User steve - MathOverflow

Weierstrass factorization with $L^2$ estimates?

2013-04-17T15:58:10Z

Let $\Omega$ be a bounded domain in $\mathbb{C}$. Let $X$ be a discrete set of points whose boundary is in the boundary of $\Omega$. Can I find an $L^2$ holomorphic function which vanishes on $X$? Can I solve the problem in weighted $L^2$ spaces?

If there are counterexamples, are precise conditions on the set $X$ known to ensure the existence of an $L^2$ solution?

I have been learning about Hormander's approach to the $\bar{\partial}$-problem, and this seems like a natural question to ask from that perspective, but I have not been able to find any work done on this.

Answer by Steve for What does a mathematician expect from mathematics education?

2013-04-19T20:26:57Z

To answer your question truthfully, I do not think most mathematicians expect anything from mathematics education as a discipline. It is not even on the radar.

The vast majority of mathematicians have never had any formal training as teachers at all. When they get jobs as educators, there is no kind of systematic support for teaching. There might be services available if you seek them out, but basically the good teachers are all ready good, and the bad teachers generally do not care enough to seek out help.

I have no idea how to change this, or even whether changing this would be a good thing. In my experience the vast majority of mathematics education research is essentially at the level of anecdotes. I think that meaningful education research is only just now starting to be possible, with the kind of huge data sets we can now gather on students taking courses on computers.

If you personally want to improve mathematics education at your university, then probably you should try to make friends with as many mathematicians as possible, and see if you could personally work with them to change their teaching habits. Try to make it fun.

Answer by Steve for functions of one complex variable: geometric theory

2013-04-03T18:31:34Z

A lot of this stuff is in Rick Miranda's book.

Smooth analgoue of Dehn Invariants?

2013-03-21T18:11:07Z

Has anyone thought about approximating smoothly embedded balls in $\mathbb{R}^3$ by polyhedra, and taking the limit of Dehn invariants to obtain a "smooth Dehn invariant"? I am guessing that you would have to be careful about allowable approximations to get a well defined invariant.

Just curious if there have been any developments along these lines.

Answer by Steve for Why is the gradient normal?

2013-03-20T16:51:13Z

This is essentially Andrew Stacey's answer, but a bit lower level. This is the story I actually try to get my calculus 3 students to understand.

Let $F: \mathbb{R}^2 \to \mathbb{R}$. Then the derivative $D_{F,p}$ is a linear map from $D_{F,p}:\mathbb{R}^2 \to \mathbb{R}$, whose matrix with respect to the standard basis is $[ \dfrac{\partial F}{\partial x} \dfrac{\partial F}{\partial y}]$.

This is the unique linear map which satisfies $F(p+h) = F(p)+D_{F,p}(h)+Error(h)$, where $\displaystyle\lim_{h \to 0} \dfrac{|Error(h)|}{|h|} = 0$. Notice that $p$ and $h$ are both vectors in $\mathbb{R}^2$.

The cool thing is about linear maps from $\mathbb{R}^n \to \mathbb{R}$ is that they look like dot products! In this case $D_{F,p}(\langle a,b \rangle) = [ \dfrac{\partial F}{\partial x} \dfrac{\partial F}{\partial y}] \begin{bmatrix}a \\ b\end{bmatrix} = \dfrac{\partial F}{\partial x}a + \dfrac{\partial F}{\partial y}b = \langle \dfrac{\partial F}{\partial x} \dfrac{\partial F}{\partial y} \rangle \cdot \langle a, b\rangle$. This alternative viewpoint on the derivative is useful, because it gives a different geometric interpretation of the derivative. We call $\langle \dfrac{\partial F}{\partial x} \dfrac{\partial F}{\partial y} \rangle$ the gradient of $F$.

Now we are interested the curve $F(x,y) = 0$. Given a point $p=(x_1,y_1)$ on this curve, the tangent direction will be the vector $h$ for which $D_{F,p}(h) = 0$, because to stay on the curve, the value of the function should not change to first order. Using the geometric interpretation in terms of dot products, we can see that $\langle \dfrac{\partial F}{\partial x} \dfrac{\partial F}{\partial y} \rangle \cdot \langle h_1, h_2\rangle = 0$, or geometrically that the gradient is perpendicular to the tangent direction!

Answer by Steve for Math Annotate Platform?

2013-02-18T02:47:55Z

Git is a version control system which is commonly used by programmers. While it is useful for managing ones own individual projects, it really starts to shine in collaborative situations. While it was designed to help programmers, it is really a great system for anyone trying writing any text based project. This includes math papers.

The advantages are numerous:

The entire history of the document is stored, so that information is never lost by writing over a document.

The contribution history is clearly documented, so you can see who is responsible for different parts of the work. This very nicely solves attribution issues.

The workflow for working together collaboratively is wonderful. If you and another person are working on different parts of the same document, or making changes to different files altogether, git will merge your work without any effort on your part. If there is a conflict between the work done, git alerts you to this, and allows you to resolve these conflicts.

It takes a little getting used to, but I now find it so essential to how I work that I use it for everything. I wouldn't dream of writing a paper without using git to help me manage versions, especially if I am working on it with another person.

Github provides a free method for people to host their papers, and allow others to collaboratively edit them, comment on them, make "bug reports", and so on, which I feel would be of great value to the mathematical community.

I think that Git really does solve all of the problems raised by the OP.

Answer by Steve for Upper bound for the order of the group of automorphisms of Riemann surfaces of genus 2

2013-02-17T23:23:27Z

http://amathew.wordpress.com/2011/11/10/automorphisms-of-compact-riemann-surfaces/#more-2954 should be of interest to you.

Fundamental motivation for several complex variables

2012-11-27T04:12:36Z

I have 3 general abstract reasons to care about complex analysis in a single variable:

The laplacian is, up to a constant multiple, the only isometry invariant PDO in the plane, and so it is abstractly very important. Holomorphic functions are intimately related to harmonic functions, so holomorphic functions are important.
Holomorphic mappings are conformal if they have nonvanishing derivatives
Many results which are real variable in nature are most easily understood in the light of complex analysis (factorization of real polynomials, radius of convergence of real power series, etc)

I currently have no such justification for several complex variables. The connection to harmonic functions mostly breaks down. Conformality breaks down. I have not seen applications to real variable phenomena.

Can anyone give me some insight into the big picture here? Where do the phenomena studied in several complex variables (domains of holomorphy and their ilk) come up in the rest of mathematics?

Answer by Steve for A question that arises in trying to make mathematically precise a well known informal statement about analytic functions

2012-11-09T20:37:05Z

If I am not misunderstanding your question, you read off the power series pretty much directly from the given data. You know f(0). You also know f'(0) by using the definition of the derivative. The higher derivatives can all be determined by using higher order difference equations http://en.wikipedia.org/wiki/Finite_difference#Higher-order_differences. Since the function is analytic the taylor series you get this way does converge on some disk...

Comment by Steve

2013-05-10T21:49:26Z

oops. I was thinking geometric surface, but algebraic curve.

Comment by Steve

2013-05-10T17:45:50Z

I am not an expert on this, but at least over C, cant you always embed into P^3? Just by taking lines through a point which is not on any chord or tangent reduces dimension by by one. Dimension the variety of chords and tangents shows you can keep finding such a point until you get down to P^3.

Comment by Steve

2013-05-03T13:47:13Z

The update is the real answer to the question. I do not really care about a proof if it does not enhance my understanding.

Comment by Steve

2013-04-20T18:45:36Z

@Scott: great link.

Comment by Steve

2013-04-20T18:39:31Z

comes down to too many intangibles. It is a real relationship with real people, and I think that those are too varied and complex to have anything like universal solutions. I have participated in wildly successful classes which had charismatic leaders who have completely opposite perspectives on how to teach.

Comment by Steve

2013-04-20T18:36:55Z

@Scott: I have high hopes for big data in education. The computer can get a huge amount more information about a student than I ever could in an even moderately sized classroom. Being able to look at the whole pattern of a students history, pinpoint what the difficulties likely are, and then address those will revolutionize everything in my opinion. Only time will tell I guess. Things like "classroom best practices" are important, and I have a lot of personal opinions about these. These opinions are subject to revision based on what I see others doing. But fundamentally, I think it

Comment by Steve

2013-04-19T22:06:14Z

I can say that I personally know some great math ed people, who are just fantastic teachers. I also think that the teacher education courses at my university really are some of the best math classes being offered. I also think Sybilla Beckman's contributions have been outstanding. But I also think that all of these are opinions.

Comment by Steve

2013-04-19T21:16:56Z

@Benjamin: I will read some of Kilpatrick's work. Thanks for recommending it. Is it okay for me to email you sometime to chat about it, whenever I get around to reading it? I guess this is really the heart of my troubles: Mathematics education research can only address implications. If I do this, then this is likely to be the outcome. That is a matter of science. It cannot decide which outcomes are the desirable ones! That is a political question, or a moral question. It seems very often that mathematics educators are trying to address the moral question. Do you agree?

Comment by Steve

2013-04-19T20:59:34Z

For example, think about teaching double digit arithmetic. There are a lot of mathematics educators who will tell you that teaching the algorithm is essentially killing any chance that the student will understand it. There are others who say that you have to learn the mechanics first, and then you have the structure to start trying to understand what is going on. Put these two people in a room, give them access to the whole internet worth of studies, and there is basically no chance that either person will budge an inch.

Comment by Steve

2013-04-19T20:56:54Z

I think one reason that I got turned off to the field is that is not really being done as a science in my opinion. Everyone involved cares deeply about education (a good thing!), but there is so much opinion. I mean, you might say "Even though students using such and such approach score lower on these tests, they are actually thinking more as evidenced by such and such". In the end, everything is subjective enough that people basically stand by what they think is the best way to teach.

Comment by Steve

2013-04-19T20:53:21Z

@Benjamin - I have read a lot of papers in mathematics education. I thought about getting a Ph.D. in mathematics education at one time. It is not an objective science, and there are huge culture wars in the field. For example, between constructivists and people who encourage rote memorization. Read en.wikipedia.org/wiki/… for example, including the criticism section.

Comment by Steve

2013-04-19T20:47:41Z

Also I would like to comment that the ultimate solution is probably economic - find a way to fund mathematicians to do research without having to teach. This way the people who want to teach can be hired on the merits of their teaching, and the people who want to do research can be hired on the basis of their research. Until this changes, I really do not see hope - there are too many researchers who are only teaching because that is how you get access to a university job, not because they want to be teachers. You will never help those.

Comment by Steve

2013-04-19T20:44:11Z

Was this downvoted for being rude? I sometimes have difficulty understanding rudeness. I did not intend any ill will with my answer, only to give my real perspective on this issue.

Comment by Steve

2013-04-17T22:41:34Z

Ah, I had not thought to use Jensen's formula. I will see if I can cook up a concrete counterexample. I was thinking of L^2 wrt lebesgue measure. I will wait a day or so to accept your answer. Thanks.

Comment by Steve

2013-04-16T21:11:21Z

This is basically how I think about the fundamental theorem...