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Jeremy Weissmann

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Jun 24, 2020, 5:55:36 PM6/24/20
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Dear all,

   A couple of things.

   (0)   Don't use polljunkie for online polls.
   (1)   I screwed up the poll somehow.
   (2)   Abstract Algebra seemed to be the clear winner, so let's go with that for now.

   The text I suggest is  Abstract Algebra: The Basic Graduate Year ,  by  Robert Ash .   It's available online here:


The reasons for this suggestion:  First, the text is free.  Second, I recall that Professor Ash has a fresh, clear expository style.  Third and finally, Professor Ash writes the following in the front matter, quite provocative from our perspective:

   Many mathematicians believe that formalism aids understanding, but I believe that when one is learning a subject, formalism often prevents understanding. The most important skill is the ability to think intuitively. [...] My writing style reflects this view. (https://faculty.math.illinois.edu/~r-ash/Algebra/Front.pdf)

And see these remarks on expository writing here:  https://faculty.math.illinois.edu/~r-ash/Remarks.pdf .

   I think it will be an interesting challenge to learn or refresh these concepts from algebra, while simultaneously seeing whether we can't make formalism work for us, where Professor Ash believed it would be a hindrance.

* * *

   I'm now open to suggestions for how to organize the reading group.  I would suggest we meet weekly through some digital platform (suggestions please?!).  Each week, someone will volunteer to guide us through a given section or subsection.  We'll divide up the exercises among participants, who can each present their work — preferably written up.

   The emphasis, as always, will be on the design of notational frameworks to carry out arguments, and on the design of the solutions to problems.  We want to lean away from solutions like,  "apply such-and-such theorem" ,  and lean towards solutions that begin by analyzing the demonstrandum.  (Of course, we might just say,  "looking at the demonstrandum, it has the exact shape as such-and-such theorem, so we can apply that here" .)

   Naturally, we'll clarify the rules of the game as we go along.

* * *

   If you're interested, why don't you respond to this email with some sense of your availability, time zone, operating system, background, etc.

   Also, if you have suggestions for what digital service to use for our group chats, let us know.

   All best,

+j


Lucas Brunno Luna

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Jun 26, 2020, 2:42:43 PM6/26/20
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Dear Jeremy,

I'm really glad for the decision to pick up abstract algebra. And you seem to have selected a good text. It may turn out not to be "ideal" from our perspective, but I reckon that we should be able to manage to learn or refresh the concepts presented in the text, even because the author himself professes to subscribe to some sensible principles regarding the quality of the texts he writes. And, should we have any trouble or find room for improvement in how the material could be better understood/taught, we shall have each other to discuss such ideas, especially in what regards the use of formalism, as you point out.



I'm now open to suggestions for how to organize the reading group.  I would suggest we meet weekly through some digital platform (suggestions please?!).  Each week, someone will volunteer to guide us through a given section or subsection.  We'll divide up the exercises among participants, who can each present their work — preferably written up.

That sounds like it could work for me. Though I wouldn't be so bold at this point to volunteer myself to act as such a guide as you describe: if I would ever do that, I would like to see how this "guiding" is supposed to look like, first.

As an aside, while this procedure could work for me and perhaps for others actively engaged in the course, do you think that we should take the trouble to share some public record of the progress of and the insights that transpire from the course, so that others could learn from our efforts, too?



If you're interested, why don't you respond to this email with some sense of your availability, time zone, operating system, background, etc.

Weekly meetings sound very reasonable to me, though I myself should be able to participate at a quicker pace, every 3-5 days, perhaps; just appoint a date and I should manage to be there, unless I must honor some other commitment.

My time zone seems to be UTC -3. (Is that how one writes it? I.e. the current universal time is 3 hours ahead of my local time.) Ideally I should be available from 13h to 20h in my local time. So, from 16h to 23h in UTC, I suppose?

My operating system is Slackware Linux, and this is my background, if that's important:

winbg.jpg


OK, I'm kidding.

My background is actually this:

wpp.jpg


Best regards to all.

Jeremy Weissmann

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Jun 26, 2020, 4:35:47 PM6/26/20
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My time zone seems to be UTC -3.

I think I’m UTC-4 (New York City) so that’s convenient!

+j


Is that how one writes it? I.e. the current universal time is 3 hours ahead of my local time.) Ideally I should be available from 13h to 20h in my local time. So, from 16h to 23h in UTC, I suppose?

My operating system is Slackware Linux, and this is my background, if that's important:

<winbg.jpg>


OK, I'm kidding.

My background is actually this:

<wpp.jpg>


Best regards to all.

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<winbg.jpg>
<wpp.jpg>

Jamie Oglethorpe

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Jun 27, 2020, 1:29:27 AM6/27/20
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MS

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Jun 27, 2020, 2:12:10 AM6/27/20
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utc +10

Regards

Mahesh Shastry

Jeremy Weissmann

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Jun 27, 2020, 2:36:36 PM6/27/20
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Dear all,

   It appears we have

Jeremy Weissmann (UTC-4)
Lucas Brunno Luna (UTC-3)
Jamie Oglethorpe (UTC+2)
Mahesh Shastry (UTC+10)

   Four is enough to start the group, provided we can find a good digital platform to work on.  Reconciling the time differences may not be easy.  Evening for me and morning for Mahesh would be ideal, but that puts Jamie in the middle of the night.  So it will have to be morning for me and Lucas, late afternoon for Jamie, and evening for Mahesh.

   I'm trying to decide if something like 6-7 AM on, say, a weekend is feasible for me.  (I have two small children.)  That would be 8-9 PM for Mahesh, I believe, which is on the later side.  Does that work?  Here's a timetable so we can compare:

Jeremy  Lucas   Jamie   Mahesh
 6 AM    7 AM   12 PM    8 PM
 7 AM    8 AM    1 PM    9 PM
 8 AM    9 AM    2 PM   10 PM
 9 AM   10 AM    3 PM   11 PM

As you can see, we don't have a lot of wiggle room.

   Let me know what seems good for you all.

+j


Lucas Brunno Luna

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Jun 27, 2020, 4:31:54 PM6/27/20
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Dear all,

Anytime in the morning would seem good to me, if that is fine with you all.

Also, as far as my suggestions for a meeting platform go, I would invite all colleagues to consider whether carrying out our course here in the mathmeth group itself would not be in line with your preferences. This is just a thought; in any case, what I have in mind is that we could work somewhat along the following lines:

The course would be divided into batches of exercises. Perhaps Jeremy could perform this whole division beforehand, if he already has an idea of how to do that, and it is not too inconvenient. Alternatively, this division would not necessarily need to be done for the whole text at first: just a few could be defined at the beginning, for a start; and then further divisions would be performed as the course goes on.

So, one of the participants would volunteer to act as the guide for a particular batch of exercises. In principle then, the first thing this guide should do would be writing up an email delineating the exercises, not neglecting to reserve some words about the concepts involved in them (preferably beforehand?); words which maybe wouldn't need to be too elaborate for a start, perhaps just pointing to places in Prof. Ash's text would suffice.

We are then left with the question of how to divide the exercises among the participants. Concerning that, I came up with the following silly idea: Depending on who's the guide for the current batch of exercises, the total number of exercises in the batch would be in principle divided equally among all participants, with all participants selecting whichever exercises they would like to solve according to a priority queue, and the exercises which remain would be left to the guide. Participants could be allocated to positions in the priority queue according to the distance of their time zones to that of the guide. So, for example, as a function of the guide, the respective priority queue for the current batch of exercises could be given by the following table:

Guide
1st
2nd
3rd
Jeremy
Mahesh
Jamie
Lucas
Lucas
Mahesh
Jamie
Jeremy
Jamie
Mahesh
Jeremy
Lucas
Mahesh
Jeremy
Lucas
Jamie

I think it would be convenient if the guide gave the current priority queue at the end of their email as a heads up, too.

Of course, as we try to solve the exercises, we could ask others for help, should we have any trouble. In the same vein, after someone presents their solutions, others could ask for clarification in case they didn't understand something, or just make whatever remarks they'd like.

And thus we would proceed until everyone is done with that particular batch of exercises, whereafter someone would volunteer to act as the guide for the next batch.

This is what I had in mind, for what it's worth.

Best wishes, everyone.

Jeremy Weissmann

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Jun 27, 2020, 5:00:22 PM6/27/20
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I haven’t read this whole thing yet, but I just wanted to say, I agree we should present anything written in the mathmeth group. But I also think we should have live discussion on some sort of digital platform so we can talk to each other.

On Jun 27, 2020, at 16:31, Lucas Brunno Luna <lucasl...@hotmail.com> wrote:


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Jeremy Weissmann

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Jun 27, 2020, 10:36:27 PM6/27/20
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I'll continue to think about the structure of our reading group.  For now I will suggest that I, as the organizer of this thing, will "assign" material to us: either problems to be solved, or theorems to be discussed.  

It goes without saying that what we should aim to present is not necessarily complete solutions, but rather our efforts at those solutions.  In particular, I would like us to be able to say something like, "I can see how this problem could be solved along such-and-such lines, but that feels ugly and I don't know how to motivate it.".

As far as I can see, Section 1.1, Groups and Subgroups, is already pretty dense, with 15 medium-challenging exercises.

Before we get there, I'd first like to invite any of us to call out any of the results in Chapter 0, Prerequisites, if we feel these would be good to review or discuss.

Finally, it seems there will be a lot of back and forth emailing pertaining to this course.  I don't want to burden the rest of the group.  Perhaps it would be better to split off into a separate group (calculational-mathematics-reading-group?), and then post occasional interesting tidbits back to the main group.  Let me know if you think that would be a better idea.

Best,

+j

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Lucas Brunno Luna

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Jun 28, 2020, 12:16:04 AM6/28/20
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Dear Jeremy,

All those suggestions sound most sensible to me and they have my full approval, for what that's worth. In particular, the decision to make a side group in which to carry the bulk of our discussions sounds like it would work just perfectly: I suppose we shall use that side group as a substitute for the live chat service?

Regarding Chapter 0 of Prof. Ash's text, I did take the time to have a look at it earlier. In the first place, I would like to comment that I can swallow the suggestion to have some set theory as a prerequisite for a text in abstract algebra, but also number theory and linear algebra? I admit, I myself am but a novice in all of these areas. In any case, that gives me the impression that our hardworking author got his priorities a little reversed there, though I grant that he might have had his own justifications. Number theory sounds like a possible area of application for the techniques/results of abstract algebra, and linear algebra strikes me as a branch of the same; it would therefore seem legitimate to introduce these areas later on in the text in connection with the main subject of abstract algebra, but to have them as prerequisites for learning it in the first place sounds a bit off.

I just hope this won't give occasion to much trouble. At any rate, as I said earlier, the text could turn out not to be "ideal" from our perspective, but I think we can manage with it.

All right, the second comment I would like to make in that connection is that, while looking at those prerequisites, I did find some stuff which I was not quite familiar with, especially in linear algebra. If it's worth the trouble, I will have another look at that chapter later and try to enumerate what exactly is still unfamiliar to me.

All the best.
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Jeremy Weissmann

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Jun 28, 2020, 12:43:47 AM6/28/20
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This is a graduate textbook on abstract algebra. As such, it presupposes a fair amount of “general” abstract mathematics (results about sets, numbers, etc) that will be drawn upon without much fanfare. 

If it feels too daunting, we can certainly find something more along the lines of an undergraduate text. 

I didn’t want to bore the participants!

+j


On Jun 28, 2020, at 00:16, Lucas Brunno Luna <lucasl...@hotmail.com> wrote:


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Jamie Oglethorpe

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Jun 28, 2020, 12:58:37 AM6/28/20
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MS

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Jun 30, 2020, 7:05:15 PM6/30/20
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Hi Jeremy 

Won't 6-7 am be too early for you, why not 7-8 am NYC time it will be 9-10 pm Sydney time 

Please let me know, what works best for everybody 

Regards

Mahesh Shastry

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