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Abstract Introduced by Gromov in the nineties, the systolic growth of a Lie group gives the smallest possible covolume of a lattice with a given systole. Issue Section:. You do not currently have access to this article. Download all figures. Sign in. You could not be signed in. Sign In Forgot password? Don't have an account? Sign in via your Institution Sign in.
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Rosenblatt, A. Furman, H. Furstenberg, D. Kleinbock, G. Margulis, Y. Shalom, A. Katok and Zimmer, between many others.
For getting a high level understanding of Lie Group and its applications in Computer Vision read the following post:. Generally it is very helpful in different applications such as analysis of rigid transformations, or even non-rigid transformations. I've seen few other applications such as analysis of DTI imaging. Sign up to join this community. The best answers are voted up and rise to the top.
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I think "why should I study X" questions are somewhat unsatisfactory; also, in your main questions, who do you mean by "people"? Number theorists? Fluid dynamics researchers? PDE people? Thus far I have no clue why Lie group are of such importance in differential geometry. Many of the question asked in differential geometry I find natural, like trying to write a Riemannian metric as the usual metric in a chart, and finding the obstruction to the existence of such local coordiantes. Or given a distribution, asking wether it is tangent to a submanifold, and finding the obstruction.
Trying to identify Vector bundles, and telling them apart. These are all questions that resonate with me.
19w Discrete Subgroups of Lie Groups | Banff International Research Station
For instance, Littlewood's conjecture, which involves Diophantine approximation, is equivalent to a problem on the dynamics of a certain Lie groups, acting on a certain homogeneous space. Einsiedler, Katok, and Lindenstrauss gave a partial answer to the conjecture via this connection to dynamics, and though I haven't read it, I imagine that the proof uses quite a lot of Lie theory and ergodic theory. So number theory is one possible answer to "why study Lie groups? It's an honest question, do you think it is a ridiculous one?
Also by "people" I mean any mathematician. Lie theory is very much a mystery to me, even though I know the basics. I'm interested in how it is being used by various branches of mathematics, and why it plays such a proeminent role. Deane Yang Deane Yang 21k 5 5 gold badges 65 65 silver badges bronze badges. It's better than any other introductory differential geometry text I've seen with regards to showing applications of Lie groups in geometry.
Emerton Emerton Amritanshu Prasad Amritanshu Prasad 4, 28 28 silver badges 41 41 bronze badges. I think Differential Galois Theory answers why some linear differential equations are solvable in known functions.. Differential Galois theory is rather algebraically defined mimicking the Galois theory of fields and algebras; the basic object is a differential field, or more generally differential ring, which is a ring equipped with a derivation.
Unfortunately, there does not seem to be developed a higher dimensional version with many derivations fitting into D-module approach , just with one derivation -- Picard-Vessiot's theory, for which there is also a Tannakian approach see Deligne's article in Gorthendieck's Festschrift. Naturally, the object of studies deserves looking for a connection. For an attempt see W. Oudshorn, M.
I got a complete answer for my question through that paper. Scott Carter Scott Carter 4, 2 2 gold badges 21 21 silver badges 33 33 bronze badges. The applications to physics are alone enough reason to study Lie groups and algebras.
Module will run
The story can be told like this: 1 classical synthetical geometry Euclidean, projective, Lie sphere geometry, etc. Faisal Faisal 8, 1 1 gold badge 34 34 silver badges 57 57 bronze badges. Charles Matthews Charles Matthews Major uses of Lie groups in Riemannian geometry are: Holonomy groups. Homogeneous and symmetric spaces, as a source of fundamental examples of Riemannian manifolds. Igor Belegradek Igor Belegradek Salvatore Siciliano Salvatore Siciliano 3, 15 15 silver badges 24 24 bronze badges. Prasad's excellent recommendation Olver's book , I would suggest you take a look at Helgason's notes In particular, it is a good idea to check out the bottom of the page here.
Here is a different one based on title and author I assume this is the article you were referring to : jstor. What questions do they ask for which Lie groups or algebras will be of any help in DS?