User xiao xinli - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T18:34:04Z http://mathoverflow.net/feeds/user/3478 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/130055/axiomatic-intersection-theory Axiomatic intersection theory Xiao Xinli 2013-05-08T05:23:57Z 2013-05-08T08:42:31Z <p>Is there an axiomatic intersection theory?</p> <p>What I expect is something like: An intersection theory is a functor from the category of schemes(or other spaces) to the category of algebras, with well-defined flat pullback and proper pushforward, and (maybe) projection formula or some other axioms.</p> http://mathoverflow.net/questions/111868/the-use-of-hall-algebras-in-physics The use of Hall algebras in physics Xiao Xinli 2012-11-09T05:00:50Z 2012-11-10T21:56:08Z <p>I once read a statement (not memorized precisely) that a certain physics quantity between two states of charge \$d_1\$ and \$d_2\$ respectively could be computed by running over the states of charge \$d_1+d_2\$ which is the extension of the original two states. Therefore we need to consider some Hall algebras on a moduli space.</p> <p>I couldn't find that literature any more, so I am not sure that this statement is correct. Could anyone help me to make clear this sort of things? Thanks a lot!</p> <p>My questions are:</p> <p>1) What is the basic physics setting of this story?</p> <p>2) Why is this "extension" important?</p> <p>3) If this is not correct, what is the correct statement/why do physicists care about Hall algebras?</p> http://mathoverflow.net/questions/75220/physicists-request-for-intuition-on-covariant-derivatives-and-lie-derivatives/75237#75237 Answer by Xiao Xinli for Physicist's request for intuition on covariant derivatives and Lie derivatives Xiao Xinli 2011-09-12T17:43:35Z 2011-09-12T17:43:35Z <p>Lie derivative is based on a Lie group (or Lie algebra) which acts on the manifold. This derivative cannot be defined just at one point because the action cannot be defined at a point even if you give explicitly the direction at that point. On the other hand, using connection, covariant derivative can be defined pointwise. I think this is the main technical difference between them.</p> http://mathoverflow.net/questions/13005/what-is-formal What is 'formal' ? Xiao Xinli 2010-01-26T03:02:53Z 2010-02-01T08:41:00Z <p>The key step in Kontsevich's proof of deformation quantization of Poisson manifolds is the so-called formality theorem where 'a formal complex' means that it admits a certain condition. I wonder why it is called 'formal'. I only found the definition of Sullivan in Wikipedia: 'formal manifold is one whose real homotopy type is a formal consequence of its real cohomology ring'. But still I am confused because most of articles I found contain the same sentence only and I cannot understand the meaning of 'formal consequence'. Does anyone know the history of this concept?</p> http://mathoverflow.net/questions/111868/the-use-of-hall-algebras-in-physics/111991#111991 Comment by Xiao Xinli Xiao Xinli 2012-11-11T03:19:56Z 2012-11-11T03:19:56Z Thank you very much! Additional question: what motivated you to use that correspondence conjecture to define the product of BPS state? http://mathoverflow.net/questions/111868/the-use-of-hall-algebras-in-physics/111900#111900 Comment by Xiao Xinli Xiao Xinli 2012-11-10T04:23:05Z 2012-11-10T04:23:05Z Thank you very much! However, what I expected is something more physics. Those two papers are two mathematics. http://mathoverflow.net/questions/75220/physicists-request-for-intuition-on-covariant-derivatives-and-lie-derivatives/75237#75237 Comment by Xiao Xinli Xiao Xinli 2011-09-12T20:20:48Z 2011-09-12T20:20:48Z Covariant derivative is the analogue of directional derivative in R^n case. So if we fix a connection and assign a direction to a point, the covariant derivative at that point is well-defined. But for Lie derivative, one direction is not enough. We have to point out the vector field. L_X(f) might not equal to L_Y(f) even if X(p)=Y(p). http://mathoverflow.net/questions/46659/the-major-families-of-quantum-groups/47919#47919 Comment by Xiao Xinli Xiao Xinli 2010-12-01T18:00:45Z 2010-12-01T18:00:45Z So actually we should focus on the categories related to the quantum group instead of the quantum group itself?