User chirag lakhani - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-20T06:30:32Z http://mathoverflow.net/feeds/user/2565 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/109643/interpolating-a-manifold-between-two-points Interpolating a "manifold" between two points Chirag Lakhani 2012-10-14T19:35:08Z 2012-10-16T15:12:25Z <p>Edit: I have reworded the question.</p> <p>This may be a basic question but I am having trouble figuring out the correct answer. I want to find a local coordinate chart that fits a d-dimensional submanifold in $\mathbb{R}^N$. I am given two points $p_1, p_2 \in \mathbb{R}^N$ and corresponding orthonormal bases $(\phi_1, \phi_2, \ldots, \phi_d)$, $(\tau_1, \tau_2, \ldots, \tau_d) \subseteq \mathbb{R}^N$ for their tangent spaces. I would like to find an algorithmic method for finding functions $(f_1(x_1,x_2,\ldots,x_d), f_2(x_1,x_2,\ldots,x_d),\ldots f_N(x_1,x_2,\ldots,x_d))$ that satisfy these conditions. </p> http://mathoverflow.net/questions/85856/places-to-learn-about-landau-ginzburg-models/85866#85866 Answer by Chirag Lakhani for Places to learn about Landau-Ginzburg models Chirag Lakhani 2012-01-17T00:40:09Z 2012-01-17T00:40:09Z <p>I think the Clay Math books have nice descriptions about Landau-Ginzburg Models. The book called <a href="http://www.claymath.org/publications/Mirror_Symmetry/" rel="nofollow">Mirror Symmetry</a> and <a href="http://www.claymath.org/publications/Dirichlet_Branes/" rel="nofollow">Dirichlet Branes and Mirror Symmetry</a> both have nice physical and mathematical descriptions of these models. Mirror Symmetry is also available as a <a href="http://www.claymath.org/library/monographs/cmim01c.pdf" rel="nofollow">pdf</a>. I would start with Chapter 13 of Mirror Symmetry.</p> http://mathoverflow.net/questions/69471/what-is-the-ring-of-invariants-of-gl-acting-on-quaternary-cubic-forms/69479#69479 Answer by Chirag Lakhani for What is the ring of invariants of GL acting on quaternary cubic forms? Chirag Lakhani 2011-07-04T16:04:20Z 2011-07-04T16:04:20Z <p>There is also a nice book by Sturmfels called <a href="http://books.google.com/books/about/Algorithms_in_invariant_theory.html?id=3lF0cyOHwgQC" rel="nofollow">Algorithms in Invariant Theory</a> that discusses the problem of finding invariants. It may be worth a look.</p> http://mathoverflow.net/questions/8872/corank-4-hypersurface-singularities Corank 4 hypersurface singularities Chirag Lakhani 2009-12-14T14:18:53Z 2011-07-03T21:29:09Z <p>A function f: ($\mathbb{C}^n$,0) $\to$ ($\mathbb{C}$,0) is considered a hypersurface singularity if the point $(0,0,\dots,0)$ is the only point in the ideal $\langle \frac{\partial f}{\partial x_1}, \frac{\partial f}{\partial x_2}, \dots, \frac{\partial f}{\partial x_n} \rangle$. The hessian matrix $(\frac{\partial^2 f}{\partial x_i x_j})$ gives rise to an invariant called the corank which is n - rank of the hessian. In Arnold's book (Singularities of Differentiable Maps V.1) there is a very algorithmic way of classifying singularities with corank 3 and less. For corank 4 and higher it's stated that they belong to a family denoted $\mathbf{O}$. </p> <p>My question is what is known about corank 4 singularities? Are there any references which discuss how to classify these singularities?</p> http://mathoverflow.net/questions/62319/learning-mathbfa1-homotopy-theory/62352#62352 Answer by Chirag Lakhani for learning $\mathbf{A}^1$-homotopy theory Chirag Lakhani 2011-04-19T23:51:51Z 2011-04-19T23:56:59Z <p>Aravind Asok has an entire website devoted to pointing out resources for learning $\mathbf{A}^1$-homotopy theory. It is organized quite well. The concept list section of the page has lots of wikipedia-like entries on topics related to $\mathbf{A}^1$-homotopy theory.</p> <p><a href="http://a1homotopy.tiddlyspot.com/" rel="nofollow">http://a1homotopy.tiddlyspot.com/</a></p> http://mathoverflow.net/questions/61515/life-after-hartshorne-the-book-not-the-person/61538#61538 Answer by Chirag Lakhani for Life after Hartshorne (the book, not the person)... Chirag Lakhani 2011-04-13T12:05:10Z 2011-04-13T12:05:10Z <p>If you are interested in complex manifolds I would recommend <a href="http://books.google.com/books?id=sWbd0rE3mhIC&amp;printsec=frontcover&amp;dq=complex+manifolds+huybrechts&amp;hl=en&amp;ei=44-lTaPBN82ztwerlqS-Ag&amp;sa=X&amp;oi=book_result&amp;ct=result&amp;resnum=1&amp;ved=0CCgQ6AEwAA#v=onepage&amp;q=complex%2520manifolds%2520huybrechts&amp;f=false" rel="nofollow">Complex Geometry: an Introduction</a> by Huybrechts. </p> <p>I also think the <a href="http://www.cs.amherst.edu/~dac/toric.html" rel="nofollow">Toric Varieties</a> by Cox, Little, and Schenck is an excellent introduction to many advanced topics in algebraic geometry. Plus you get to learn a bunch of combinatorics in the process!</p> http://mathoverflow.net/questions/57975/reference-request-for-manifold-learning Reference request for manifold learning Chirag Lakhani 2011-03-09T17:59:24Z 2011-04-10T05:26:12Z <p>I am interested in learning about manifold learning (no pun intended) and would like to know of some references that discuss the subject from a more geometric perspective. By manifold learning I mean the idea of studying high dimensional data using techniques from geometry. I'm interested in knowing how topics from differential geometry and topology such as Hodge theory and Morse theory can be used to study questions in manifold learning. I thought I would ask if people have any recommendations for papers or books that explain these topics more from a more geometric perspective.</p> <p>Update: I expect that there is no mythical survey paper that explains all aspects of manifold learning to someone that knows about geometry and topology. Specifically, I would be interested in knowing of some survey papers which explain how tools from Riemannian geometry would be useful in manifold learning. Perhaps how such tools can be used for nonlinear dimensionality reduction.</p> http://mathoverflow.net/questions/57975/reference-request-for-manifold-learning/61175#61175 Answer by Chirag Lakhani for Reference request for manifold learning Chirag Lakhani 2011-04-10T01:17:07Z 2011-04-10T01:17:07Z <p>I came across a nice video lecture by Niyogi that gives a nice survey of manifold learning. I thought I would share in case anyone else was interested.</p> <p><a href="http://videolectures.net/mlss09us_niyogi_belkin_gmml/" rel="nofollow">http://videolectures.net/mlss09us_niyogi_belkin_gmml/</a></p> http://mathoverflow.net/questions/31065/applications-of-non-reductive-git Applications of non-reductive GIT Chirag Lakhani 2010-07-08T15:33:02Z 2010-07-08T20:00:04Z <p>Geometric invariant theory works well when the algebraic group $G$ acting on a variety is reductive. There has been recent work by Doran and Kirwan <a href="http://arxiv.org/abs/math/0703131" rel="nofollow">here</a> and <a href="http://arxiv.org/abs/0801.4607" rel="nofollow">here</a> to find a canonical method of constructing GIT quotients for non-reductive groups. My question is what are potential applications for their work? One specific application they mention is constructing moduli of hypersurfaces in toric varieties. I would be interested in knowing of other applications.</p> http://mathoverflow.net/questions/23817/what-is-known-beyond-the-tangent-cone-for-hypersurface-singularities What is known beyond the tangent cone for hypersurface singularities? Chirag Lakhani 2010-05-07T03:19:01Z 2010-05-07T04:11:41Z <p>If [1:0:....:0] is an s-fold singularity of a degree $r$ hypersurface $F$ in $\mathbb{P}^n$ then the hypersurface can be written as $F=x_0^{r-s}g_s(x_1,...,x_n) + x_0^{r-s-1}g_{s+1}(x_1,...,x_n) + ... +g_r(x_1,...x_n)$. After we dehomogenize it is known that the initial term $g_s(x_1,...x_n)$ is the tangent cone at the singularity in $\mathbb{C}^n$. My question is known about the higher order terms such as $g_{s+1}(x_1,..,x_n)$? Do they admit a some geometric interpretation? </p> <p>I know that the common locus of $g_s=g_{s+1}=...=g_{s+h}=0$ give the set of points whose line through the origin has intersection multiplicity s+h+1 with the hypersurface. I would like to find a geometric interpretation of just the hypersurface $g_{s+1}=0$.</p> http://mathoverflow.net/questions/2755/a-learning-roadmap-for-representation-theory/18088#18088 Answer by Chirag Lakhani for A learning roadmap for Representation Theory Chirag Lakhani 2010-03-13T19:17:08Z 2010-03-13T19:17:08Z <p>I ran across an excellent book by Lakshmibai and Brown called <a href="http://www.ams.org/bookstore?fn=20&amp;arg1=hinseries&amp;ikey=HIN-40" rel="nofollow">Flag Varieties: an Interplay of Geometry, Combinatorics, and Representation Theory</a>. It seems like an excellent book for an algebraic geometer who is interested in representation theory and algebraic groups.</p> http://mathoverflow.net/questions/4172/where-does-a-math-person-go-to-learn-statistical-mechanics/17201#17201 Answer by Chirag Lakhani for Where does a math person go to learn statistical mechanics? Chirag Lakhani 2010-03-05T18:32:50Z 2010-03-05T18:32:50Z <p>I've only briefly looked at it but the book <em><a href="http://books.google.com/books?id=1WlFhiz5UjwC&amp;dq=Mehran+Kardar&amp;client=firefox-a&amp;source=gbs_navlinks_s" rel="nofollow">Statistical Physics of Particles</a></em> by Mehran Kardar seems pretty good. It starts with an introduction to the relevant probability theory and then moves on to the basics of statistical mechanics. There is also a subsequent book <em><a href="http://books.google.com/books?id=nTxBhGX01P4C&amp;printsec=frontcover&amp;dq=Mehran+Kardar&amp;client=firefox-a&amp;cd=2#v=onepage&amp;q=&amp;f=false" rel="nofollow">Statistical Physics of Fields</a></em>.</p> http://mathoverflow.net/questions/109643/interpolating-a-manifold-between-two-points/109676#109676 Comment by Chirag Lakhani Chirag Lakhani 2012-10-20T14:07:49Z 2012-10-20T14:07:49Z Thanks! The use of algebraic topology is really interesting, even if it provides a nonexistent solution! http://mathoverflow.net/questions/109643/interpolating-a-manifold-between-two-points/109824#109824 Comment by Chirag Lakhani Chirag Lakhani 2012-10-20T14:05:33Z 2012-10-20T14:05:33Z Thanks! This looks like an interesting construction. I will have to play around with this and see if I can get nice explicite formulas for my case. http://mathoverflow.net/questions/109643/interpolating-a-manifold-between-two-points Comment by Chirag Lakhani Chirag Lakhani 2012-10-14T23:19:13Z 2012-10-14T23:19:13Z I mean edit the question. http://mathoverflow.net/questions/109643/interpolating-a-manifold-between-two-points Comment by Chirag Lakhani Chirag Lakhani 2012-10-14T23:18:43Z 2012-10-14T23:18:43Z Sorry I have tried to edit the equation. I would like to just find the functions in one chart. The problem I find is what is an appropriate polynomial form to fit given this data. There is ambiguity in how I choose my coordinates and then I need to fit the right polynomial given my constraints. I just wasn't sure if there was a canonical way of doing this type of thing (something like a generalization of a spline). http://mathoverflow.net/questions/57975/reference-request-for-manifold-learning/61184#61184 Comment by Chirag Lakhani Chirag Lakhani 2011-04-10T15:14:12Z 2011-04-10T15:14:12Z Thanks! I didn't know about the Topology and its Applications books. Looks interesting. http://mathoverflow.net/questions/57975/reference-request-for-manifold-learning/59441#59441 Comment by Chirag Lakhani Chirag Lakhani 2011-03-25T14:29:10Z 2011-03-25T14:29:10Z Thanks for that website and for the Smale, Niyogi, and Weinberger reference it looks very interesting. http://mathoverflow.net/questions/57975/reference-request-for-manifold-learning Comment by Chirag Lakhani Chirag Lakhani 2011-03-09T22:31:26Z 2011-03-09T22:31:26Z Thanks for your comments I will try to be more precise. http://mathoverflow.net/questions/57975/reference-request-for-manifold-learning Comment by Chirag Lakhani Chirag Lakhani 2011-03-09T18:19:05Z 2011-03-09T18:19:05Z Sorry I should be more clear about manifold learning. I mean the idea of studying high dimensional data using techniques from geometry. http://mathoverflow.net/questions/31065/applications-of-non-reductive-git/31074#31074 Comment by Chirag Lakhani Chirag Lakhani 2010-07-08T17:22:45Z 2010-07-08T17:22:45Z Thanks. My advisor mentioned that he had conversations with Hain about this paper. Looking forward to seeing a paper on it if they are in fact working on this problem. http://mathoverflow.net/questions/31065/applications-of-non-reductive-git/31069#31069 Comment by Chirag Lakhani Chirag Lakhani 2010-07-08T17:20:11Z 2010-07-08T17:20:11Z Thanks! I never knew that \widetilde{g} was considered a Grothendieck-Springer resolution. You are correct that Doran-Kirwan also use this construction, which they call a reductive envelope. http://mathoverflow.net/questions/23817/what-is-known-beyond-the-tangent-cone-for-hypersurface-singularities/23819#23819 Comment by Chirag Lakhani Chirag Lakhani 2010-05-09T20:34:09Z 2010-05-09T20:34:09Z Thanks! I never thought about it in terms of blowups. http://mathoverflow.net/questions/2755/a-learning-roadmap-for-representation-theory/18088#18088 Comment by Chirag Lakhani Chirag Lakhani 2010-03-14T17:33:53Z 2010-03-14T17:33:53Z No problem. I really enjoyed their treatment of representations of algebraic groups.