User koushik - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-20T08:16:43Z http://mathoverflow.net/feeds/user/30081 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/128254/possible-directions-in-noncommutative-geometry/128256#128256 Answer by Koushik for Possible directions in noncommutative geometry Koushik 2013-04-21T16:39:03Z 2013-04-21T16:39:03Z <p>1)You look into the book "Noncommutative Dynamics and E0 semigroups" by William Arveson 2) There is an approach to attack multivariate operator theory through algebraic geometry.You may look to "Operator Theory and Complex Geometry" by douglas for an introduction 3)You may look into brown,douglas,fillimore's paper in essential normality and K-Homology which was a precursor to connes approach 4)or look into connes's book</p> http://mathoverflow.net/questions/127253/complex-dynamics-in-several-variables complex dynamics in several variables Koushik 2013-04-11T14:44:46Z 2013-04-11T21:23:41Z <p>Dear mathematicians, I want to know how much advance there has been in complex dynamics of several variables. I am at present reading Carleson's book on Complex Dynamics on one variables.Curious to know about several variables case.t Specifically,what are the best papers in this area.</p> http://mathoverflow.net/questions/126091/invariant-subspace-equivalent-form invariant subspace equivalent form Koushik 2013-03-31T10:53:16Z 2013-03-31T10:53:16Z <p>"This was until a beautiful â€˜index-theoreticâ€™ characterisation of quasitriangularity was obtained by Apostol, Foias and Voiculescu, which had the unexpected consequence that if an operator or its adjoint is not quasitriangular, then it has a non-trivial invariant subspace" A Glimpse at Hilbert Space Operators :Paul R. Halmos in Memoriam by Sheldon Axler ,Peter Rosenthal,Donald Sarason</p> <p>Can this be used to say something on the invariant subspace problem(I mean translating it to an equivalent form). The only problematic thing here is that the subspace is "not" closed.Otherwise,it would have already given a nice equivalent form.</p> http://mathoverflow.net/questions/114493/corona-theorem-in-several-variables/126063#126063 Answer by Koushik for Corona Theorem in several variables Koushik 2013-03-31T01:40:30Z 2013-03-31T01:40:30Z <p>No corona theorem fails for several variables.<a href="http://av.fields.utoronto.ca/slides/11-12/coronaproblem/sawyer/download.pdf" rel="nofollow">look here for a counterexample</a></p> http://mathoverflow.net/questions/124135/maxwells-equations-and-hodge-theory maxwell's equations and hodge theory Koushik 2013-03-10T10:06:28Z 2013-03-10T10:57:58Z <p>How is Hodge theory of harmonic forms related to maxwell's equations.Atiyah says that Hodge was directly motivated by considerations of maxwell's equations while commenting on donaldson.</p> http://mathoverflow.net/questions/124119/is-c-infty0-1-or-s-separable/124124#124124 Answer by Koushik for Is $C^{\infty}[0,1]$ or $S$ separable? Koushik 2013-03-10T03:30:58Z 2013-03-10T03:30:58Z <p>$C^{\infty}[0,1]$ certainly is.take polynomials with rational coefficients.so by wierstrass approximation theorem ,we are done.</p> http://mathoverflow.net/questions/122581/blaschke-condition-on-upper-half-plane Blaschke condition on upper half plane Koushik 2013-02-21T23:22:05Z 2013-03-07T22:37:46Z <p>if f is in $H^{1}$ the Hardy space with integrable norm, and if f is not identically zero, then the zeroes of f (certainly countable in number) satisfy the Blaschke condition.Can any "Blaschke condition" be defined if hardy space is considered on the upper-half plane instead of unit disk. Curious to know if this carries over by the conformal equivalence that maps upper-half plane to unit disk and vice-versa.</p> http://mathoverflow.net/questions/123523/examples-of-functions-in-hardy-space-and-bergman-space examples of functions in hardy space and bergman space Koushik 2013-03-04T11:55:35Z 2013-03-05T10:51:02Z <p>What are the basic examples of functions in <a href="http://en.wikipedia.org/wiki/Hardy_space" rel="nofollow">hardy space</a> and <a href="http://en.wikipedia.org/wiki/Bergman_space" rel="nofollow">bergman space</a> that I can test the theorems against.I want to know how to produce examples for each parameter. Edited: I don't want trival examples that belong to the space for all parameters.i need examples so as to distinguish spaces for different parameters</p> http://mathoverflow.net/questions/123302/a-problem-in-functional-analysis-that-erdos-solved-in-2-lines a problem in functional analysis that erdos solved in 2 lines Koushik 2013-03-01T07:09:00Z 2013-03-01T20:08:44Z <p><a href="http://math.stackexchange.com/questions/261685/paul-erdoss-two-line-functional-analysis-proof" rel="nofollow">http://math.stackexchange.com/questions/261685/paul-erdoss-two-line-functional-analysis-proof</a> .</p> <p>does anyone know about what the problem was and what was his solution.</p> <p>[Edit by quid:] please follow the link before trying to answer, there is already quite a bit of relevant information behind the link. [End edit]</p> http://mathoverflow.net/questions/122115/problems-from-the-scottish-book problems from the scottish book Koushik 2013-02-18T00:30:08Z 2013-02-26T17:14:13Z <p>Which of the problems from the <a href="http://en.wikipedia.org/wiki/Scottish_Book" rel="nofollow">Scottish Book</a> (<a href="http://kielich.amu.edu.pl/Stefan_Banach/pdf/ks-szkocka/ks-szkocka3ang.pdf" rel="nofollow">pdf of English version</a>) by Stefan Banach are still open? I know that one of the problems was solved by Per Enflo for which he got a live goose from Stanislaw Mazur.</p> http://mathoverflow.net/questions/122793/spectacular-applications-of-functional-analysis-in-resolutions-of-apparently-unre spectacular applications of functional analysis in resolutions of apparently unrelated problems Koushik 2013-02-24T12:54:09Z 2013-02-24T16:55:18Z <p>What are some of the spectacular applications of functional analysis to apparently unrelated problems.One that comes to my mind is Per Enflo's resolution of Hilbert's 5th problem.There are also reformulations of RH in Hilbert Space.I would be happly to hear about it's nice applications in geometry,number theory,topology,etc. specially in context to solving conjectures.</p> http://mathoverflow.net/questions/101169/not-especially-famous-long-open-problems-which-higher-mathematics-beginners-can/122677#122677 Answer by Koushik for Not especially famous, long-open problems which higher mathematics beginners can understand Koushik 2013-02-23T00:43:44Z 2013-02-23T10:03:21Z <p>Showing that if product of $n$ Toeplitz operators is again a Toeplitz Operator.this is open and quite elementary to state. <a href="http://www.mathnet.or.kr/mathnet/thesis_file/WYLee.pdf" rel="nofollow">http://www.mathnet.or.kr/mathnet/thesis_file/WYLee.pdf</a></p> http://mathoverflow.net/questions/122175/diffm-and-connectedness Diff(M) and connectedness Koushik 2013-02-18T14:37:53Z 2013-02-19T00:08:05Z <p>Can anything be said about connectedness of a smooth manifold M from some property of Diff(M) in an analogous way Like C(X) has no idempotents iff X is connected.</p> http://mathoverflow.net/questions/121618/dreams-of-mathematicsramannujan-others dreams of mathematics(ramannujan) others? Koushik 2013-02-12T16:03:48Z 2013-02-12T16:08:48Z <p>"Ramanujan credited his acumen to his family Goddess, Namagiri of Namakkal. He looked to her for inspiration in his work,[84] and claimed to dream of blood drops that symbolised her male consort, Narasimha, after which he would receive visions of scrolls of complex mathematical content unfolding before his eyes.[85] He often said, "An equation for me has no meaning, unless it represents a thought of God." from wiki. Mendeleev discovered periodic table in his dream. has there been any other such recorded incidences.I find it quite amazing how one could dream such complicated formulas in dream. I myself have had dreams on mathematics but never was it that I discovered something new.So I am curious to know.Do people here have had such experiences.I would be glad if they share their experiences. Since some of the best minds of mathematics are here,I bate there would be quite a people who can contribute.</p> <p>I am not sure by the way if this question is appropriate for this forum.</p> http://mathoverflow.net/questions/121168/group-of-diffeomorphisms-of-a-manifold group of diffeomorphisms of a manifold Koushik 2013-02-08T10:18:03Z 2013-02-08T19:13:22Z <p>How much has been the group of diffeomorphisms of a manifold " been studied. I got this information from wiki. " Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra is (more or less) the Witt algebra, which has a central extension called the Virasoro algebra, used in string theory and conformal field theory. Very little is known about the diffeomorphism groups of manifolds of larger dimension. The diffeomorphism group of spacetime sometimes appears in attempts to quantize gravity." </p> <p>.What more is known about this?Has this group been calculated for the standard manifolds.Since this group is a big group,so what are the better ways of studying this object.</p> http://mathoverflow.net/questions/19505/riemannian-geometry-introductory-text/121181#121181 Answer by Koushik for Riemannian Geometry Introductory Text Koushik 2013-02-08T12:12:53Z 2013-02-08T12:12:53Z <p>You may look into Novikov's 3 volumes of differential geometry</p> http://mathoverflow.net/questions/120819/open-problems-in-seiberg-witten-theory-on-4-manifolds open problems in Seiberg-Witten Theory on 4-Manifolds Koushik 2013-02-05T01:28:44Z 2013-02-07T06:17:25Z <p>What are some of the open problems in Seiberg-Witten Theory on 4-Manifolds.I tried googling but couldn't any. I tried googling it, but couldn't find any resources.The places where I can a survey or review of them would be welcome. </p> http://mathoverflow.net/questions/120936/applications-of-gauss-bonnet-theorem applications of gauss-bonnet theorem Koushik 2013-02-06T04:51:13Z 2013-02-06T04:51:13Z <p>In wikipedia,I was pretty amazed to find a proof of <a href="http://en.wikipedia.org/wiki/Fundamental_theorem_of_algebra" rel="nofollow">fundamental theorem of algebra</a></p> <p>using <a href="http://en.wikipedia.org/wiki/Gauss-Bonnet_theorem" rel="nofollow">gauss bonnet theorem</a>.I think given how central it is to mathematics with its far reaching generalizations like Riemann-Roch Theorem and more,I am wondering if there are more.I would also be happy to see striking applications of its generalizations.</p> http://mathoverflow.net/questions/120612/trichotomies-in-mathematics/120646#120646 Answer by Koushik for Trichotomies in mathematics Koushik 2013-02-03T03:44:12Z 2013-02-03T04:14:52Z <p>The classification theorem of closed surfaces states that any connected closed surface is homeomorphic to some member of one of these three families: the $sphere$; the connected sum of $g$ $tori$, ; the connected sum of $k$ $real$ $projective$ $planes$. this is a simple example of the trichotomy.sphere can be taken as $0$ tori. so $SPHERE$ serves the middle column. </p> http://mathoverflow.net/questions/120485/fourier-analytic-proofs fourier analytic proofs Koushik 2013-02-01T03:46:15Z 2013-02-02T16:40:21Z <p>While searching through Mathoverflow, I found out a fourier analytic proof of the Isoperimetric Inequality.Also, by google search I found a fourier analytic proof of Quadratic Reciprocity theorem.I know of the fourier analytic approach used by combinatorialists like Ben Green. But what are the other fourier analytic proofs of some of the well known classical theorems other than what I have mentioned above specially those which admit a starkly different proofs.</p> http://mathoverflow.net/questions/119916/bass-stable-rank-of-multiplier-algebra bass stable rank of multiplier algebra Koushik 2013-01-26T05:17:36Z 2013-01-28T08:17:50Z <p>Is <a href="http://www.math.hut.fi/~kmikkola/research/art/MS-stablerank-CAOT.pdf" rel="nofollow">bass stable rank</a> known for <a href="http://en.wikipedia.org/wiki/Multiplier_algebra" rel="nofollow">multiplier algebra</a> of <a href="http://internetanalysisseminar.gatech.edu/sites/default/files/ias_lecture1.pdf" rel="nofollow">besov-sobolev space</a>?</p> <p><strong>note added by YC 28-01-13:</strong> the original version of this question seemed originally to be motivated by C-star questions (see the comments).</p> http://mathoverflow.net/questions/119648/geometry-and-quantization Geometry and quantization Koushik 2013-01-23T12:14:06Z 2013-01-24T15:28:06Z <p>I know that lots of effort is being put into quantization of geometry(NCG).This effort of course comes with the idea of operator algebra being a powerful machinery. Has any effort been given in the other direction.I mean to make some analogue of geometrical structures in Operator Algebras. This question may be a bit absurd and arises from one of my "RANDOM" thoughts. Precisely I mean "what are the differntial geometry ideas that are applicable in operator algebras"</p> http://mathoverflow.net/questions/119261/a-question-on-completeness-of-a-metrizable-group a question on completeness of a metrizable group Koushik 2013-01-18T12:08:18Z 2013-01-18T14:54:22Z <p>How to show that If G is a metrizable group. K is a closed normal subgroup of G, K and G/K are complete then G is complete.</p> <p>( I have alreaady asked in <a href="http://math.stackexchange.com/questions/281171/metrizable-group" rel="nofollow">math stacks</a> but with no answer. I don't know if it is suitable for this forum.)</p> http://mathoverflow.net/questions/118208/intuitive-meaning-of-double-commutant-theorem Intuitive meaning of Double Commutant Theorem Koushik 2013-01-06T16:02:01Z 2013-01-06T18:05:44Z <p>Is there any intuitive explanation of the Double Commutant Theorem for Von Neumann Algebras? By intuitive I mean in terms of Quantum Mechanics. For example, duality of states and observables in the case of the Gelfand-Naimark Theorem. <a href="http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem" rel="nofollow">http://en.wikipedia.org/wiki/Von_Neumann_bicommutant_theorem</a></p> http://mathoverflow.net/questions/118054/nuclear-space-problem Nuclear Space problem Koushik 2013-01-04T13:38:21Z 2013-01-04T13:38:21Z <p>I need to show that if X is compact,then C(X) is nuclear.Also is the condition X is metrisable necessary. I am at present attending a conference "Recent Aadvances in Operator Theory". This problem was given by Adam Skalski,Warshaw in the conference</p> http://mathoverflow.net/questions/97788/on-connection-between-knot-theory-and-operator-algebra/117470#117470 Answer by Koushik for On connection between Knot theory and Operator algebra Koushik 2012-12-29T02:50:32Z 2012-12-29T02:50:32Z <p>you may looks into David Evans monumental book on quantum symmetries on operator algebras</p> http://mathoverflow.net/questions/116771/on-topology-of-p-adic-numbers On topology of p-adic numbers. Koushik 2012-12-19T09:16:27Z 2012-12-19T09:36:29Z <p>This may be a stupid question.But I am stuck with it.Is Q_p(the p-adic) connected under the usual topology?I was confounded with this problem while trying to construct a counter-example related to my master's thesis.</p> http://mathoverflow.net/questions/128254/possible-directions-in-noncommutative-geometry Comment by Koushik Koushik 2013-04-21T16:49:05Z 2013-04-21T16:49:05Z i think so but I being a newbie myself I couldn't resist to pour my knowledge over http://mathoverflow.net/questions/128254/possible-directions-in-noncommutative-geometry/128256#128256 Comment by Koushik Koushik 2013-04-21T16:47:51Z 2013-04-21T16:47:51Z it's my personal opinion but I think the second approach may go a long way in the future.the heart of the approach is &quot;arveson conjecture&quot; which relates homogeneous varieties in the unit ball of $C^n$ with essential normality of d-shift operators vaguely http://mathoverflow.net/questions/126759/great-non-medalists/126760#126760 Comment by Koushik Koushik 2013-04-07T07:24:01Z 2013-04-07T07:24:01Z no one mentioned arne beurling his contributions have been very beautiful if not revolutionary http://mathoverflow.net/questions/126091/invariant-subspace-equivalent-form Comment by Koushik Koushik 2013-04-03T00:56:33Z 2013-04-03T00:56:33Z peter rosenthal communicated to me that the invariant subspace is closed http://mathoverflow.net/questions/126091/invariant-subspace-equivalent-form Comment by Koushik Koushik 2013-04-01T04:52:59Z 2013-04-01T04:52:59Z the worst thing i am not being able to find the paper.the book gives no reference for this paper http://mathoverflow.net/questions/124404/phd-in-operator-algebras-and-non-commutative-geometry/124410#124410 Comment by Koushik Koushik 2013-03-14T04:06:25Z 2013-03-14T04:06:25Z evans works in operator algebra and TQFT http://mathoverflow.net/questions/124404/phd-in-operator-algebras-and-non-commutative-geometry/124410#124410 Comment by Koushik Koushik 2013-03-14T01:05:57Z 2013-03-14T01:05:57Z penn-state in usa paul baum is there,vanderblit jones,kasparov,connes spends few months there.In UK there is lancaster university(martin lindsay,many others check their website),glassgow(joachim zacharius), cardiff university(David E.Evans) these are all I know. http://mathoverflow.net/questions/101169/not-especially-famous-long-open-problems-which-higher-mathematics-beginners-can/122677#122677 Comment by Koushik Koushik 2013-03-11T02:20:20Z 2013-03-11T02:20:20Z yes. most of the things are not known about toeplitz operator. I find most of operator theory like number theory.beautiful,devoid of much applications and abound in difficult problems http://mathoverflow.net/questions/124119/is-c-infty0-1-or-s-separable/124124#124124 Comment by Koushik Koushik 2013-03-10T04:28:23Z 2013-03-10T04:28:23Z possibly that can be taken care of by taking truncated taylor series and having rational coefficients http://mathoverflow.net/questions/123959/nonreducing-invariant-subspace-of-a-seperable-hilbert-space-example Comment by Koushik Koushik 2013-03-08T16:55:39Z 2013-03-08T16:55:39Z i don't want 2 diml examples. i want infinite diml. as i have edited http://mathoverflow.net/questions/123953/what-is-your-opinion-from-this-please Comment by Koushik Koushik 2013-03-08T10:31:45Z 2013-03-08T10:31:45Z bogus and maddenning question http://mathoverflow.net/questions/123873/how-to-find-the-tensor-product-of-modules-that-we-dont-know-a-basis-for-them Comment by Koushik Koushik 2013-03-07T15:20:16Z 2013-03-07T15:20:16Z who are you?someone from isi?i say this because the same question was raised in our class http://mathoverflow.net/questions/123737/weak-topology-of-operator Comment by Koushik Koushik 2013-03-06T13:39:48Z 2013-03-06T13:39:48Z why do you ask here? do google search http://mathoverflow.net/questions/123302/a-problem-in-functional-analysis-that-erdos-solved-in-2-lines/123369#123369 Comment by Koushik Koushik 2013-03-02T03:53:55Z 2013-03-02T03:53:55Z both problem much less the solution is given there.i may be very much possible that someone here might be knowing that. http://mathoverflow.net/questions/123302/a-problem-in-functional-analysis-that-erdos-solved-in-2-lines Comment by Koushik Koushik 2013-03-02T03:51:06Z 2013-03-02T03:51:06Z @todd, that's also what I think.I am curious to know how a 30 page solution can be reduced to 2 lines.