most recent 30 from http://mathoverflow.net 2013-05-22T03:27:22Z http://mathoverflow.net/feeds/user/23911 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/97788/on-connection-between-knot-theory-and-operator-algebra Arnab 2012-05-23T20:10:06Z 2012-12-29T02:50:32Z

<p>What exactly is the connection between knots and operator algebra? I heard that Jones established such a connection while discovering the celebrated Jones Polynomial. </p> <p>Now Jones Polynomial is probably understood out of that context on its own, but what was it with Operator algebras in this space ? Can someone explain in plain English?</p> <p>As is probably very evident, I am a complete newbie to this area.</p>

http://mathoverflow.net/questions/108002/is-there-any-alternative-characterization-of-sparsity-of-a-signal-in-compressed-s Arnab 2012-09-24T19:51:55Z 2012-09-24T23:31:46Z

<p>The starting assumption for compressed sensing (CS) is that the underlying signal is sparse in some basis, e.g., there are a maximum of $s$ non-zero Fourier-coefficients for an $s$-sparse signal. And real life experiences do show that the signals under consideration are often sparse.</p> <p>The question is - given a signal, before sending out the compressively-sampled bits to the receiver and let her recover to the best of her abilities, is there a way to tell what its sparsity is, and if it is a suitable candidate for compressed sensing in the first place? </p> <p>Alternatively, is there any additional/alternative characterization of sparsity that can tell us quickly whether CS will be useful or not. One can trivially see that the sender could do exactly what the receiver will do with some randomly chosen set of measurements, and then try to figure out the answer. But is there any alternate way to resolve this question ? </p> <p>My suspicion is that something like this must have been studied, but I couldn't find a good pointer.</p>

http://mathoverflow.net/questions/107424/knot-security-when-to-trust-your-life-with-a-knot/107535#107535 Arnab 2012-09-19T08:04:56Z 2012-09-19T08:04:56Z

<p>Louis Kauffman's masterful book "Knots and Physics" has some thoughts on it. Especially the introductory chapter, and then later a small chapter called "The Theory of Hitches". Worth a reading IMO..</p>

http://mathoverflow.net/questions/24526/good-papers-books-essays-about-the-thought-process-behind-mathematical-research/107036#107036 Arnab 2012-09-12T18:27:30Z 2012-09-12T18:27:30Z

<p><a href="http://www.amazon.com/History-Algebraic-Differential-Topology-1900/dp/0817649069/ref=sr_1_1?ie=UTF8&amp;qid=1347474254&amp;sr=8-1&amp;keywords=history+of+algebraic+and+differential+topology" rel="nofollow">http://www.amazon.com/History-Algebraic-Differential-Topology-1900/dp/0817649069/ref=sr_1_1?ie=UTF8&amp;qid=1347474254&amp;sr=8-1&amp;keywords=history+of+algebraic+and+differential+topology</a></p> <p>I am currently reading this book, A history of Algebraic and Differential Topology, by Jean Dieudonne, partially in the same spirit of the question. Very good reference, but more importantly throws very good light on the development of the subject, not just in the mind of one mathematician, but across many of them, and across decades.</p>

http://mathoverflow.net/questions/106951/degree-of-faces-in-a-regular-graph Arnab 2012-09-11T20:15:47Z 2012-09-12T08:29:03Z

<p>Is there any known result on the maximum degree of faces in regular-and-planar graphs ? In particular, is anything known about maximum degree of the faces in a 4-regular planar graph? By degree of a face I mean the number of edges forming it.</p>

http://mathoverflow.net/questions/97991/computational-complexity-of-knot-polynomials Arnab 2012-05-25T21:41:07Z 2012-05-26T00:28:21Z

<p>What's known about computational complexity of different types of knot invariant polynomials?</p> <p>For example, Evaluating Jones Polynomial is known to be #P hard. Is there any reference that surveys such complexity results on other knot polynomials?</p>

http://mathoverflow.net/questions/107034/counting-graphs-on-n-vertices-by-chromatic-number Arnab 2012-09-12T20:16:39Z 2012-09-12T20:16:39Z

@alexander: absolutely. implicit in this process will be somehow quotienting with the isomorphic sets - because for every graph, you can get the size of its isomorphic class - but that's where it will get messy I think.

http://mathoverflow.net/questions/107034/counting-graphs-on-n-vertices-by-chromatic-number Arnab 2012-09-12T18:16:11Z 2012-09-12T18:16:11Z

It seems that one should be able to count this for a given number N of vertices. compute a partition into different colors, and then compute how many ways edges can be formed between these partitions. The computation may be a bit messy, but seems doable. There may be some elegant way of getting it, of course, that I am not aware of.

http://mathoverflow.net/questions/106951/degree-of-faces-in-a-regular-graph Arnab 2012-09-12T18:07:05Z 2012-09-12T18:07:05Z

Thanks much to both of you, Brendan, and Joseph. I understand that it is difficult to say much without the additional information such as connectivity. However, in the same vein, I am thinking it might be possible say a bit more than what's observed here; say, for instance, the number of faces with unbounded degree has an upper bound, or something like that.

http://mathoverflow.net/questions/106951/degree-of-faces-in-a-regular-graph Arnab 2012-09-11T23:31:47Z 2012-09-11T23:31:47Z

This is one abstracted piece of a bigger problem of course, and at least on the surface I don't have any additional information.

http://mathoverflow.net/questions/106844/compressed-sensing-of-a-riemann-surface Arnab 2012-09-10T23:01:16Z 2012-09-10T23:01:16Z

I kept it as general as possible to see if someone can point out anything relevant in this connection. Didn't mean for it to be a problem statement. Let me rephrase it, more concretely, and ask it as anothr question (will be somewhat different, but more pointed). I guess this can be closed. Thanks to both of you, Igor, and Yemon.

http://mathoverflow.net/questions/97991/computational-complexity-of-knot-polynomials/98006#98006 Arnab 2012-05-27T19:45:52Z 2012-05-27T19:45:52Z

Prof. O'Rourke, that is a beautiful modern reference; thanks very much.

http://mathoverflow.net/questions/97788/on-connection-between-knot-theory-and-operator-algebra Arnab 2012-05-24T17:52:10Z 2012-05-24T17:52:10Z

Yes, 'celebration' has a different meaning in mathematics, I suppose. Thanks very much for sketching the path to the connection. I read up a bit on Temperley-Lieb and it is slowly emerging for me. Daniel, thanks a lot for pointing out the invariance under Markkov moves. No surprise, but cannot be overstated that abstract math often becomes alive while representing physical systems.

http://mathoverflow.net/questions/97788/on-connection-between-knot-theory-and-operator-algebra Arnab 2012-05-23T21:49:28Z 2012-05-23T21:49:28Z

Yemon, I agree with you. In fact, I probably used it a bit loosely here. I know decently about Hilbert Spaces and Von Neumann Algebras; just started learning braids. Let me amend my words - instead of 'plain english', let me ask - what are the concepts I need to learn to understand this connection?