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I am a PhD student at the University of Maryland working on tilings and their relationship to harmonic analysis and noncommutative geometry.


Mar
23
comment What are the applications of operator algebras to other areas?
Yet I have consistently encountered people outside operator algebras (and NC geometry in particular) who ask "Well what's it all for? Do I really have to learn it?" to the point where I begin to feel guilty if a result I can prove is "internal" to NC geometry. I don't know whether people in number theory or analysis feel the same way. I was attempting to explain why NC geometry may inspire this more than other fields, but perhaps I'm wrong and it doesn't.
Mar
23
comment What are the applications of operator algebras to other areas?
It is sweeping, presumptuous, and opinionated, and I did not mean to put words in anyone's mouth. I don't really work on any of these areas myself, this is just my ambient sense of the mathematical community at large. It's hard to understand and describe why mathematicians find a certain subject or result (un)interesting.
Mar
23
comment What are the applications of operator algebras to other areas?
While I certainly agree that operator algebras are intrinsically interesting, I have encountered some negativity towards the subject from mathematicians who work in other fields. In the original question, I understood the author as asking for an explanation of this negativity. But perhaps you have never encountered it and it is just my personal misfortune (or perhaps this kind of negativity towards other fields is more common than I expected, though I hope that isn't the case).
Mar
22
awarded  Nice Answer
Mar
22
comment What are the applications of operator algebras to other areas?
Here's one of the original papers of GPS which classifies minimal $\mathbb{Z}$ actions on a Cantor set up to orbit equivalence using K-theory of associated C^*-algebras: dspace.library.uvic.ca:8080/bitstream/handle/1828/2617/….
Mar
22
comment What are the applications of operator algebras to other areas?
Edited to include a few dynamics references and a note on quasicrystals.
Mar
22
revised What are the applications of operator algebras to other areas?
added 438 characters in body
Mar
22
answered What are the applications of operator algebras to other areas?
Jan
9
answered quasicrystal and penrose tiling, mathematical introduction
Dec
29
comment Commutative spectral triples not coming from manifolds
You might have a look at this paper arxiv.org/abs/1010.0156 which constructs spectral triples for $C(X)$ where $X$ is a compact metric space. Their particular examples come from tiling spaces and dynamics, where $X$ is a Cantor set.
Dec
7
comment Can any Delone set be approximated by a model set?
I make no comment on what the spelling ought to be, nor do I have any power over how it is generally used, as this type of choice is made over time, often somewhat arbitrarily, by the community at large...
Dec
7
comment Can any Delone set be approximated by a model set?
The vast majority of recent papers in the theory of tilings use the spelling Delone, although the spelling Delaunay can be found in some older papers. Examples: arxiv.org/abs/1411.0578, arxiv.org/abs/1401.3725, arxiv.org/abs/1407.1787.
Dec
7
comment Can any Delone set be approximated by a model set?
I think that's right? I've seen both spellings in papers before.
Dec
6
revised Can any Delone set be approximated by a model set?
added 216 characters in body
Dec
6
comment Can any Delone set be approximated by a model set?
Edited to include complete definition of $\Lambda_W.$ A cut and project set (or model set) is any set of the same form as $\Lambda_W$ above, where we can vary the lattice $L,$ the subspace $H,$ and the window $W,$ so long as $H$ remains in irrational position w.r.t. $L.$ A Meyer set is any subset of a model set.
Dec
6
revised Can any Delone set be approximated by a model set?
added 135 characters in body
Dec
6
asked Can any Delone set be approximated by a model set?
Sep
29
answered Projective modules over noncommutative tori?
Sep
26
comment Morita Equivalence of Full Corners in $C^*$-algebras
Thanks Alain - suppose I'm given a projection in $\mathcal{A}$ or in $M_n(\mathcal{A}).$ Based on this method, is it easy to construct a projection in $M_k(\mathcal{B})$ that it gets mapped to under this isomorphism?
Sep
26
asked Morita Equivalence of Full Corners in $C^*$-algebras