User hyyy - MathOverflow

The discrete theory of compressible fluids dynamics

2013-04-27T13:50:22Z

I am working on the discrete theory of compressible fluids dynamics, i.e., numerically solving and simulating the compressible fluids , we are interested in the way using discrete exterior calculus, my question is: Is there any work on the discrete theory (especially using discrete exterior calculus) of compressible fluids?

Software to numerically solve partial differential equation

2013-03-22T15:03:57Z

When we use software to numerically solve differential equation, for example, using finite difference, finite element or finite volume methods, etc., is it possible that people input differential equation , and then the software can automatically transfer the mathematical equation to something that the software can recognize? The background of my question is that there is a kind of language for expressing variational form of PDE, called Unified Form Language (UFL) used in FEniCS, an excellent software for solving differential equations using Finite Element Method, there are some form complier which can generate C++ code based on the UFL inputed, then solve them using finite element method. So I was wondering if there are something even more close to mathematical particial differential equation that we just input these then the software can solve them, because this is more human-friendly. And is there any relationship of my question to Symbolic Computing, or what is the principle underlying in Symbolic Computing?

Questions on Discrete Exterior Calculus in numerial computing

2013-03-16T08:00:28Z

I have several questions about the Discrete Exterior Calculus (DEC) in the numerical method for solving partial differential equation in physics:

(Discrete Exterious Calculus is the newly developed subject mainly used in numerical computing, one reference is, for example, Hirani's PhD thesis: Discrete Exterior Calculus)

Has any kind of convergence property been proved?I mean, under what conditions, the numerical solution of DEC scheme will converge to the the actual solution of PDE. I checked many literature and didn't see any material concerning the convergence property, because I am doing engineer problem in computer and if we can't gurantee it will converge then the precision will be a problem.
What is the current status of using DEC to numerically solve equations/simulation in fluid mechnanics, elasticity and electromagnetism, respectively? Should anyone give me some relevant papers, I have found some but just don't know if I missed anything.

Thanks for any help!

Deligne-Mumford space defined in complex geometry category

2010-07-04T20:20:55Z

Is Deligne-Mumford space could also be defined in the complex geometry context? I check wiki, it says we can similarly define Riemann surface with nodes and stability condition, I am wondering if there is any reference providing more details about this aspect. Thanks!

Website for temporary instructor/lecturer positions

2011-01-01T10:57:06Z

Is there any website for temporary instructor/lecturer positions in mathematics? Or do we have to check school's websites one by one?

homotopy between solutions of Maurer-Cartan equation

2010-12-07T09:32:13Z

If $S_0, S_1$ are two solutions of Maurer-Cartan equation $dS+\frac{1}{2}{S,S}=0$ for a dg-Lie algebra $g$, do we have a suitable concept of homotopy between $S_0$ and $S_1$?

Why is this a local constant sheaf

2010-12-06T14:57:29Z

If a group $G$ acts on a topological space $M$, and a representation of $G$ on a vector space $V$, why $M \times_G V$ is a local constant sheaf over $M/G$?

What's the current state of Yang Mills Mass Gap question?

2010-03-06T02:14:33Z

What's the current state of Yang Mills Mass Gap question, is there any place that does this problem? Especially I want to know if there is any progress (out of that mentioned in the introduction article by Witten and Jaffe). Is it too hard for a mathematician? Thanks!

Where can I find material that introduce homotopy (co)invariant

2010-11-01T07:50:57Z

Thanks for pointing out any reference.

Question related to the moduli space of Riemann surfaces and a fibration

2010-08-08T04:15:42Z

If $M_{g}$ is the moduli space of Riemann surface of genus $g$, $M^1_{g}$ is the moduli space of Riemann surface of genus $g$ with one boundary, how can we show that the natural map:

$M^1_{g} \rightarrow M_{g}$

the fiber over a point $S \in M_{g}$ is the sphere bundle associated to the tangent bundle of Riemann surface $S$?

I don't have appropriate coordinates that is probably why I can't show it.

I am asking this question because I want to show that if $M^{m,b}_{g,n}$ is the moduli space of Riemann surfacw of genus $g$ and $b$ boundary components, and with $n$ interior marked points, $m=(m_1,m_2,\ldots,m_{b})$ marked points on the boundary, $m_i$ marked points on the $i$th boundary component, then the homology dimension

$H_i(M^{m,b}_{g,n},Q)=0,$ for $i\ge 6g-7+2n+3b+m$ possibly excludes some bad low dimensional cases.

I learned one way is tp assoicate it to mapping class group (especially we got a virtual cohomological dimension of mapping class group by Harer"on virtual cohomological dimension of mapping class group of oriented surface") and homology of mapping class group is the same as homology of the above moduli space, but I still don't understand how it works. Thank you for help.

stability of Riemann surface with boundary

2010-08-01T04:53:11Z

The stability condition of Riemann surface with boundary, that is, a Riemann surface of genus $g$,$h$ boundary components, and with $n$ marked points in the interior,$m=(m_1,m_2,\ldots,m_{h})$marked points on the boundary, here $m_i$ is the number of marked points on the $i$th boundary component, such a surface is called stable if the automorphism group is finite. I am wondering if the stability condition is also equivalent to the euler characteristic, $2-2g-n-h-m/2$, is negative?

Homology dimension of the mapping class group of a surface with boundary

2010-07-30T05:22:22Z

There is a result on the dimension bound for ${M_{g,n}}/S_n$, (the moduli space for Riemann surfaces of genus $g$ with $n$ marked points) that is $H_{i}({M_{g,n}}/S_n)=0$, for $i\ge 6g-7+2n$ except $(g,n)=(0,3),(0,2),(0,1),(0,0),(1,1)$. This result (see Costello: Gromov-Witten potential associated to a TCFT) can be derived from the virtual cohomology dimension of the mapping class group (see J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface, Invent. Math). I am wondering if there is such a theorem for surfaces with boundary. For example, is there any similar result for mapping class groups of orientable surfaces with boundary(and marked points if necessary)? Then can we get a result similar to the above dimension bound for moduli spaces of Riemann surfaces with boundary and marked points? I just want to know if such a result already exists.

Can we use reduced homology so that we don 't need exclude those low dimensional cases? I mean if it is natural?Thanks!

Spectral sequence for reduced homology

2010-07-31T13:38:46Z

In the Serre spectral sequence, is it true that we can replace homology by reduced homology? That is: If $f:X\rightarrow B$ is a Serre fibration,with $F$ the fiber, then if $\tilde E^2_{pq}=\tilde H_p(B,\tilde H_q F) \longrightarrow \tilde H_{p+q}(X)$?

I think it is fine: we use complexes involving "augmentation", then filtration, do the usual things in the spectral sequence, and finally we get a sequence that converges to the homology of the original complex. Only now it becomes reduced homology. But I am not precisely sure.

I ask this question because I want to know the answer of another problem (asked by me) "homology dimension of mapping class group of surface with boundary". (I am sorry I don't know how to insert a link). I need some help for that problem. Thanks!

What's the chain level Gromov-Witten theory

2010-06-30T02:43:57Z

I think I heard there is such a theory, but I just can't find reference.So I am asking if there really has such a theory and reference if yes. Thanks firstly!

different "derived structure" in derived algebraic geometry

2010-07-26T08:27:00Z

In derived algebraic geometry there are several different setting,i.e., sometimes we use $E_{\infty}$ring,somethings we use dg-algebra,... It is for different situations. But could someone give some examples illustrating under what problem we use relevant "derived structure" ($E_{\infty}$ ring, dg-algebra...)? Some motivations?

website for jobs in applied or industrial mathematics (or financial math)

2010-07-21T01:08:10Z

What are the websites for general position in applied or industrial mathematics(or financial mathematics) related jobs (that is if we have to find a non academic job temporarily) ?Thanks!

Is there any disadvantage from non-academic job turn to academic job in math

2010-03-28T01:31:33Z

If you get your PhD in math , and then work for 1 or 2 years in a non-academic institution and then turn to apply for postdoc or tenure-track position in math like usual, is there any disadvantage (I mean for your application for postdoc or tenure-track position)?

An appendix: I just want to make sure whether or not I can't or it is difficult to get reference letter, take conference or give talks (in the future) because you are not in academic institution. This is the most important for someone (like me) who will returen to academic job.(but for some reasons he can't now)

Computation of homology groups of $M_{g,n}$

2010-07-18T18:46:06Z

First some definitions: $\bar{M_{g,n}}$ is Deligne-Mumford space, i.e., the moduli space of stable nodal complex projective curves of genus $g$ with $n$ marked points. It is a complex orbifold, $\partial \bar{M_{g,n}}$ is the locus in $\bar{M_{g,n}}$ corresponding to nodal curves (with singularity). Do we have any relationship between the homology groups $H_{*}(M_{g,n},Q)$ and the homology groups of $\bar{M_{g,n}}$, $\partial \bar{M_{g,n}}$, the pair $(\bar{M_{g,n}},\partial \bar{M_{g,n}})$ (relative homology); here $M_{g,n}$ is $\bar{M_{g,n}}\setminus \partial \bar{M_{g,n}}$ the locus of smooth curves?

The point is for any pair of compact oriented manifolds $(X, Y), Y\subset X$, can we calculate the homology groups of $X\setminus Y$ from those of $X, Y$ and the relative homology groups $(X,Y)$ (it is not an excision case)?

This is a problem I find on page 23 of the paper: Costello, "Gromov-Witten potential associated to a TCFT", (although there it is $\bar{M_{g,n}}/S_n$, modulo the action of permutation of marked points, but it is not a big deal).

One more question is: is $\bar{M_{g,n}}/S_n$ orbifold?

To differently gluing of two Riemann surfaces with boundary we get different surfaces

2010-07-01T15:33:16Z

If $M,N$ are two Riemann surfaces with boundary, then we can glue them along one of each of their boundary component, which is $S$, to form a new Riemann surface with boundary, but for different gluing we may form different Riemann surfaces with boundary, for example, there may be a $S$ twist, intuitively, it is just we rotate one $S$ an angle then glue it with another surfaces, but my question is how can we show the resulting two surfaces (twisted gluing and untwisted gluing) are different Riemann surface with boundary (their differential structures are the same because we can regard it as a kind of connected sum)? Thanks!

algebraic geometry and complex geometry in dimension 2

2010-07-04T22:01:57Z

Even if in dimension 2, complex structure is equivalent to algebraic structure for surfacs, but when studying deformation theory or moduli theory for surface, they are different, for example, the concept of the family of Riemann surface and the family of complex smooth algebraic curve are different, but it seems that we can study the deformation theory (or moduli question) for Riemann surface in the language of algebraic geometry (or just consider it is an algebraic geometry object), I don't know if it is true or it is my misunderstanding. So my question is if the deformation theory for Riemann surface is the same as the deformation theory for smooth complex algebraic curve? Thank you for any answer for this question which confuses me a long time!

How can we show the spaces $M_{g}(n)$ and $M_{g, n}$ are homotopy equivalent?

2010-07-05T23:31:35Z

How can we prove that the moduli space,$M_{g}(n)$, of genus $g$ Riemann surface with $n$ boundary components is homotopy equivalent to $M_{g,n}$, that is ,the moduli space of genus $g$ Riemann surface with $n$ punctures? Thanks! (It is very intuitive, but it seems that I can't make it)

Do the virtual fundamental classes satisfy functorial properties?

2010-06-30T22:02:37Z

In Gromov–Witten theory, if the symplectic virtual fundamental classes constructed by B.Siebert satisfy functorial properties, i.e., if $f\colon X\to Y$ is an appropriate map between symplectic manifolds $X$ and $Y$, then $f_*\colon [X]^{\rm vir}=[Y]^{\rm vir}$ ? In his paper constructing symplectic GW invariant, I didn't see he mentions this, so does anyone knows anything about this? Thanks!

Why are people interested in defining GW invariant in algebraic geometry category

2010-07-04T21:23:41Z

Originally it is in symplectic geometry. Is it just curosity or any other special reason? Thank you for clarifying.

Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is hausdorff

2010-07-05T00:39:48Z

Is there a reference showing that the space $\bar{M_{g,n}}$ is a closed oriented orbifold and it is Hausdorff? Note: here $\bar{M_{g,n}}$ is not the Deligne-Mumford space in the usual algebraic geometry; it is the moduli space for smooth or nodal Riemann surfaces with genus $g$ and $n$ marked smooth points such that it satisfies the stability condition. Thanks!

Is it possible a variety be a manifold with boundary

2010-07-04T19:26:30Z

As a complex affine variety or projective variety, is it possible it is a manifold with boundary?

Number of non-Abelian groups of order $2^n$

2010-07-02T23:41:24Z

Related to A000679 (Number of groups of order $2^n$), how many non-Abelian groups of order $2^n$ are there?

a question on Costello's theorem

2010-06-30T03:01:42Z

Costello's theorem,"open TCFT=Calabi Yau A-infinity category",he also mentions when applied to Fukaya category we can recover Gromov-Witten theory, but I see that it needs some assumption,also even if the assumption is true,I just see that the TCFT associated to Fukaya category is the same as the one associated to Gromov-Witten invariant, BUT can we from that TCFT to RECOVER Gromov-Witten invariants? (I mean the opposite direction) Because otherwise,it is not A model.

I appreciate if there is any answer.

recommendation letter for teaching

2010-06-26T00:06:40Z

I need a recommendation letter on my teaching. I want to ask the instructor in last semester for which I was a TA, but I don't know how his impression for my teaching. So do I need to ask him for his opinion about my teaching before letting him write the recommendation letter?

Are the two B model constructions equivalent?

2010-06-24T15:38:30Z

Now we have two B model constructions: Kontsevich-Baranikov (for genus 0) and Costello (for any genus). My question is, are they equal in genus 0? Or now which one is the one we want? Thanks!

Do we have a pullback operation on simplicial chains?

2010-06-22T22:08:10Z

Do we have a pullback operation on singular simplicial chains,that is if f:X-->Y is a continuous map between topological space X and Y,and C is a singular simplicial chain on Y,then do we have a singular simplicial chain on X which is the pullback of C along f?

Comment by HYYY

2013-03-22T14:52:21Z

@Artur Palha:Thanks!I am working on numerically solve equations in fluid mechanics, elasticity and electromagnetism using DEC, and develop software for this, however, it seems that the solution is still not completely found. What's your opinion?Thanks!

Comment by HYYY

2010-12-07T07:30:28Z

@Emerton, thanks!but how can we see the action of $G$ on $V$ from this local system? If we choose a point $x$ in $M$,do we get a section $c\in V$ of this local system over the point $[x]$ in $M/G$?And then if we choose another point $y$ which in the same orbit of $x$,assume $yg=x$,then we get another section $c'\in V$ over $[y]=[x]$,is that $gc=c'$?

Comment by HYYY

2010-11-14T10:43:14Z

@Sinha, Could you explain why taking equivariant homomorphisms from chains on $S^\inf_{+}$ to a given $V$ yields $V[[t]]$? Thanks!

Comment by HYYY

2010-11-14T09:52:10Z

In Costello's work "The Gromov-Witten potential associated to a TCFT", it says the pullback operation is not well-defined in simplicial chain, but it is defined for homotopy coinvariant, that's what I am studying.

Comment by HYYY

2010-09-27T11:40:53Z

By the way, is it still true for X being a manifold?

Comment by HYYY

2010-09-17T21:58:42Z

if that is already in equivariant homotopy theory...?

Comment by HYYY

2010-09-17T21:57:01Z

@Dan, hi,do you know if that result has been written down somewhere except in the Tohoku paper?

Comment by HYYY

2010-08-28T13:49:24Z

Does that mean we can totally change geometic problems into algebraic ones, like we did in algebraic geometry? Is there any difference between these?Thank you!

Comment by HYYY

2010-08-20T10:27:16Z

Kevin, do you read that paper"The Gromov-Witten potential associated to a TCFT", I have several questions?

Comment by HYYY

2010-08-09T22:35:22Z

Dear Andy, the mapping class group in Birman exact sequence should be pure mapping class group,i.e.,it fixes each puncture individually, then the corresponding moduli space should be the moduli space of Riemann surface with ordered puncture right?

Comment by HYYY

2010-08-08T08:43:15Z

Thank you,Andy and algori, is the classifying space of mapping class group the same as the corresponding moduli space?Thanks!

Comment by HYYY

2010-08-07T10:50:53Z

Hi,Oscar,thank you for your help. but could you show me why that is a sphere bundle assoicated to the tangent bundle? and is that fibration just a fiber bundle?Thanks!

Comment by HYYY

2010-07-31T14:43:16Z

Hi,Tim Perutz, it is not valid.

Comment by HYYY

2010-07-30T13:10:28Z

Hi,Oscar, according to your argument, it seems that I can show the homology of $M^{b,m}_{g,n}$ (here, there could be marked points on the boundary,for which $m=(m_1,m_2,\ldots,m_b), m_i$ is the number of marked points on the $i$th boundary component) vanishes in degrees at least $6g-7+2n+3b+m$. However, when $6g-7+2n+3b+m=0$,$H_0$ will not be 0. Is this result true for other cases except the cases for $6g-7+2n+3b+m=0$? In fact, for the homology dimension for closed surface we need to exclude the case $(g,n)=(0,3)$ because $H_0$ is also not 0. How to deal with such things? Thanks!

Comment by HYYY

2010-07-23T01:25:07Z

Thanks!Kevin. but I thought Sigma model is only one kind of topological (or conformal) field theory.