User changwei zhou - MathOverflow

Answer by Changwei Zhou for japanese/chinese for mathematicians?

2012-07-28T15:15:10Z

As a native Chinese speaker I would suggest the author at least try to make friends with Chinese math students who can check if his understanding of the paper he interested is accurate - is Lemma A really about Statement B is not true?, etc. Chinese is a very flexible language and sometimes the meaning has to be discerned from the proper context. So to have a dictionary and google translate probably is not enough, and since ordinary Chinese people do not know mathematical terminology that well, you should consult professionals. Alternatively, many papers have author's email, so I guess if you drop him or her an email the author will be happy to provide a brief note on the contents of the paper.

I had not read Chinese math papers for a while since I graduated from high school, so I expect someone who did his undergraduate math studies in China might be more helpful. My impression is the math papers I used to read were either too difficult to understand or written in such a way impossible to understand clearly what the author is really talking about. My Chinese classmates told me they often encounter similar difficulities. So presumably for a non-native speaker he or she will find the situation even more difficult, since sometimes the proof style, tex format, definitions, etc are all different. For example, sometimes Chinese people invent a name for foreign mathematicans using characters with a similar pronounciation. For not so well-known young mathematicans there is no orthodox translation, so you might be puzzled to look up who 西尔弗曼(Joe Silverman) is.

It is not impossible to train yourself to speak/read Chinese like a native speaker in a few years, and there are remarkable math work done in Chinese still yet to be translated. But this cost of energy and time seems neither what you wanted nor practical in real life.

What is the importance of $\pi_{i}G$?

2012-07-22T02:12:43Z

I often see conditons like $\pi_{2}G\not=0$ in reading old papers on Lie groups(no, my memory is wrong, they asked if $\pi_{1}G$ is free). I want to ask why we need this condition and how the higher dimensional homotopy group matters. I have `read' most of available introductory graduate level textbooks on representation theory of Lie group and Lie algebra like Fulton and Harris, and I seldom(if ever) see higher diemsional homotopy groups enters into the discussion. I remember when I ask this to my undergraduate advisor, he responded that this is a "common condition" so we should not worry about it. Given the fact that most Lie groups are of huge dimension, it seems reasonable not to worry about if it is 2-connected. But now thinking in retrospective, I am wondering if the matter is this simple. Also, for practical purposes is there a practical way to compute it by its representations? For example $Sp(14,\mathbb{C})$ or $SO(5)$?

Answer by Changwei Zhou for Has mathoverflow yet led to mathematical breakthroughs?

2012-05-14T01:02:13Z

David Speyer's answer to my question has helped me to finish my Senior Project paper. But it is more at the level of graduate level homework exercise or student research than serious breakthrough. In case anyone wishes to look at it, I proved David's basis really works as well as provided computational proof for some other small rank examples.

Reference Request: Basis in terms of ring of symmetric polynomials

2012-02-08T23:11:58Z

As part of the result of solving the problem I am working on, my advisor and I translated the task of finding a basis for $R(T_{sl_{\mathbb{C}}(n)})$ in terms of $R(sl_{\mathbb{C}}(n))$ into the following problem:

Finding a basis for $\mathbb{Z}[x_{1},x_{2},...x_{n+1}], \text{with condition} \prod x_{i}=1$ in terms of $$\mathbb{Z}[x_{1}+x_{2}..+x_{n+1}, \sum_{i\le j} x_{i}x_{j}, \sum_{i\le j\le k} x_{i}x_{j}x_{k},...]$$

The later ring is obviously a ring of elementary symmetric polynomials. We are wondering if anyone from ring theory has already worked out this earlier. My advisor suggest I should look up online resources before I ask in here, but I could not find anything particularly relevant (I do not know if I am being lazy). I already searched sciencedirect, arxiv, wolfram-mathworld, wikipedia, etc. It seems to me that people are working with the basis of the ring of symmetrical polynomials itself instead of this problem.

I am more willing to work out this myself instead of learning from reading others, but I think I should have an accurate bibilography to acknowledge other's work. The associated question is I do not know what is the usual practice in such situation, for every trivial question in a research project may be something already well-known to experts. For example I do not wish to ask similar questions for type B,C,D, etc's representation ring in here. I wish to apologize in advance if this turned out to be something well-known or mundane.

ADDED:

For example, the well-known $SU(2)$ case has a torus whose representation ring is isomorphic to Laurent series in one variable, and $R(SU(2))\cong \mathbb{Z}[w+w^{-1}]$, which is the above problem when $n=2$. I am not familiar with ring theory and module theory (I know Hungerford's section in rings and some Hilton&Stammach,etc, but these seem quite unrelated).

ADDED:

Now voting to close. I might have seen this in Fulton&Harris so it should be well-known.

ADDED:

I finished the proof that David Speyer's basis is correct. I extended the result by low-dimensional coincidences to a few $SO_{2n}$ cases.

Is $R(su_{4})\cong R(so_{6})$?

2012-04-13T01:15:43Z

This is one of small the unsettled questions I had in my senior project. I want to prove for type $D$ we have $R(T)$ is a free module over $R(G)$ by finding a basis. I think we should have,$R(G)\cong R(g)$, $R(T)\cong R(h_{g})$, but since $SO_{6}$ is not simply connected this probably does not work and I have to "ascend" to spin groups, but I do not know how.

Define the representation ring of a lie algebra to be the formal sums of its characters, it is not hard to show that $$R(su_{4})\cong \mathbb{Z}[x+y+z+w,xy+yz+zx+wz+wy+wz,xyz+yzw+xzw+xyw]/(xyzw-1)$$ and $$ R(h_{su_{4}})\cong \mathbb{Z}[x,y,z,w]/(xyzw-1)$$

a typical basis of $R(h_{su_{4}})$ over $R(su_{4})$ consists of $x^{i}y^{j}z^{k}, 0\le i\le 3, o\le j\le 2, 0\le k\le 1$.

I proved that the weight lattice of $su_{4}$ and $so_{6}$ are isomorphic, and their Weyl group are both isomorphic to $S_{4}$. So $R(h_{so_{6}})$ should be a free module over $R(so_{6})$ with rank 24 as well. But I found I could not use this to find a basis for $R(h_{so_{6}})$ over $R(so_{6})$, because we have:

$$R(so_{6})\cong \mathbb{Z}[x+y+z+x^{-1}+y^{-1}+z^{-1},x^{\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}+x^{\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}+x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{\frac{1}{2}}+x^{-\frac{1}{2}}y^{\frac{1}{2}}z^{-\frac{1}{2}},x^{-\frac{1}{2}}y^{-\frac{1}{2}}z^{-\frac{1}{2}}+x^{\frac{1}{2}}y^{-\frac{1}{2}}z^{\frac{1}{2}}+x^{-\frac{1}{2}}y^{\frac{1}{2}}z^{\frac{1}{2}}+x^{\frac{1}{2}}y^{\frac{1}{2}}z^{-\frac{1}{2}}]$$

the first is the standard representation with weights $\pm L_{i}$, the second and the third are the spin representations one obtain from clifford algebra or "ascend" to spin group(can be found at Fulton&Harris, Chapter 23.2 or here). As one commentator noted I am not clear about the relationship between $R(so_{6})$ and $R(h_{so_{6}})$.

and $$R(h_{so_{6}})\cong \mathbb{Z}[x,y,z,x^{-1},y^{-1},z^{-1}]$$ because we know the two diagonal submatrices in $so_{6}$ must be skew-symmetric. From $A+D^{T}=0$ we conclude $T$ is isomorphic to $S^{1}\times S^{1}\times S^{1}$. Thus we conclude this.

I thought it would be a simple change of variable to prove the two cases are just the same, but I found the isomorphism between $R(so_{6})$ and $R(su_{4})$ does not extend nicely to an isomorphism between $R(h_{so_{6}})$ and $R(h_{su_{4}})$. So I believe I must be confused. My advisor suggested me that maybe there is some subtly in $Spin_{6}$, but I still do not know how to estbalish an isomorphism or to find the basis right away.

Answer by Changwei Zhou for Math blog directory

2012-04-08T07:22:18Z

I read Shuyun Wu's and Danny Calegari's blogs occasionally, as a rising graduate student they are "high quality" for me...

I would add a few blogs I seldom read like Zachery Abel's, Gil Kalai's, etc. Generally you can "discover" a wide range of math blogs simply by following links in anyone of them...

Answer by Changwei Zhou for What do heat kernels have to do with the Riemann-Roch theorem and the Gauss-Bonnet theorem?

2012-03-27T22:11:34Z

I am not able to say much regarding how to prove RR by Atiyah-Singer theorem, I know Gauss-Bonnet follows naturally from Atiyah-Singer theorem. Here is a heat kernel proof of Atiyah-Singer in case you are still looking for relevant material.

Answer by Changwei Zhou for Readings for an honors liberal art math course

2012-03-03T00:14:12Z

I think it would help to achieve a synthesis of mathematics and history. I do not mean how the development of mathematics influenced history(or, on the other hand how various historical phenomenon influenced mathematics), but how mathematicans acted as historical figures. It would be interesting to know how mathematicans communicated with each other, how cooperation helps to engender beautiful mathematics and how inevitable disagreements follow from different personality and world-view. Controversies in mathematics, though rare, helps to clarify issues that made further development possible.

It is not possible to make all students get used to mathematical thinking, for many of them must come from a varied background having little to do with science. Instead one should focus on the human side of the story to let them believe mathematics is a subject that focused on original(not magic or mechnical) contributions, sharp individual insights, and the mathematican community is no different from other professional groups of natural science. Instead of showing 'how spectacular math is!' and drill the students with introductory level texts that marvel them, one should help them realize how mathematical research is being produced in real life. For example, it is often being misunderstood that mathematicans do not make experiments; that sitting all day in front of a computer doing huge amount of calculations is the typical way of making progress. One should make the student understand the process of constructing a theory that might help us understand some underlying structure.

It is equally important to let the students understand that mathematical proofs is not so different from other forms of formal logic employed in real life. From my experience mathematicans often suffer from the shallow opinion by the public that they are being too critical on details and could not see the whole picture behind the main argument. It important to let them realize that while insisting on consistency and simplicity, mathematical proofs are ultimately human-mind products like the process of using rational reasoning to reach a conclusion in other fields.

One may organize a reading course by mimicking Carl Siegel's seminar at a much lower level. There are plenty of historical manuscripts, important papers, etc that has already been translated into English and worth discussing in a class. Manuscripts I can come up with are:

Abel's proof of the impossibility of solving the quintic equation in radical;

Riemann's speech "On the hypotheses which lie at the foundation of geometry";

Siegel's book "Topics in Complex Function Theory", Vol I should suffice for the purpose.

E.T.Bell's book "Men of Mathematics".

Hao Wang's book "A Logical Journey: From Gödel to Philosophy".

Answer by Changwei Zhou for eBook readers for mathematics

2010-07-05T03:08:21Z

Update: http://www.technologyreview.com/blog/mimssbits/27622/?ref=rss

It seems we finally got something having color....

Answer by Changwei Zhou for Do names given to math concepts have a role in common mistakes by students?

2012-02-28T01:52:11Z

I think it takes sometime for starters to realize the tensor product between representations and tensor product between modules, despite the similarity on the surface that we can treat $g$-representations as $g$-modules. Usually a confusion appears when a concept which makes perfect sense in one area was re-defined or used in a more subtle way in the other area, which might be counter-intuitive in some sense. A concrete example comes up in my mind is Segal's paper on representations of compact Lie groups, where a lot of definitions are rather ad-hoc in modern literature but makes perfect sense when one read his paper careful enough.

Self-similarity of Riemann's "non-differentiable" function

2011-12-13T03:05:54Z

I hope it doesn't seem inappropriate for me to raise on MO an unanswered question from MSE, indeed a question actually posed there by someone other than myself.

I want to ask the following:

1) Consider the function $$f(z)=\sum^{\infty}_{n=1}\frac{z^{n^{2}}}{n^{2}}.$$ By the original post it is highly likely that it has fractal behavior on the circle $|z|=1$. Lacking access to Maple, I do not have the means to generate such a graph so that I may enlarge it myself to check. The commentators found the following:

A: This "fractal" behavior seems to appear in a wide range of complex functions. Alex Jordan noted this holds for any function of the type $\sum^{\infty}_{n=1}\frac{z^{g(n)}}{g(n)}$ where $g$ grows fast enough.

B: The imaginary part is essentially the Weierstrass function. And Riemann's "nowhere differentiable" function appears as well.

C: Slight modification (consider $f(z)=z$ [???], etc) generates similar fractal behavior.

D: $f'(z)$ seems to be essentially the well-studied Jacobi elliptic function $f(z)=\frac{1}{2}+\frac{1}{2}\theta_{3}(0,z)$.

2) I know that complex dynamics has been well studied over the past two decades, but it is not my specialty and my knowledge of the field does not suffice for understanding this problem.

I want to ask:

E: Is this something essentially new? For this function in particular, it there any association with classical elliptic functions? (I do not know much about analytical number theory.)

F: Since this function is not analytic in most of the boundary points, is there a way to describe the boundary behavior in terms of the zeros of $f'(z)$, $f''(z)$..etc?

G: Is there a way to calculate the Hausdorff dimension of it?

I do not know if this question rises to research level.

Reference Request: Steinberg's 1975 paper "On a paper of Pittie"(retrieved)

2012-02-01T22:19:26Z

I am currently work on a senior project trying to prove for semisimple Lie groups, $R(T)$ is a free module over $R(G)$ by computing an explicit basis for all the A,B,C,D cases. The canoical reference is a paper by Pittie( H.V. Pittie: Homogeneous vector bundles on homogeneous spaces, Topology II (1972) 199-203), but I could not find it online or in any books available in the library. Steinberg generalized Pittie's statement in his paper (Robert Steinberg, On a theorem by Pittie, Topology Vol. 14. pp. 173-177. Pergamon Press, 1975, Printed in Great Britain. Received 1 October 1974).

Since they already proved this in the past, I would like to see their papers before I finish my project, even if at some monetary cost. But I could not access either of them. Not knowing their work would not hinder my research, for I work in a much more elementary level than they did, but I think their work might be related to my eventual results and I should acknowledge in case they proved some formula I proved again on my own. So I want to ask where I can find them in paper or electronically. I can read parts of Steinberg's paper via google books, but I would like a pdf file or something (so I may check).

ADDED:

With advisor's help and the links provided by all the people below, I retrieved the two papers.

ADDED:

Received Steinberg's replying email. He notes "A correction should be made on p.175, line 6 ( which starts with "Consider now ") by putting the exponent "n sub a" on the item over which the product is being taken.The paper by Pittie appears in Topology, vol. 11, 1972, pp. 199-203, and, if I remember correctly, does not contain an explicit basis for the quotient. " This is important so I put in here.

Answer by Changwei Zhou for Textbook recommendations for undergraduate proof-writing class

2011-04-22T17:00:31Z

http://www.amazon.com/Proofs-Fundamentals-Course-Abstract-Mathematics/dp/0817641114

This is written by my professor Ethan Bloch. It is slightly overpriced, though.

homotopy groups for good rings

2011-03-08T02:47:45Z

I think this question should already be abound in literature but the only place I find is from this article:

http://math.uchicago.edu/~lxiao/files/notes/Fundamental%20Groups.pdf

which seems to be elaborating this definition:

http://en.wikipedia.org/wiki/%C3%89tale_fundamental_group

but unfortunately as I do not understand much algebraic geometry I do not how to make use of this definition.

I am thinking about extending classical Bott periodicity to arbitrary rings that is good enough (UFD, for example). By extending I mean that I want to measure infinite matrices of entires in a ring $R$ with determinant 1 by the "one point compactification" of $R^{n}$ via introducing some topology. Hence in the classical case we can measure $U$ by $S^{n}$. I want to ask:

1): Is this possible? ( I thought about it over a bus trip but do not know how to establish universal bundles if the base ring is discrete, so I am stuck in here).

2): Is there any previous such constructions? What are their properties?

I feel there must be something well-known because Bott-periodicity theorem is a very old theorem. I do not know whether this is more appropriate for MO or for stack exchange, but I decided to put in here.

Is this set equidistributed?

2011-03-19T09:47:46Z

For $n\in \mathbb{N}$ numbers $I_{n}=(1,2,3..n)$ and prime $p$, we define operation $(1,2,3..n)$ to $A=(a_{1},a_{2}...a_{p-1})$ as follows:

We arrange the $n$ numbers in a circle, then we eliminate the first number, the $p$th number, the 2$p$th number, etc, until there is only $p-1$ numbers left and the process terminated. We identify this subset as $A$.

My question is, for given $p$, does $A$ being equidistributed in $I_{n}$ with $n\rightarrow \infty$? I feel that "equidistributed" in arbitrarily set seems to be not well defined. In this one I want at least for a subset of $I_{n}$ of the form $S=(s,s+1...s+t-1)$. $|S\cap A|\rightarrow \frac{t}{n}*(p-1)$ with $n\rightarrow \infty$. I do not know whether this is possible. A few simple cases (like $p$=3, $n$=2011) is already in need of programming and the result seemed to be very random, I feel "intuitively" this should be true, but I do not know how to prove it.

There is some confusion which is obvious from the comment. I mean a circular process that eliminate a certain number, jump $p-1$ numbers in between, and then terminate the next number. This process will stop at the place there are $p-1$ numbers left.

An example: $n=20$, $p=5$, we have $(1,2,3,4,6,7,8,9,11,12,13,14,16,17,18,19)$ in the first elimination process. Then we have $(1,2,3,4,7,8,9,11,13,14,16,17,19)$ in the second round elimination process, and $(1,2,3,7,11,14,16,17,19)$ in the third round, and $(2,3,7,11,16,17,19)$ in the fourth round, finally yielding $(2,7,11,17)$ in the end.

Answer by Changwei Zhou for Which topics/problems could you show to a bright first year mathematics student?

2011-03-13T02:41:23Z

I will write from the perspective of an undergraduate student. I think one reason undergraduate math is not interesting is the technical jargon laid before undergraduate students in the learning process. Usually one had to understand why one has to comes up with such a particular method, a concept, or a theory to understand the subject. The utility of any math they learned usually can only be manifested by sample calculations or problems.

But there is something much greater than this. For a really bright first year math student, I would assume normal course material is insufficient, and textbook+lecture+office hour do not solve his or her problems. Nevertheless the technical tool available to him or her does not enable him or her to do real research. I remember when I was a high school student I asked a math graduate student if there are two different holomorphic structures on a four dimensional manifold. At that time I know some complex analysis and some differential manifolds, but they are not sufficient to give an answer to this. Even now I do not know a yes or no answer in a form I can understand. A really bright first year math student is very likely to ask strange questions he or she cannot solve, and it stuck him or her so much that he or she decided to quit math. Equally likely is he or she found Putnam problems too technical, problem solving too time consuming, and in the end learning math becomes mastering a machine. I still remember the days I was stucking with my problems in reading Hatcher, at that time I was serious considering of transfering to history. And I am hardly a bright math student.

The solution to such a problem should be to guide him or her to read some master's work. If he or she can understand how a working mathematican did to tackle a problem, developing a math concept, or expanding a theory, it would be very beneficial because it helps the student to achieve a higher level of mathematical maturity unavailable by normal curriculum. Abel once replied "I read the masters, not their pupils". Clearly a lot of past mathematical work had been discarded or being forgotten, yet the professor can always choose something relatively more readable. If this is not possible, the professor should at least devise an individualized class with good supplementary reading.

What I never understand is why the professor will adopt a useless undergraduate level textbook in his or her class instead of reading good math articles, or even lecture notes. I would hope the professor can pointed out where the material is insufficient, where the results had been updated, and where better strategies had been developed. A good list of sample problems coupled with good mathematical reading can change people. Sadly I did not had a chance to have such a class in US. As a result I had to go to Moscow, where I rediscovered math. But I hope future students would have a better fate.

Answer by Changwei Zhou for Which mathematicians have influenced you the most?

2010-04-07T01:24:39Z

Bernhard Riemann.

The idea of the Riemann Surface and manifolds stroke me when I was a high school student.

Deeper meanings of barycentric subdivision

2010-07-08T12:33:19Z

I just want to ask if there is any deeper motivation or clear geometric "sense" behind the barycentric subdivision. Some friend asked me about this a few months ago, looking back the section at Hatcher, I still feel quite confused. I remember one friend told me combinatorically one can do this from posets back to posets, but this does not give me any way to "understand" it properly. In some books (Bredon, for example), the author use excision property as one of the axioms, I'm wondering "where they came from, why they make any sense?".

Answer by Changwei Zhou for Examples of "folk theorems"

2010-08-08T07:54:43Z

In Fudenberg's book Game Theory, the following was listed as a folk theorem:

The folk theorem for repeat games assert that if players are sufficiently patient then any feasible, individual rational payoffs can be enforced by an equilibrium. Thus, in the limit of extreme patience, repeated play allows any payoff to be an equilibrium outcome.

Answer by Changwei Zhou for Distribution of fractional parts of n^{3/2}

2010-08-02T01:31:45Z

I think you can try Weyl's criterion on this.

Answer by Changwei Zhou for Undergraduate Derivation of Fundamental Solution to Heat Equation

2010-07-28T05:17:50Z

I think you can find a solution of this from Elias Stein's book Fourier Analysis. Sorry I don't have the file available. But using fourier analysis to solve this may be like using a hammer to kill a fly.

Answer by Changwei Zhou for Jordan Curve Homotopy

2010-07-21T03:43:43Z

I am not sure if this is right.

Since both curves are Jordan curves, you can "enlarge" $C_{1}$ by affine transformations such that it has no intersection with $C_{2}$ since the diameter of $C_{i}$ is bounded(since $S_{1}$ is compact, its image is also compact, therefore closed and bounded).Then you just need some sort of radial homotopy $C_{rt}=rC_{1}(t)+(1-r)C_{2}(t)$ with $r\in [0,1]$. This will be the homotopy from $C_{1}$ to $C_{2}$ you need.

How to solve Diophantine equations in $F_{p}$?

2010-04-06T21:24:25Z

For example, how to solve the equation $\sum^{p-1}_{i}x_{i}^{2}=0$ in $F_{p}$? This is not a homework problem. I think it should have a definite answer, so not an open problem. I just don't know how to solve it.

Answer by Changwei Zhou for Careers advice for Ph.D.s without current postdocs or university jobs

2010-07-20T11:27:11Z

I think someone in Dr. H's situation should try to find a position in a liberal arts college. There are several reasons he should consider this.

First and foremost, a teaching position in a liberal arts college involves generally more time and energy than a teaching position in a large university. Myself went to a liberal arts college, and from my experience most of the professors in my math department have to deal with all kinds of wierd students and all types of daily matters involving computer fixing, room cleaning, and driving cars. With such a working load, it can be assumed that Dr. H would have better teaching ability after the two years. As far as I know, liberal arts colleges typically has some limits in the minimal teaching time one has to be a school, like three days a week for example. That will be more demanding than university teaching jobs.

Second, probably unlike university, in liberal arts college it is quite common that one has to direct student's researchs in the summer or winter holidays. I don't know whether the faculities get paid from this, but generally one has the chance to spent the holidays off with a group of students on a project on selected. The drawback is undergrads typically don't understand the material well, however this may be a good place to try various toy problems.

Third, a liberal arts college can be a good place to solve the two body problem in case you do have one. In my college, it is really common to see a professor and his or her's couple become companions in the department.

Answer by Changwei Zhou for Modern algebraic geometry vs. classical algebraic geometry

2010-07-13T10:36:10Z

I think one should work on modern ones. You can always pick up classical ones if you become suddenly interested in the realization of the abstract ideas. For books of course you need to read Hartshrone. I would say Eisenbud is not appropriate if you don't have other books to read. Anyway enjoy the books you read, and think hard before throwing away the books.

Is this a covering space?

2010-06-23T10:27:27Z

In Hatcher's book Page 79 I was asked to provide two-sheeted covering space $Y \rightarrow X_{1}$ such that the composition $Y \rightarrow X_{1} \rightarrow X$ of the two covering spaces is not a covering space. In here the space $X$ is the shrinking wedge of circles, and $X_{1}$ is placing infinite such spaces onto the line.

(See the figure in the book) http://www.math.cornell.edu/~hatcher/AT/ATpage.html

The example I imagined is this one:

I use $Y$ the same as $X_{1}$, but I make the mapping $Y\rightarrow X_{1}$ like this: I map and the second circle to the first circle, and map the rest to themselves. Then if $Y \rightarrow X_{1} \rightarrow X$ is a covering map, according to the defintion the inverse of a neighborhood in $X$, they must be disjoint; but in here they simply coincide.

I don't know whether this really works as he required. Mostly because the original space is sufficiently bad (not locally path connected) therefore I expect Hatcher would need me to utilize this condition. Also, I want to ask if one can assert that if $X$ is locally path connected, then $Y$ must be a covering space of X. I'm thinking about this because in the next page problem 16, Hatcher asked the reader to prove this:

"16. Give maps $X\rightarrow Y\rightarrow Z$ such that both $Y\rightarrow Z$ and the composition $X\rightarrow Z$ are covering spaces, show that $X\rightarrow Y$ is a covering space if $Z$ is locally path-connected...."

Sorry that this is a purely homework level question.

Answer by Changwei Zhou for What should be offered in undergraduate mathematics that's currently not (or isn't usually)?

2010-06-16T04:05:23Z

As a college student myself, I wish to study these classes when I was in my college, but they are not offered(I took most of these in Moscow instead):

Algebraic topology.
Real analysis (graduate level)
Complex analysis (graduate level)
Measure theory, geometric measure theory.
Commutative algebra and homological algebra (at least Ext, Tor, etc)
Riemann Surfaces
An intro course in algebraic geometry
Algebraic number theory.
Classical Mathematical Physics
Some intro course in ODE, dynamical systems (like Smale's horseshoe), and PDE.
Combinatorical game theory.
Elliptic curves.

Answer by Changwei Zhou for When is (n choose k) < (n+j choose k-1) for fixed j?

2010-05-19T03:38:20Z

I think some computation can be done like this(I guess that's the tedious math you mentioned):

Mark $C^{n}_{k}=P$;

and $C^{n+j}_{k-1}=Q$, we have $\frac{P}{Q}$ equal

$$\frac{(n!)}{(n-k)!k!}*\frac{(n+j-k+1)!(k-1)!}{(n+j)!}$$

Mark $n-k=S$, we have the above to be

$$\frac{(n!)}{S!k}*\frac{(S+j+1)!}{(n+j)!}=k^{-1}\frac{n!}{(n+j)!}\frac{(S+j+1)!}{S!}=k^{-1}\frac{\frac{n!}{(n+j)!}}{\frac{(S)!}{(S+j+1)!}}$$

Therefore we only need to consider the function $$A(m,j)=\frac{(m+j)!}{m!}$$ For changing $j$ we have $A(m,j+1)=A(m,j)*(m+j+1)$, for changing $m$ we have $$\frac{A(m,j)}{A(m+1,j)}=\frac{m+1}{m+j+1}<1$$

Therefore $A(m,j)$ increased by changing $m$ as well. Therefore we have the equation:

$$\frac{P}{Q}=\frac{S+j+1}{k}*\frac{A(n,j)}{A(s,j)}=(\frac{n+j+1}{k}-1)\frac{A(n,j)}{A(s,j)}$$

We consider the situation when we change $j$ or $k$. $P_{j},P_{k}$ means $P$'s value when we concern about $j$ or $k$. If we change $j$, we have $$\frac{P_{j}}{Q_{j}}/\frac{P_{j+1}}{Q_{j+1}}=\frac{1}{(S+j+2)}<1$$, moreover it has no lowerbound other than $0$ with $j$ increased. Therefore $\frac{P}{Q}$ is increasing with $j$ increased irrespective of $k$. Hence $P< Q$ will hold eventually when increasing $j$.

We consider the situation when we change $k$. In this situation $$j>\frac{P_{k}}{Q_{k}}/\frac{P_{k+1}}{Q_{k+1}}=\frac{(S+j+1)(k+1)}{(S+j)k}*\frac{S+j+1}{S+1}>1$$, therefore $\frac{P}{Q}$ decreases with one increases $k$ while fixing $j$.

The points $(j,k)$ such that $P=Q$ are satisfies this equation:

$$\frac{(n+j-k+1)(n+j-k)...(n-k+1)}{(n+j)(n+j-1)...(n+1)*n)}=\frac{k}{n}$$.This cannot hold if $k$ is a prime. I don't know how to push further at this point.

Answer by Changwei Zhou for Geometrical meaning of Grassmann Algebra

2010-04-22T20:57:59Z

I think a good introductory book is Federer's book "Geometric measure theory", I remember the first chapter is on Grassmann algebra. Another more accessible book is "The Road to Reality" by Roger Penrose, you can check the chapter on Grassmann algebra and Clifford algebra.

Answer by Changwei Zhou for Examples of common false beliefs in mathematics.

2010-05-06T00:10:37Z

My example is $G_{1}$ isomorphic to $G_{2}$'s subgroup and $G_{2}$ isomorphic to $G_{1}$'s subgroup implies $G_{1}$ and $G_{2}$ are isomorphic...

Comment by Changwei Zhou

2013-01-23T22:35:34Z

@Pete: This does not make sense; someone more experienced is not entitled to provided more accurate information than someone inexperienced but can give the right advice. I consider the fact I was an undergraduate in a liberal arts colleges to be an advantage, not the reverse. It is two years now, and I still think my advice makes a lot of sense to someone in Dr H's position.

Comment by Changwei Zhou

2012-08-01T19:56:03Z

No need - I asked a HW question from Hatcher two years ago at here as well.

Comment by Changwei Zhou

2012-08-01T19:24:22Z

nothing personal but your question is more appropriate in mathstackexchange, this is a homework problem and not suitable at here.

Comment by Changwei Zhou

2012-07-30T06:05:00Z

@Pete: Do you mean the lower derivatives may be wild? Do you have a good counter-example?

Comment by Changwei Zhou

2012-07-22T07:19:49Z

I am really embarrassed, my memory is not accurate; the papers asked whether $\pi_{1}G$ is free instead. Sorry for raisng the wrong question at here.

Comment by Changwei Zhou

2012-07-22T02:34:50Z

Thanks. I feel surprised.

Comment by Changwei Zhou

2012-07-22T02:27:13Z

You mean simple ones, or any? I did read Bott's collective papers but never seen this theorem. I shall check the link.

Comment by Changwei Zhou

2012-07-22T02:21:42Z

I see the problem, corrected.

Comment by Changwei Zhou

2012-07-20T02:32:01Z

would it be more appropriate to put this into a comment?

Comment by Changwei Zhou

2012-07-20T02:27:50Z

Thanks David! Very nice response.

Comment by Changwei Zhou

2012-06-15T23:00:29Z

It is in page 36.

Comment by Changwei Zhou

2012-06-10T17:18:52Z

Yes. This is great.

Comment by Changwei Zhou

2012-05-16T02:24:59Z

In the view of the downvote I should justify myself that this article gives a proof of RR via Atiyah-Singer index theorem, which is why I cannot say much in the answer I wrote.

Comment by Changwei Zhou

2012-05-14T07:21:59Z

I was confused as well. Maybe my "breakthrough" is too trivial to be put in here...

Comment by Changwei Zhou

2012-04-15T01:02:21Z

@Gjergji Zaimi: Could you answer me? (or should I email you?)