User anthony pulido - MathOverflow

Request for errata for Automorphic Forms on GL(2)

2013-02-15T22:35:06Z

Hello everyone,

I'm currently proofreading Hervé Jacquet's and Robert Langlands's Automorphic Forms on GL(2) for future republication. I was wondering if some of you had noticed mistakes in it that we might correct.

Mathematical errors, misspellings, typos---any mistake, no matter how small, is welcome!

Answer by Anthony Pulido for What would you want to see at the Museum of Mathematics?

2010-12-27T07:47:36Z

How about leading them through an interesting problem, like a geometry IMO problem or, if that is asking too much, a Mathcounts problem? It could be set up on square tiles, the left most of which would contain the problem, with the following tiles showing the steps of the solution. It should be a problem that can be written such that viewers see a surprise toward the end, thereby possibly giving a glimpse into why mathematicians enjoy so much what they do. Although a Mathcounts problem would no doubt be accessible, a very beautiful IMO problem could be inspiring. Very likely, it would be entertaining for both children and their parents.

One might also include multiple solutions to a problem to dispel the notion that for each problem only one solution exists.

One can see examples of interesting presentations and ideas for problems at Rusczyk's Mathcounts channel at

http://www.youtube.com/user/mathcountsfdn

Using the same format from above, one could present a suitable Putnam problem and show its connection to research. This is discussed in Kedlaya, Poonen, and Vakil's book.

Finally, this response might be related to Kevin Lin's. and ein's.

Answer by Anthony Pulido for Most harmful heuristic?

2010-10-03T01:11:01Z

From Keith Devlin's article

http://www.maa.org/devlin/devlin_06_08.html

"Multiplication is repeated addition."

This is true when multiplying natural numbers, but is a special case of a scaling operation in the reals. We know it is also a rotation in the complexes, but that should probably be left out at the beginning, although it might interesting to think about how one would include them at the beginning.

Devlin also mentions "exponentiation is repeated multiplication."

Answer by Anthony Pulido for Thinking and Explaining

2010-09-14T03:49:50Z

Professor Thurston,

In my own study, I've struggled with giving meaning to mathematical objects for many years. It's likely I've learned more slowly than most, because I find it difficult to move on unless I've found something I can rely on. Usually, when I try to explain mathematics, I will give the person I'm speaking to a simple problem to give him or her the experience of doing mathematics. Almost always, I will ask to show that $\pi > 3.$ The experience is usually familiar. One person has asked me, "how is this different from finding the answer to any question?"

Another example I like to give people is one of the several graphical proofs of the Pythagorean theorem. The one I present is #9 from this site, since it seems to be the simplest:

http://www.cut-the-knot.org/pythagoras/index.shtml

I've always had trouble explaining things like group without giving simple geometric examples, like the dihedral groups. That the turns of the Rubik's cube is offered as an example of a group is well known. Conjugation, for instance, immediately makes sense. Elohemahab Solomon offered this video

http://mathoverflow.net/questions/24794/does-a-connections-blog-podcast-exist-for-math/27919#27919

where Mr. du Sautoy, (at about 12:00) in the context of groups of symmetry, compares the founding of the concept of group with that of number. To summarize Mr. du Sautoy, the symmetry a group measures is analogous to the quantity number measures. For example, we have chairs and tables, which are different, but if we have 3 of each, the quantity is the same. If we have two walls of the Alhambra with different looking designs, but have the same symmetry, then one group acts on the designs. He discusses $S_3$ and $\mathbb Z_6.$

I accepted complex numbers for a long time, but after experience with math past calculus, and the requirement for rigor slowly pervaded my thinking, I couldn't accept them as more than ad hoc constructions. Finally, I was satisfied with their construction as elements of the field $\mathbb R[x]/(x^2+1).$ I talked about this in my answer to the question:

http://mathoverflow.net/questions/30156/demystifying-complex-numbers/37817#37817

I now think of complex numbers as an abstract construction from things I have intuition about, that is, real numbers and polynomials. The usual interpretations, as vectors in the complex plane, vectors in $\mathbb R^2$ that one can multiply, are still of course, extremely useful in understanding them.

It's likely you know this already: it looks like Hilbert coined the term "ring" from Mr. Gleason's interpretation of cyclic groups.

http://en.wikipedia.org/wiki/Ring_(mathematics)#History

A distribution is something, with whose definition I've worked successfully, but still have little sense of. I know they are generalized functions, but my lack of understanding might just be lack of experience.

Answer by Anthony Pulido for Demonstrating that rigour is important

2010-09-06T06:49:42Z

The fundamental lemma is an example that most believed and on whose truth several results depend. According to Wikipedia, Professor Langlands has said

... it is not the fundamental lemma as such that is critical for the analytic theory of automorphic forms and for the arithmetic of Shimura varieties; it is the stabilized (or stable) trace formula, the reduction of the trace formula itself to the stable trace formula for a group and its endoscopic groups, and the stabilization of the Grothendieck–Lefschetz formula. None of these are possible without the fundamental lemma and its absence rendered progress almost impossible for more than twenty years.

and Michael Harris has also commented that it was a "bottleneck limiting progress on a host of arithmetic questions."

Answer by Anthony Pulido for Demystifying complex numbers

2010-09-05T19:38:57Z

This answer doesn't show how the complex numbers are useful, but I think it might demystify them for students. Most are probably already familiar with its content, but it might be useful to state it again. Since the question was asked two months ago and Professor Zudilin started teaching a month ago, it's likely this answer is also too late.

If they have already taken a class in abstract algebra, one can remind them of the basic theory of field extensions with emphasis on the example of $\mathbb C \cong \mathbb R[x]/(x^2+1).$

It seems that most introductions give complex numbers as a way of writing non-real roots of polynomials and go on to show that if multiplication and addition are defined a certain way, then we can work with them, that this is consistent with handling them like vectors in the plane, and that they are extremely useful in solving problems in various settings. This certainly clarifies how to use them and demonstrates how useful they are, but it still doesn't demystify them. A complex number still seems like a magical, ad hoc construction that we accept because it works. If I remember correctly, and has probably already been discussed, this is why they were called imaginary numbers.

If introduced after one has some experience with abstract algebra as a field extension, one can see clearly that the complex numbers are not a contrivance that might eventually lead to trouble. Beginning students might be thinking this and consequently, resist them, or require them to have faith in them or their teachers, which might already be the case. Rather, one can see that they are the result of a natural operation. That is, taking the quotient of a polynomial ring over a field and an ideal generated by an irreducible polynomial, whose roots we are searching for.

Multiplication, addition, and its 2-dimensional vector space structure over the reals are then consequences of the quotient construction $\mathbb R[x]/(x^2+1).$ The root $\theta,$ which we can then relabel to $i,$ is also automatically consistent with familiar operations with polynomials, which are not ad hoc or magical. The students should also be able to see that the field extension $\mathbb C = \mathbb R(i)$ is only one example, although a special and important one, of many possible quotients of polynomial rings and maximal ideals, which should dispel ideas of absolute uniqueness and put it in an accessible context. Finally, if they think that complex numbers are imaginary, that should be corrected when they understand that they are one example of things naturally constructed from other things they are already familiar with and accept.

Reference: Dummit & Foote: Abstract Algebra, 13.1

Answer by Anthony Pulido for Good books on theory of distributions

2010-08-27T16:43:08Z

I agree with Johannes's comment, but despite this, one book that might fit your criteria is Theory of distributions by M.A. Al-Gwaiz. I haven't looked at it for some months, but it made the following standard texts more accessible:

Friedlander and M. Joshi's Introduction to the Theory of Distributions.
Hörmander's The Analysis of Linear Partial Differential Operators.

A book that I haven't looked at thoroughly, but you might find interesting, is Guide to Distribution theory and Fourier transforms by Robert S. Strichartz. I once took a class with the author, whose verbal explanatory style is complete and who is also a clear writer.

Answer by Anthony Pulido for Value of "of course" in the mathematical literature

2010-08-24T02:39:16Z

Although this is my own experience, it might give an answer to your first question "if something is obvious, then it is obvious that it is obvious (so why include it at all?)." There are times when I will write, "clearly," or "of course," when I feel leaving them out might insult the reader's intelligence. If we imagine your example

If d divides a and d divides b, then of course d also divides a+b,

(probably without the italics) in an advanced context, that is, as a prelude to invoking an important theorem or stating something not so obvious that depends on that simple fact, then I might look at it that way.

Answer by Anthony Pulido for What would be your suggestion of textbooks in Lie groups and Galois theory?

2010-08-17T12:02:22Z

Ian Stewart's Galois Theory is a nice introductory text to Galois theory. Recently, however, I've been doing exercises from chapters 13 and 14 from Abstract Algebra by Dummit and Foote. This might be a faster introduction. Joseph Rotman's short book Galois Theory is also introductory, but fast and very readable.

Community experiences writing Lamport's structured proofs

2010-08-16T05:30:53Z

About two years ago, I came across this paper by Lamport

http://research.microsoft.com/en-us/um/people/lamport/pubs/lamport-how-to-write.pdf

on writing proofs hierarchically. It changed how I wrote all of my proofs for about six months and identified the gaps in my understanding and knowledge extremely well. These days, I won't use it for simpler proofs, but I find it indispensable when I want to thoroughly understand long and complex ones.

I think this is potentially a wonderful pedagogical tool, in that all steps and assumptions are organized and easily referred to, and is also useful for self-checking. Some discussion on the blog evaluating Deolalikar's claimed proof of $P\neq NP$, to give one example, Professor Tao's apposite remarks on the need for precision, (August 15, 2010, 3:05 PM - I hope he doesn't mind my quoting him)

One thing this illustrates is the importance of setting out precise definitions. I feel that if Deolalikar had written down a precise definition of what it meant for a solution space to be polylog parameterisable, the difficulties would have been found a lot sooner, and in particular probably by Deolalikar himself, well before he finished the preprint to share with others.

reminded me of Lamport's essay. Lamport comments on something similar from his own experience:

The style was first applied to proofs of ordinary theorems in a paper I wrote with Martín Abadi. He had already written conventional proofs—proofs that were good enough to convince us and, presumably, the referees. Rewriting the proofs in a structured style, we discovered that almost every one had serious mistakes, though the theorems were correct. Any hope that incorrect proofs might not lead to incorrect theorems was destroyed in our next collaboration. Time and again, we would make a conjecture and write a proof sketch on the blackboard—a sketch that could easily have been turned into a convincing conventional proof—only to discover, by trying to write a structured proof, that the conjecture was false. Since then, I have never believed a result without a careful, structured proof. My skepticism has helped avoid numerous errors.

Has anyone had experience with this style of writing proofs?

History of the Normal Basis Theorem

2010-08-14T13:31:54Z

The Normal Basis Theorem: If $E/F$ is a finite Galois extension, then there exists $a \in E$ such that the orbit of $a$ under the action of $\mathrm{Gal}(E/F)$ is a basis for $E$ as a vector space over $F.$

Who discovered this?

I've looked through the collected works of Frobenius and Dedekind, which are the earliest works I've seen referring to it, but it looks like the theorem led Dedekind to what is called the group determinant, and he doesn't give a reference. (p. 433 of Dedekind's Gesammelte Werke, v. 2, via Curtis's Pioneers of Representation Theory. See KConrad's answer below.) Among others, I've also looked at some of the correspondence of Hasse and Noether. The works are in German, which is second language to me, so it's possible I missed something. Needless to say, I've searched using Google to no avail. If anyone knows something, I'd be very grateful.

Comment by Anthony Pulido

2013-02-17T18:37:46Z

In case, my email address is the usual Google mail one with userID anthony.pulido.

Comment by Anthony Pulido

2013-02-17T16:50:56Z

Hi Joël, not yet, but I will. Thanks! What is your full name, if you don't mind my asking? If it's OK, when I write to him I would like to mention that I learned of his comments from a former student of his.

Comment by Anthony Pulido

2013-02-16T23:52:43Z

Yes, I've just corrected the question text. Thank you, wccanard!

Comment by Anthony Pulido

2010-12-27T08:03:01Z

Actually, I would like to see that. One can see a meter stick anywhere, but seeing an original bar would concretize the notion that a great deal of thought went into the meter. It's at the BIPM. I'm not sure how willing they might be to let it go or loan it: en.wikipedia.org/wiki/Metre

Comment by Anthony Pulido

2010-09-20T23:56:05Z

I've been looking forward to live-TeXing since I read your answer and had the chance to start today. It's just as you said! Thanks!

Comment by Anthony Pulido

2010-09-20T05:44:27Z

I've found this extremely useful. I don't use it for collaboration, but it has allowed me to often leave my laptop at home and use the abundant campus computer clusters.

Comment by Anthony Pulido

2010-09-15T00:03:43Z

@Bill: Thank you very much for your clear-headed and encouraging words. I agree that we need the variety of styles for the reason you mentioned. I'll certainly remember it.

Comment by Anthony Pulido

2010-09-08T05:16:02Z

These statements are difficult to understand in isolation, like Emerton said, and therefore not obvious, like Victor said, if I understand you correctly. According to Emerton, it had a precise formulation in the 80s, and Victor claims it didn't, that it evolved in such a way that one can't say it was formulated precisely until recently? Victor, if you have the time, it might help me understand you to know how your comments relate to the short history here: netera.ca/seminars/math/fl-white.pdf

Comment by Anthony Pulido

2010-09-08T05:15:31Z

It's possible Professor Gowers was asking for an example of a proof that gave new unexpected knowledge, not something that was known to be an essential ingredient for a body of existing work, unless I missed something, which is possible, and indeed likely, and the proof of the FL indeed has already led to more.

Comment by Anthony Pulido

2010-09-08T05:08:19Z

The question was, which statements were widely believed to be true, whose eventual proof led to more than just knowledge of their truth. From considering both your comments, I think I gave an example of precise statements that were believed to be true because they were the results of a plausible principle, whose proof led to more than knowledge of their truth.

Comment by Anthony Pulido

2010-09-08T05:07:28Z

Thank you both for your enlightening comments. I've learned a great deal and have something to think about. Emerton... Matthew? I can't be sure that you remember me. I met you in the spring of 2009, I believe, at the IAS. I accompanied you to Nassau St. afterwards, along which way we talked about various things. Thank you again for the nice time and thank you both again for your comments. I hope my answer wasn't an excessive stretch.

Comment by Anthony Pulido

2010-09-06T22:55:15Z

I was under the impression that the fundamental lemma, at least, is a set of statements clear enough to be amenable to proof attempts. I think we have on page 3 here arxiv.org/abs/math/0404454 and Theorems 1 and 2 here arxiv.org/abs/0801.0446 precise statements for the various fundamental lemmas... It's possible I'm misunderstanding you.

Comment by Anthony Pulido

2010-09-06T22:45:10Z

Thank you, Victor. I'm not proposing that the Fundamental Lemma is obvious, but it seems that is was accepted as likely to be true, because others based new work on it. PC below gives the example of Skinner and Urban, and Peter Sarnak says here time.com/time/specials/packages/article/…, that "It's as if people were working on the far side of the river waiting for someone to throw this bridge across," ... "And now all of sudden everyone's work on the other side of the river has been proven."

Comment by Anthony Pulido

2010-08-27T16:53:53Z

The other question does look like good place to start. I have one recommendation that I think isn't included in the responses to it, which I posted below. I'm still a bit new here, so I'm not sure how things are run. Shall I delete my answer here and post it over there? Sify, I don't want to do this prematurely since it might inconvenience you. Might there be a difference between the two questions?

Comment by Anthony Pulido

2010-08-23T22:52:35Z

I can't be sure whether Dedekind looked only at Q or also at finite extensions. Curtis's translation is, of course, accurate, and in it Dedekind states that he looked at normal fields (Normalkörper) but doesn't specify the base field. He does say "arbitrary normal field" (ein beliebiger Normalkörper), but that is still ambiguous. I've looked in previous articles of his, but haven't found the answer. It might just be that my German isn't strong enough.