal contribution belongs to Ahlfors 1950 supplying an existence-proof of circle maps, as well as an analysis of an allied function-theoretic extremal problem. Works by Yamada 1978--2001, Gouma 1998 and Coppens 2011 suggest sharper degree controls than available in Ahlfors' era. Accordingly, our partisan belief is that much remains to be clarified regarding the foundation and optimal control of Ahlfors circle maps. The game of sharp estimation may look narrow-minded "Absch\"atzungsmathematik" alike, yet the philosophical outcome is as usual to contemplate how conformal and algebraic geometry are fighting together for the soul of Riemann surfaces. A second part explores the connection with Hilbert's 16th as envisioned by Rohlin 1978.Comment: 675 pages, 199 figures; extended version of the former text (v.1) by including now Rohlin's theory (v.2