Math + Tech

Emerging Applications of Topology

Introduction

Suppose there is an intricate map of four landmasses connected by seven bridges, and people want to walk across each bridge once and only once to end up at the same location as they started. Since all possible paths only depend on the bridges’ connectivities, separations, and intersections, their inessential features like angles and lengths can be removed, therefore simplified. This is how Leonard Euler (1707-1783) developed his famous resolution (Martin 2).

In fact, that is the very essence of topology: abstracting systems to truly focus on their fundamental spatial properties. This discipline is also called “rubber-sheet geometry” because objects may be deformed and stretched while still maintaining their qualitative properties (“Topology”). Topologically continuous shapes, such as a sphere, can be legitimately transformed like rubber into any other shape of its class, such as a cube, without tearing or gluing. Thus, topologists are jokingly identified as mathematicians who can't tell the difference between a coffee mug and a donut; both objects are homeomorphic through continuous deformation, a notion of topological equivalence.

Most research on topology has been done in the 20th century, making the study a relatively recent branch of mathematics. Today, topology is applied everywhere: interconnecting computers with local area networks, examining how enzymes unknot DNA, and even finding the universe’s shape with the Ricci flow (O’Shea 176). Scholars are consistently demonstrating how interesting topological structures materialize in real-life settings and how its implementations can profoundly help optimize scientific solutions. This article will show topology as a preeminent mathematical discipline that is and will continue gaining importance through practical applications over diverse academic fields.

Types of Topology

Before immersing oneself in the applications of topology, it would be appropriate to discuss the primary subfields of such a broad study. Throughout history, topology has been generally divided into four branches, as listed below. Understanding each type of topology and its basic terms is crucial to appreciating how such abstract concepts can be diversely applied to real-life problems.

General Topology: Serving as the most foundational branch to establish all other branches, general topology focuses on the constructional analysis of topological spaces and its immanent concepts. It introduces the dominant concept of manifolds; manifolds are topological spaces that locally resemble the Euclidean space. One-dimensional manifolds include lines and circles, whereas two-dimensional manifolds are called surfaces. General topology is also named Point-Set Topology because local properties in connectivity, compactness, and continuity are primarily studied. In order to define distances in such spaces, metric numbers are assigned to form metric spaces, which generalize the most universal topological spaces (Rusin 54).

Algebraic Topology (previously Combinatorial Topology): By using algebraic objects (e.g. groups, rings) and abstract algebra, algebraic topology often renders topological problems into algebraic problems for more convenient solutions. Such solutions necessitate the search for algebraic invariants to quantitatively classify topological spaces of geometric figures; predominantly used invariants include homotopy and homology, both procedures introduced by Henri Poincaré (1854-1912). Algebraic topology was also called Combinatorial Topology until topological invariants of spaces were no longer considered as combinatorial decompositions. Combinatorial topology regarded objects as combinations of many simple pieces (Barr 192).

Geometric Topology: Specifically assigning geometric structures to topological spaces, geometric topology is the study of geometrical relations in low-dimensional manifolds. This particular branch originated in order to distinguish spaces which were homotopy equivalent yet not homeomorphic, hence the inception of useful topics in local flatness, orientability, surgery theory, and knot theory (which will be examined further in Section 3.2.). One key theorem in geometric topology is Thurston’s geometrization conjecture, which states that every 3-manifold can be cut into pieces of 8 possible geometries. This then led to Grigori Perelman’s proof of the Poincaré Conjecture by using advanced geometric concepts like Ricci flows (Collins 88).

Differential Topology: As the name indicates, differential topology focuses on differentiable manifolds—topological manifolds with smoothness at each point to allow calculus—and its differentiable functions. Thus, certain obstructions arise when dealing with topological equivalences and deformations because differential topology only defines manifolds with smooth structures. For instance, although topologically equivalent (homeomorphic), the cube and the sphere are not differentiably equivalent (diffeomorphic). Studying topological properties of vector fields in electricity and magnetism often give utility to differential topology (Hare 1).

Fields of Applications

Finally, the heart of the matter: the emerging applications of topology. Below are multiple areas of study that heavily depend on effective topological implementations. Most of what’s discussed below are relatively modern fields utilizing topological algorithms more than ever before. Each topic below, although concise, will together demonstrate the vast scale of applied topology in today’s academia.

Network Science When interconnecting networks that share information in data packets, whether it’s at home or an office, optimizing the network topology of the paths’ layout is critical. Networks are defined by physical topologies (the geometric positioning of multiple network components like cable installation and device location) and logical topologies (how nonphysical data are transmitted within a network). Alike the local area networks—LANs—used in office buildings and computer laboratories, graphical network topologies help map the interconnection of nodes, the endpoints of branches in computer workstations. Such structural topologies matter immensely because the distance traveled by data packets determine the speed of communication, the costs of cabling wires, and even the prevention of connection failures when cyberattacks defy security policies (Rangan 6).

There are distinct advantages and disadvantages to selecting different communication network topologies. For instance, ring topologies can easily span longer distances than other networks, but a performance failure of one computer can topple the whole network—hence the difficulty of adding or removing nodes. On the other hand, mesh topologies offer supreme protection of malfunctioning computers with surplus links, but the necessitated amount of cabling is excessive and costly. Therefore, schools and offices must seek the most suitable and economic type of network topology, inevitably utilizing optimally compacted structures through analyses in general topology (Pandya 24).

Computational Biology Large polymeric molecules, like human DNA, entangle and knot themselves due to their long lengths, and because such structural variations influence the manufacture of their products, biologists began seeking ways to topologically categorize different knot structures. Furthermore, once biologists realized that many viruses attack cells by knotting DNA molecules with scissions and stitches, the need to characterize each virus’ knot signature in diseased cells augmented. Soon enough, knot theory, a chief topic in geometric topology, helped bring about solutions by offering insight into visualizing how compressed DNA in the genes must be quickly unknotted by enzymes that facilitate replication or transcription. Knot theory allowed biologists to discover the topological mechanisms of enzymes and viruses (Brown 1).

Computational biology, or bioinformatics, mainly develops systematic algorithms from an unprecedented abundance of biological data. And in order for computational biologists to thoroughly study macromolecules and protein structures, topological methods are regularly used to understand and answer fundamental biological questions. Other biological fields even apply differential topology to identify multivariate interactions, which were recently instrumental in neuroscience to deduce the complex interactions of neurons. Many more challenges in biology necessitate the powerfully invariant, fundamentally qualitative tools that only geometric topology can provide (Darcy et al. 68).

Robotics Modern research in robotics show that topology has an indispensable role to play in making complex ideas of constructing robots a reality: robotics takes advantage of topology through what’s called “configuration spaces”. A configuration space describes a robot’s diverse set of possible positions as a manifold, and such a concept is useful because a dominant problem in robotics is motion planning—searching for optimal paths in the configuration space for the robot to move in recognized spaces of the physical system. Analyses in algebraic and geometric topology are then used to seek the most favorable path between two particular points in the configuration space, illustrating the robot’s optimal joint movements at desired locations. When finding configuration spaces like the graphic representation below, one would ask what its homology groups are—a measure of the hole structure of a topological space—and whether this configuration space is homeomorphic to another vector space (Farber 12).

Topological robotics is a modern field at the crossroads of topology, engineering, and computer science, studying both theoretical and applied problems. Today, countless robots like automated guided vehicles (AGVs) are coordinated in factories to move along specialized wires topologically embedded in the floor’s guidance system. Moreover, spatial layouts in urban spaces are greatly dependent on systematic configuration spaces to optimize movement distances between locations (Barton et al. 511).

Econometrics Ever since the Great Depression, economists started focusing on global macroeconomic networks, interdependent on the economic systems of heterogeneous entities (countries). Such recurrent financial crises called for a more comprehensive understanding of the highly interconnected global economic systems. Thus, by incorporating topological analyses in econometrics, a mathematical branch in economics to describe economic systems, scholars were able to identify potential long-run trends from “seemingly disparate characteristics” of socioeconomic topologies (Klinedinst et al. 3).

Economic systems are often viewed as an evolving network, in which countries interact with other countries by strengthening profitable connections and terminating overpriced channels. As shown in the diagram below, many countries today maintain multiple trade relations, and its topological connectivity profile indicates the expected impact of a spreading financial crisis. For example, the more direct trade channels a nation has, the more amplified and accelerated its economic crises will be, mainly due to entangled domino effects from other linkages (Lee et al. 1).

Theorems in differential topology and algebraic topology facilitated the development of many crucial concepts in economics, namely the Nash equilibrium—a solution concept in game theory established by John F. Nash, Jr.—which was ultimately proved by the Brouwer fixed-point theorem in topology (Dilkina et al. 42- 43). Many other economic theories, such as the microeconomic general equilibrium theory, largely depend on topological theorems. Moreover, analyzing different topological networks of economic systems can provide mathematical insight into how the society is financially functioning. For instance, by producing empirical models of labor markets which connect individuals’ employment situations, economists found that the differences in the topology of such networks greatly influence the inequalities in wages as well as the duration and correlation of their unemployment rates (Klinedinst et al. 11).

Potential Future Applications

Considering the growingly extensive applications above and the recent outburst of research in applied topology, it is sensible to state that this momentum will be maintained, if not further accelerated, to numerous more fields in the future. Below is a coverage of several projected topological applications and their current progress for what lies ahead.

Currently, there are plans being made in areas such as quantum mechanics and the topological quantum field theory to compute advanced topological invariants and decide whether certain spaces are homeomorphic or not (Jinsong et al. 1583). Similarly, cosmologists are using differential topology to further describe the space-time structure of the universe. Cosmic topology is especially useful when answering connectivity questions: Is the universe spatially closed or open? and Does its spacetime structure have holes or handles? Radically high-speed photography with inkjet printers are also in progress using topological analyses (Libii 198), and recent fMRI research on stimulus-response associations of the brain’s prefrontal cortex are examining neural networks through effective topology tests (Masunaga et al. 609).

Along with the growth of computer science, a brand new field called Computational Topology is currently being established to apply topological techniques when developing algorithms for problems in data and shape analysis. Much research is being done and will soon resolve various topological problems such as homology cycle extractions, manifold reconstructions, and topology inferences. Such growth in these fields are triggering more synergistic work between mathematics, computer science, and engineering (Basener 51).

Nowadays, computer graphics and topological designing capabilities are rapidly advancing, and they’re expected to expand in the future after computer scientists program more algorithms and systems that draw better-approximated surfaces. Additionally, many research projects today are exploring how industrial designs in the future can be fully optimized through topological analyses (Del Tin 2).

Conclusion

This article introduced topology, a relatively modern branch of mathematics studying the core spatial properties of shapes and surfaces, discussed its basic types, and showed the sheer diversity and immensity in applications of such an abstract field. Through the compact survey of topological applications in sections above, this article proves topology’s ascendancy as well as its auspicious prospects.

Topology is truly omnipresent—and that much more functional. Jean Piaget, a renowned Swiss psychologist and philosopher, once said “A child[’s] … first geometrical discoveries are topological.” Still, despite its apparent ubiquity, topology is not universally taught in primary and secondary schools, often underrated due to its seemingly esoteric terminology. Yet as this article demonstrates, topology is undoubtedly a leading field not only in mathematics but also in network science, biology, physics, robotics, economics, design, and fundamentally any scientific domain. It is therefore rightful to deem topology as a prevalent language of modern mathematics and its widespread applications as the powerful tools to abstract and optimize today’s world.

Bibliography

Barr, Stephen. Experiments in Topology. New York: Thomas Y. Crowell Company, 1964. 1-206. Print.

Barton, Kevin R., Deltcho Valtchanov, and Colin Ellard. "Seeing Beyond Your Visual Field: The Influence Of Spatial Topology And Visual Field On Navigation Performance." Environment and Behavior 46.4 (2014): 507-529. Professional Development Collection. Web. 17 Aug. 2014.

Basener, William F. "Module Presentations for use with Topology and Its Applications." Topology and Its Applications. Ed. William F. Basener. National Science Foundation, 17 June 2004. Web. 17 Aug. 2014.

Brown, Ronnie. Pop Math. Ed. Cara Quinton. The London Mathematical Society, 1996. Web. 17 Aug. 2014.

Collins, Graham P. "The Shape Of Space." Scientific American 291.1 (2004): 86-93. Health Source - Consumer Edition. Web. 17 Aug. 2014.

Darcy, Isabel K., and De Witt Sumners. "Applications of Topology to DNA." Banach Center Publications 42 (1998): 65-75. Web. 12 Dec. 2014.

Del Tin, Laura. Delft University of Technology. Ed. Matthijs Langelaar. TUDelft, 2009. Web. 18 Aug. 2014.

Dilkina, Bistra, Carla P. Gomes, and Ashish Sabharwal. "The Impact of Network Topology on Pure Nash Equilibria in Graphical Games." 22nd Conference on Artificial Intelligence (2007): 42-49. Web. 20 Nov. 2014.

Farber, Michael. "Topology and Robotics." Universidad de Málaga. Málaga. 29 Oct. 2007. Web. 25 Nov. 2014.

Hare, Kathryn. "What is Topology?." Pure Mathematics. Ed. Lis D'Alessio. University of Waterloo, n.d. Web. 11 Sept. 2014.

Jinsong, Zhang, et al. "Topology-Driven Magnetic Quantum Phase Transition In Topological Insulators." Science 339.6127 (2013): 1582-1586. Professional Development Collection. Web. 17 Aug. 2014.

Kahle, Matthew. "Configuration Spaces: Combinatorics." Wake Forest University. Winston-Salem. 9 Feb. 2013. Web. 30 Nov. 2014.

Klinedinst, Mark, and Charles Rock. "Topology of Economic Systems." Journal of Economic Literature (2009): 2-31. Web. 20 Dec. 2014.

Lee, Kyu-Min, Jae-Suk Yang, Gunn Kim, and Jaesung Lee. "Impact of the Topology of Global Macroeconomic Network on the Spreading of Economic Crises." PLoS ONE 6.3 (2011): 1-9. Web. 20 Oct. 2014.

Libii, Josue Njock. "The Illustrated Topology Of Liquid Drops During Formation." Physics Education 39.2 (2004): 197-200. ERIC. Web. 17 Aug. 2014.

Lynn, G. (1999) Animate form: a book and interactive CD-ROM. Architectural Press, Princeton University. Web. 17 Oct. 2014.

Martin, Francisco. "The Seven Bridges of Königsberg." University of Massachusetts. Hampshire County. Jan. 2010. Web. 8 Oct. 2014.

Masunaga, Hiromi, et al. "Neural Substrates Of The Topology Test To Measure Fluid Reasoning: An fMRI Study." Intelligence 36.6- (2008): 607-615. ERIC. Web. 17 Aug. 2014.

O’Shea, Donal. The Poincaré Conjecture: In Search of the Shape of the Universe. New York: Walker Publishing Company, Inc., 2007. 1-201. Print.

Pandya, Kartik. "Network Structure or Topology." International Journal of Advance Research in Computer Science and Management Studies 1.2 (2013): 22-27. Web. 27 Nov. 2015.

Rangan, Prahalad. "Polymorphic Attacks and Network Topology: Application of Concepts from Natural Systems." ProQuest LLC (2010). ERIC. Web. 17 Aug. 2014.

Rusin, Dave. "54: General Topology." The Mathematical Atlas. Northern Illinois University, 14 Jan. 2000. Web. 18 Nov. 2014.

“Topology.” Funk and Wagnalls New World Encyclopedia (2014): 1p. 1. Funk & Wagnalls New World Encyclopedia. Web. 17 Aug. 2014.