User priyavrat deshpande - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-21T17:40:08Z http://mathoverflow.net/feeds/user/7494 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43799/discrete-morse-theory-and-existence-of-minimal-complex Discrete Morse theory and existence of minimal complex Priyavrat Deshpande 2010-10-27T14:13:13Z 2013-04-12T13:02:10Z <p>A minimal complex is a CW complex whose only cells are the homology cells. </p> <p>Is there some sort of criterion on CW complexes about existence of minimal complexes?</p> <p>Actually I am working on a problem of understanding homotopy type of certain spaces (see: <a href="http://mathoverflow.net/questions/43711/how-to-show-that-a-space-has-the-homotopy-type-of-wedge-of-spheres" rel="nofollow">http://mathoverflow.net/questions/43711/how-to-show-that-a-space-has-the-homotopy-type-of-wedge-of-spheres</a>)</p> <p>My hope was to use discrete Morse theory (acyclic matching of face poset to be precise) and find the minimal complex. But then I don't know if the existence of the minimal complex is always guaranteed. </p> http://mathoverflow.net/questions/95403/connected-covering-spaces-of-a-homotopy-colimit Connected covering spaces of a homotopy colimit Priyavrat Deshpande 2012-04-27T22:50:18Z 2012-05-01T17:01:13Z <p>Let $\mathcal{D}: C\to Top$ be a diagram of spaces (spaces are "nice", and $C$ is small). Let $X$ denote the homotopy colimit of $\mathcal{D}$ (which is connected) and $\pi(C)$ be the free groupoid on $C$ (i.e., fundamental groupoid of the geometric realization). </p> <p>Is it possible to characterize (or express) connected covers of $X$ in terms of (connected) groupoid covers of $\pi(C)$?</p> <p>For example, if $C$ is acyclic and $\mathcal{D}$ assigns points then it can be done using the Groethendieck construction. </p> <p>Is there a general recipe that explains such a construction? Can it be done for diagrams of categories ?</p> http://mathoverflow.net/questions/1243/how-to-write-math-well/43894#43894 Answer by Priyavrat Deshpande for How to write math well? Priyavrat Deshpande 2010-10-27T22:41:29Z 2010-10-27T22:41:29Z <p>Currently I am reading the book "<strong>A primer of Mathematical writing</strong>" by <strong>Steven Krantz</strong>. I have found it extremely useful. It is helping me with my thesis and 2 papers. The book covers everything from grammar, writing papers, CV, grants, job applications, books, book reviews, referee report, expositions also technical aspects like bibliography, index, appendix and even time management. </p> <p>The book is published by AMS, here is the amazon <a href="http://www.amazon.com/Primer-Mathematical-Writing-Disquisition-Appreciated/dp/0821806351" rel="nofollow">link</a>.</p> http://mathoverflow.net/questions/43861/online-introduction-to-lattice-theory/43872#43872 Answer by Priyavrat Deshpande for Online introduction to Lattice Theory? Priyavrat Deshpande 2010-10-27T20:38:50Z 2010-10-27T20:38:50Z <p>This is also pretty good</p> <p><a href="http://boole.stanford.edu/cs353/handouts/book1.pdf" rel="nofollow">http://boole.stanford.edu/cs353/handouts/book1.pdf</a></p> <p>short and sweet.</p> http://mathoverflow.net/questions/43844/algorithm-to-find-an-acyclic-matching-on-a-poset Algorithm to find an acyclic matching on a poset Priyavrat Deshpande 2010-10-27T18:09:30Z 2010-10-27T18:44:20Z <p>Are there algorithms / theorems to find an acyclic matching on the Hasse diagram of a poset. </p> <p>I am particularly interested in the face poset of a regular CW complex. Also, how to decide if the given acyclic matching is perfect (impossible to add one more vertex). Is there some structure on the set of all acyclic matchings ?</p> http://mathoverflow.net/questions/43711/how-to-show-that-a-space-has-the-homotopy-type-of-wedge-of-spheres How to show that a space has the homotopy type of wedge of spheres ? Priyavrat Deshpande 2010-10-26T18:40:22Z 2010-10-27T18:39:51Z <p>Let me try and put the question in context. I am studying certain subsets of the tangent bundle of a sphere. I also have a regular CW complex which is a deformation retract of such a subset. Hence I have a description of the cells and the information that tells me which cell is in the boundary of which other cell. Fortunately this cell complex is a homotopy colimit of a diagram of spaces. As a result I can compute the cohomology groups (but not the product). </p> <p>All my examples concerning $S^1$ and $S^2$ show that these subsets have the homotopy type of wedge of copies of $S^1$ and $S^2$ respectively. Hence I am trying to prove that this is the case in all dimensions. In this process the only thing I was able to prove that there is a retraction from these subsets to the underlying sphere. </p> <p>So I would like to know about various methods to show that a space is a wedge of spheres. </p> <p>I understand that this question might sound vague and the information too little.</p> http://mathoverflow.net/questions/42986/computing-homology-of-very-large-posets/43060#43060 Answer by Priyavrat Deshpande for Computing homology of very large posets Priyavrat Deshpande 2010-10-21T15:49:21Z 2010-10-21T15:49:21Z <p>I agree with Jim Conant, discrete Morse theory should help. Acyclic matching of (the Hasse diagram of) the poset is a good candidate for a discrete vector field (see D. Kozlov's book Combinatorial Algebraic Topology, chapter 11). </p> <p>However, it does not guarantee that one will get a minimal complex (all the cells are homology cells). If your posets are not "nice" enough then finding a (perfect) acyclic matching will be as hard as finding a recursive (co)atom ordering.</p> http://mathoverflow.net/questions/43844/algorithm-to-find-an-acyclic-matching-on-a-poset/43847#43847 Comment by Priyavrat Deshpande Priyavrat Deshpande 2010-10-27T18:49:28Z 2010-10-27T18:49:28Z Thank you very much. http://mathoverflow.net/questions/43711/how-to-show-that-a-space-has-the-homotopy-type-of-wedge-of-spheres/43722#43722 Comment by Priyavrat Deshpande Priyavrat Deshpande 2010-10-27T14:16:49Z 2010-10-27T14:16:49Z Thanks for the reply. I tried using acyclic matching but then I realized that I don't know whether the spaces under consideration admit the minimal complex (only cells are the homology cells). I have asked this question here: <a href="http://mathoverflow.net/questions/43799/discrete-morse-theory-and-existense-of-minimal-complex" rel="nofollow" title="discrete morse theory and existense of minimal complex">mathoverflow.net/questions/43799/&hellip;</a> http://mathoverflow.net/questions/43711/how-to-show-that-a-space-has-the-homotopy-type-of-wedge-of-spheres Comment by Priyavrat Deshpande Priyavrat Deshpande 2010-10-27T00:58:12Z 2010-10-27T00:58:12Z I can express $\pi_1$ as a colimit of fundamental groups of the spaces in a diagram, other than that I don't know anything in general. In case of the examples I could do by hand, I Was able to directly observe the wedge product hence I didn't bother to calculate the colimit and verify at the level of $\pi_1$. http://mathoverflow.net/questions/43711/how-to-show-that-a-space-has-the-homotopy-type-of-wedge-of-spheres Comment by Priyavrat Deshpande Priyavrat Deshpande 2010-10-26T22:26:32Z 2010-10-26T22:26:32Z Computation of $\pi_1$ doesn't really help because in higher dimensions $S^1$ is absent.