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What’s the Shape of the Universe? Mathematicians Use Topology to Study the Shape of the World and Everything in it

admin — Fri, 28 Feb 2025 19:23:41 +0000

What’s the Shape of the Universe? Mathematicians Use Topology to Study the Shape of the World and Everything in it admin Fri, 02/28/2025 - 14:23

When you look at your surrounding environment, it might seem like you’re living on a flat plane. After all, this is why you can navigate a new city using a map: a flat piece of paper that represents all the places around you. This is likely why some people in the past believed the earth to be flat. But most people now know that is far from the truth.

You live on the surface of a giant sphere, like a beach ball the size of the Earth with a few bumps added. The surface of the sphere and the plane are two possible 2D spaces, meaning you can walk in two directions: north and south or east and west.

What other possible spaces might you be living on? That is, what other spaces around you are 2D? For example, the surface of a giant doughnut is another 2D space.

Through a field called geometric topology, mathematicians like me study all possible spaces in all dimensions. Whether trying to design secure sensor networks, mine data or use origami to deploy satellites, the underlying language and ideas are likely to be that of topology.

The Shape of the Universe

When you look around the universe you live in, it looks like a 3D space, just like the surface of the Earth looks like a 2D space. However, just like the Earth, if you were to look at the universe as a whole, it could be a more complicated space, like a giant 3D version of the 2D beach ball surface or something even more exotic than that.

A doughnut, also called a torus, is a shape that you can move across in two directions, just like the surface of the Earth. YassineMrabet via Wikimedia Commons, CC BY-NC-SA

While you don’t need topology to determine that you are living on something like a giant beach ball, knowing all the possible 2D spaces can be useful. Over a century ago, mathematicians figured out all the possible 2D spaces and many of their properties.

In the past several decades, mathematicians have learned a lot about all of the possible 3D spaces. While we do not have a complete understanding like we do for 2D spaces, we do know a lot. With this knowledge, physicists and astronomers can try to determine what 3D space people actually live in.

While the answer is not completely known, there are many intriguing and surprising possibilities. The options become even more complicated if you consider time as a dimension.

To see how this might work, note that to describe the location of something in space – say a comet – you need four numbers: three to describe its position and one to describe the time it is in that position. These four numbers are what make up a 4D space.

Now, you can consider what 4D spaces are possible and in which of those spaces do you live.

Topology in Higher Dimensions

At this point, it may seem like there is no reason to consider spaces that have dimensions larger than four, since that is the highest imaginable dimension that might describe our universe. But a branch of physics called string theory suggests that the universe has many more dimensions than four.

There are also practical applications of thinking about higher dimensional spaces, such as robot motion planning. Suppose you are trying to understand the motion of three robots moving around a factory floor in a warehouse. You can put a grid on the floor and describe the position of each robot by their x and y coordinates on the grid. Since each of the three robots requires two coordinates, you will need six numbers to describe all of the possible positions of the robots. You can interpret the possible positions of the robots as a 6D space.

As the number of robots increases, the dimension of the space increases. Factoring in other useful information, such as the locations of obstacles, makes the space even more complicated. In order to study this problem, you need to study high-dimensional spaces.

There are countless other scientific problems where high-dimensional spaces appear, from modeling the motion of planets and spacecraft to trying to understand the “shape” of large datasets.

Tied Up In Knots

Another type of problem topologists study is how one space can sit inside another.

For example, if you hold a knotted loop of string, then we have a 1D space (the loop of string) inside a 3D space (your room). Such loops are called mathematical knots.

The study of knots first grew out of physics but has become a central area of topology. They are essential to how scientists understand 3D and 4D spaces and have a delightful and subtle structure that researchers are still trying to understand.

Knots are examples of spaces that sit inside other spaces. Jkasd/Wikimedia Commons

In addition, knots have many applications, ranging from string theory in physics to DNA recombination in biology to chirality in chemistry.

What Shape Do You Live On?

Geometric topology is a beautiful and complex subject, and there are still countless exciting questions to answer about spaces.

For example, the smooth 4D Poincaré conjecture asks what the “simplest” closed 4D space is, and the slice-ribbon conjecture aims to understand how knots in 3D spaces relate to surfaces in 4D spaces.

Topology is currently useful in science and engineering. Unraveling more mysteries of spaces in all dimensions will be invaluable to understanding the world in which we live and solving real-world problems.

This article is republished from The Conversation under a Creative Commons license. Read the original article.

Summary sentence

Whether trying to design secure sensor networks, mine data or use origami to deploy satellites, the underlying language and ideas are likely to be that of topology.

Summary

Whether trying to design secure sensor networks, mine data or use origami to deploy satellites, the underlying language and ideas are likely to be that of topology.

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Fri, 02/28/2025 - 12:00

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John Etnyre, Professor of Mathematics, Georgia Institute of Technology

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Shelley Wunder-Smith
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Fri, 02/28/2025 - 14:22

Mathematics, Physics Use Moths and Origami Structures for Innovative Defense Research

bwaye3 — Mon, 25 Apr 2022 16:30:03 +0000

Mathematics, Physics Use Moths and Origami Structures for Innovative Defense Research bwaye3 Mon, 04/25/2022 - 12:30

Georgia Tech has received two Department of Defense (DoD) 2022 Multidisciplinary University Research Initiative (MURI) awards totaling almost $14 million. The highly competitive government program supports interdisciplinary teams of investigators developing innovative solutions in DoD interest areas. This year, the DoD awarded $195 million to 28 research teams across the country.

Georgia Tech’s MURIs are both primarily within the School of Physics. First, Simon Sponberg, a Dunn Family Associate Professor of Physics and Biological Sciences, leads a team discovering how animals strategically use sensing and cognition to make decisions in complex environments. The project, Fast, Lexicographic Agile Perception Integrates Decision and Control in a Spike-Resolved, Sensorimotor Program (FLAP), specifically addresses the core DoD topic area of understanding neural systems integration for competent autonomy in decision and control.

“We have all these great, sophisticated algorithms for processing big data, but an animal doesn't have time to process a million samples of its environment and then figure out what’s a predator,” said Sponberg.

Studying moths for their agile, sophisticated flying and complex sensing abilities, the team will record electrical activity in the brain to determine how the moths make decisions and use natural language processing techniques to see how a moth derives meaning from sensory cues and movements. The goal is to develop an information processing framework that enables quick, flexible decision-making that could facilitate the next generation of autonomous bio-inspired systems and better integrate living systems with engineered technologies

The interdisciplinary nature of the team makes complex research possible. Half the team is made of experimentalists: Sponberg specializes in sensors connected to motor systems with precisely timed signals; Jeff Riffell, a professor at the University of Washington, studies how the nervous system processes sensory signals to control behavior; and as a vision neuroscientist at Florida International University, Jamie Theobald, determines how animals parse complex environments. The other half of the team will build the framework: Duke Professor Vahid Tarokh models complex datasets, Georgia Tech School of Mathematics Assistant Professor Hannah Choi focuses on neural networks, and Cornell Professor Silvia Ferrari ties it all together as a control theorist embedding control in neural structures.

“MURIs were originally training grants for the DoD to develop the next generation of scientists who would make progress,” said Sponberg. “This funding will allow us to have postdocs and graduate students across all six labs and disciplines working together tightly and creating a community.”

For the second MURI, Programming Multistable Origami and Kirigami Structures via Topological Design, Georgia Tech Assistant Professor Zeb Rocklin is part of a team exploring a new class of origami- and kirigami-inspired flexible, lightweight structures capable of transitioning between many stable shapes to perform different tasks or adapt to changing environmental conditions. These structures could be used in a range of applications, from multifunctional robots and collapsible antennae to rapidly assembled bridges and temporary structures, and force protection elements like origami-inspired bulletproof shields.

The team combines experts in mathematics, physics, material science, mechanics, robotics, numerical modeling, and computation, including Harvard University Professors Katie Betoldi, Jennifer Lewis, L. Mahadevan, and Robert Wood, as well as University of Pennsylvania Associate Professor Eleni Katifori

The researchers will develop mathematical models to characterize and design the complex mechanical behavior of multi-stable origami and kirigami structures; new scale-spanning manufacturing processes that efficiently integrate actuation and sensing; and experimental test beds to serve as a platform for evaluation and optimization of design concepts.

"This project benefits from Georgia Tech's ability to develop tight, powerful connections between engineering advanced technologies and developing universal, mathematically rigorous physical theories,” Rocklin said. “We'll be starting from concepts that anyone can get a sense of by looking at or feeling a piece of origami and using robotics and multifunctional 3D printing to create complex, flexible and robust dynamical structures that can do things nobody has ever seen before."

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Georgia Tech professors receive nearly $14 million in MURI Funding

Summary sentence

Georgia Tech has received two Department of Defense (DoD) 2022 Multidisciplinary University Research Initiative (MURI) awards totaling almost $14 million.

Summary

Georgia Tech has received two Department of Defense (DoD) 2022 Multidisciplinary University Research Initiative (MURI) awards totaling almost $14 million.

Dateline

Mon, 04/18/2022 - 12:00

tess.malone@gatech.edu

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Tess Malone, Research writer/editor

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Atlanta, GA

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Fri, 04/22/2022 - 13:15

School of Physics Uses Moths and Origami Structures for Innovative Defense Research

bwaye3 — Tue, 19 Apr 2022 16:42:57 +0000

School of Physics Uses Moths and Origami Structures for Innovative Defense Research bwaye3 Tue, 04/19/2022 - 12:42

Subtitle

Georgia Tech professors receive nearly $14 million in MURI Funding

Summary sentence

Georgia Tech has received two Department of Defense (DoD) 2022 Multidisciplinary University Research Initiative (MURI) awards totaling almost $14 million.

Summary

Georgia Tech has received two Department of Defense (DoD) 2022 Multidisciplinary University Research Initiative (MURI) awards totaling almost $14 million.

Dateline

Mon, 04/18/2022 - 12:00

tess.malone@gatech.edu

Contact

Tess Malone, Research writer/editor

Location

Atlanta, GA

Associated importer

Keywords

go-researchnews

go-bio

School of Biological Sciences

School of Mathematics

go-neuro

School of Physics

Biophysics

_for_math_site_

News room topics

Science and Technology

Mercury ID

657428

Source updated

Tue, 04/19/2022 - 10:49

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What’s the Shape of the Universe? Mathematicians Use Topology to Study the Shape of the World and Everything in it

The Shape of the Universe

Topology in Higher Dimensions

Tied Up In Knots

What Shape Do You Live On?

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Mathematics, Physics Use Moths and Origami Structures for Innovative Defense Research

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School of Physics Uses Moths and Origami Structures for Innovative Defense Research

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