7/3/2023 0 Comments Positively curved space![]() ![]() With these tools, researchers can better explore the topsy-turvy world of hyperbolic space. Their new toolbox includes what they call a “dictionary between discrete and continuous geometry” to help researchers translate experimental results into a more useful form. The research builds on Kollár’s previous experiments (link is external) to simulate orderly grids in hyperbolic space by using microwave light contained on chips. In a recent paper in Physical Review A, a collaboration between the groups of Kollár and JQI Fellow Alexey Gorshkov, who is also a physicist at the National Institute of Standards and Technology, presented new mathematical tools to better understand simulations of hyperbolic spaces. But scientists can still mimic hyperbolic environments to explore how certain physics plays out in negatively curved space. Even a two-dimensional, physical version of a hyperbolic space is impossible to make in our normal, “flat” environment. One type of non-Euclidean geometry that is of interest is hyperbolic space-also called negatively-curved space. Physicists are interested in new physics that curved space can reveal, and non-Euclidean geometries might even help improve designs of certain technologies. Non-Euclidean geometries are so alien that they have been used in videogames and horror stories (link is external) as unnatural landscapes that challenge or unsettle the audience.īut these unfamiliar geometries are much more than just distant, otherworldly abstractions. These environments overturn core assumptions of normal navigation and can be impossible to accurately visualize. (Credit: Springer Nature Produced by Princeton, Houck Lab) On the right is a circuit that simulates a similar hyperbolic grid by directing microwaves through a maze of zig-zagging superconducting resonators. In the appropriate hyperbolic space, each heptagon would have an identical shape and size, instead of getting smaller and more distorted toward the edges. To fit the uniform hyperbolic grid into “flat” space, the size and shape of the heptagons are distorted. On the left is a representation of a grid of heptagons in a hyperbolic space. In such a world, four equal-length roads that are all connected by right turns at right angles might fail to form a square block that returns you to your initial intersection. Or it could expand so that they forever grow further apart. Space might contract so that straight, parallel lines draw together instead of rigidly maintaining a fixed spacing. If you could explore non-Euclidean environments, you would find perplexing landscapes. Spaces that have different geometric rules than those we usually take for granted are called non-Euclidean (link is external). “But, any place where there's actually a laboratory is very weakly curved because if you were to go to one of these places where gravity is strong, it would just tear the lab apart.” “We know from general relativity that the universe itself is curved in various places,” says Assistant Professor JQI Fellow Alicia Kollár, who is also a Fellow of the Joint Quantum Institute and the Quantum Technology Center. But studying how physics plays out in a curved space is challenging: Just like in real estate, location is everything. And in curved space, normal ideas of geometry and straight lines break down, creating a chance to explore an unfamiliar landscape governed by new rules. ![]() 119 (2) 261 - 307, October 2021.Thanks to Einstein, we know that our three-dimensional space is warped and curved. "The stable converse soul question for positively curved homogeneous spaces." J. Citation Download Citationĭavid González-Álvaro. Part of the research for this publication was conducted in the framework of the DFG Research Training Group 2240: Algebro-Geometric Methods in Algebra, Arithmetic and Topology. The SNSF-Project 200021E-172469 and the DFG-Priority programme Geometry at González-Álvaro received support from: the The stable converse soul question (SCSQ) asks whether, given a real vector bundle $E$ over a compact manifold, some stabilization $E \times \mathbb$, there is essentially one stable class of real vector bundles for which our method fails. ![]()
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