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Some stuff I found interesting about number theory research

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Re: Some stuff I found interesting about number theory research

#101

Post by mmusson »

rwallace wrote: post #85

> I suggested that perhaps this is because the famous people are the only ones who are fast enough to actually do everything you're "supposed to do" and still be productive. Conjecture: it is instead or also because famous people are the only ones who are given enough slack to be able to take the time to actually do everything you're supposed to do. Proposed test to distinguish between the conjectures: do famous math…

Some of the famous people just have knack for asking the right questions and then answering them.

I am reminded of a joke that goes something like, if you hit a roadblock in your math problem, the best way to solve it is to get Terry Tao interested in it.

Re: Some stuff I found interesting about number theory research

#102

Post by jaksmit »

Here is a reply to this tweet thread, posted on behalf of an incarcerated person (christopherhavensmath.com):

"I don't quite agree with much of what is said, but it does also have some truth to it.

The fact is that many people publishing thesis papers are not publishing in major journals and so often have many errors. As well, one can publish a paper that is never refereed by experts in that field at all. Those are disreputable papers. It is our responsibility as mathematicians to pay attention of where the work that we build from comes from. When publishing work, there is a system which ranks the integrity of the academic journals, and my method involves beginning at the best quality journal that is relevant to your work.. the referees attack the paper and send you feedback, possibly rejecting your paper. Then you make adjustments according to their feedback and either resubmit to them, repeating the process, or you submit to the next best journal. After being rejected so many times, your article strengthens more and more until the best possible journal accepts your work.

This method assures that the quality of your work reflects the quality of the publishing journal. My first paper went down the list three notches and landed on a high level journal. But the process of submission was brutal and lasted over two years. There are no gaps in our work (Myself and my 3 coauthors). Rather there are gaps in the understanding of the concepts that immediately surround our work.

When we submit work, often we continue studying in that field and learn rather quickly that our submitted work could have been much better in the sense that you wish you would have included things that are obvious upon countless reviews of your own work. In this sense, good work can have gaps, but that does not necessarily mean that the submission contained errors.

But of course there are many papers out there published through websites or some other means that have zero academic value. The best thing to do is make sure that any research paper you use or reference comes from a reputable journal.

There is one thing to keep in mind... probably the most important part. We research for many reasons. Part of it for me is that research represents the pursuit of beauty. If you're doing math for the reason of adding another notch to your ladder towards academic elitism, then you're doing it for the wrong reason. We are the chasers of mathematical beauty. When we find such beauty the hard part is expressing it flawlessly in the form of human knowledge."

Re: Some stuff I found interesting about number theory research

#103

Post by ska »

smallnamespace wrote: post #99
ska wrote: post #90

Earlier quoted context omitted.

There is a fundamental tension here because a) doing this stuff properly takes a long time and b) lots of undergraduate math teaching is "service" teaching, meant to get students up to a level where they can do the manipulations needed in other courses. This is why there is typically separate Calculus and Real Analysis streams in undergraduate - the former is needed for all the hard sciences and engineering, the latt…

Knowing and using a thing and proving it are (usually) different skills.

True; but so are understanding a thing well and being able to use it.

I wasn't really talking about proof-based vs. not though - it's true that an analysis course will be more proof based (than Calculus) because that is also a skill you are expected to be developing.

However most calculus graduates, even the ones at the top of their class, still have at best a somewhat superficial understanding of the set of real numbers and functions on it, regardless of whether or not they can regurgitate an epsilon-delta proof.

This isn't a knock on calculus courses, there is an opportunity cost to the time. By the time a continuous math student hits measure theory and understands why they need a(nother) different definition of an integral, most calculus students have been happily chugging along calculating things they need, blissfully ignorant of the "problems" with the integrals they barely remember being defined.

Re: Some stuff I found interesting about number theory research

#104

Post by klyrs »

soheil wrote: post #84

> except that for programmers it's possible (and advantageous IMO) to deeply understand way more than they actually do, which seems less clearly true in number theory. But that's not true for programmers either. How often do we look at the compiler or interpreter and check each opcode to make sure the right transistor gets turned on at the right time? The fact that number theory papers are 50-100 pages long may be an…

I'd draw a distinction between interpreted and compiled languages. It's good to know what an interpreted language does under the hood, and I'd say "look as often as you hit an unexpected performance issue." With compilers, I don't care how the sausage is made... but you'd better believe I'm going over the generated assembly for hotspots.

Re: Some stuff I found interesting about number theory research

#105

Post by Biganon »

jaksmit wrote: post #97

If anyone is interested in number theory and would like to help some incarcerated people that are working on a number theory paper: I'm a director of the Prison Math Project: http://prisonmathproject.org/ We have a few advanced mathematicians in jail who don't have access to computers. They need help with formatting their work in LaTeX and stuff like that to make it ready for publication (credit offered). They have p…

Very interesting project. BTW the Photos link leads to a 404 error and a Hello world.

Re: Some stuff I found interesting about number theory research

#106

Post by uyt »

williamstein wrote: post #7
benrbray wrote: post #4

Earlier quoted context omitted.

I do wonder where all the mathematicians / academics hang out online. Now that I'm out of (compsci) grad school, I find myself without like-minded peers to study math with. Currently working my way through Riehl's category theory book, but it's a bit tough without anyone to talk to! I know there's MathOverflow, and some people in the FP community hang out on IRC. What's the academic equivalent of HN? GitHub? I'm gues…

One answer to "What's the academic equivalent of HN?" is "going to week-long conferences and workshops", which often involves a bunch of presentations of papers (with Q&A) and discussions long into the night. Before the pandemic, an academic (such as me) could easily spend the entire summer just going from one conference to another. With the pandemic, there's now a lot of live video chat, and seminar series where eve…

The Lean community uses zulip which seems to be a nice compromise between chat and forum/email threads

https://leanprover.zulipchat.com/login/

Re: Some stuff I found interesting about number theory research

#107

Post by squid_demon »

jkhdigital wrote: post #45
williamstein wrote: post #6

I did a Ph.D. in number theory, published a few dozen research papers, and have programmed a lot and this post sounds about right to me. I did CS as an undergrad, before doing a math Ph.D., and remember being very surprised that math papers weren't a lot more wrong than they actually are (since computer software is so often full of bugs, and all it takes is one single bug to completely invalidate an entire paper). Wh…

Are you familiar with homotopy type theory? One of its proponents, Fields medalist Vladimir Voevodsky, has stated that his contributions are part of a “personal mission” to bring mathematics into a new age of formal verification [1]. I’d be curious to know how this compares to LEAN. [1] https://www.ias.edu/ideas/2014/voevodsky-origins

Homotopy type theory is a mathematical foundation, like set theory, or Martin Lof's type theory. This is different from a proof assistant like Coq or Lean. In fact, Lean has a library which implements part of HoTT (see https://github.com/leanprover/lean2/blob/master/hott/hott.md). In short, there is no comparison between Lean and HoTT.

Re: Some stuff I found interesting about number theory research

#108

Post by ummwhat »

lcq92 wrote: post #37
shuckles wrote: post #9

Earlier quoted context omitted.

During an undergraduate research program, I discovered a gap in a theory in PDEs which I’m assuming nobody else did because I was the only one who tried to follow each step (just getting that far was the entire purpose of the effort!). My advisor said it would be best to fix the gap before publishing about it because multiple students entering the job market depended on the result being true. That was an eye-opening…

Can you share what the gap was?

I could tell you, but it might cost some people their job.

Re: Some stuff I found interesting about number theory research

#109

Post by privong »

caddemon wrote: post #59
hyperbovine wrote: post #41

Earlier quoted context omitted.

> He responded that it was because people secretly "proved" everything to themselves in multiple ways, but only wrote up one proof. I don't do pure math, but I write the occasional theory paper, and this resonates. So much ends up on the cutting room floor--usually you proved the key result three or four different ways before finding a proof that is actually incisive/aesthetically pleasing/whatever to justify signing…

As a student working on some theory papers, I wish this kind of thing would go in an Appendix somewhere. Is it left out because it's not worth the effort to include compared to the number of people that would actually read it? Or is it something that runs the risk of negative perception by those established in the field?

> As a student working on some theory papers, I wish this kind of thing would go in an Appendix somewhere.

For what it's worth, this kind of behavior isn't limited to theory papers. Papers presenting analysis and model development also generally only show the "finished" result, omitting most or all of the bad models, training models, development models, etc. that were used to get to the final result. Same thing with observational papers in astronomy and related fields. There's a lot of work done that builds the authors' confidence in the correctness of the result, but that doesn't lend itself to a clean "story" for the paper or perhaps also ends up being supportive but not necessary to demonstrate a result.

I remember being surprised as a student, to learn that papers aren't narrations of the path to a discovery, but rather a narrative to describe the idea in a compelling (and hopefully clear) fashion.

Re: Some stuff I found interesting about number theory research

#110

Post by caddemon »

privong wrote: post #109
caddemon wrote: post #59

Earlier quoted context omitted.

As a student working on some theory papers, I wish this kind of thing would go in an Appendix somewhere. Is it left out because it's not worth the effort to include compared to the number of people that would actually read it? Or is it something that runs the risk of negative perception by those established in the field?

> As a student working on some theory papers, I wish this kind of thing would go in an Appendix somewhere. For what it's worth, this kind of behavior isn't limited to theory papers. Papers presenting analysis and model development also generally only show the "finished" result, omitting most or all of the bad models, training models, development models, etc. that were used to get to the final result. Same thing with…

Oh for sure, I started out doing molecular biology so I dealt with missing methodology details all the time there - certainly some things make sense to leave out, but there were also cases where it took trial and error to replicate because details on intermediate steps were left out.

That is what it is, but in the case of theory I think the alternative proofs could actually be really educational. The exact details of a biology technique (and hyperparameter searching type stuff for that matter) don't have the same inherent interest to me.

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