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Stream: practice: communication

Topic: starting Zulip conversations

David Egolf (he/him) (Mar 12 2026 at 19:12):

Especially for newcomers (to category theory and to this zulip), I think it can be very difficult to start and sustain discussion on this zulip. I wonder if we could make this process easier. One might hope that, by making it easier to start and sustain good conversations, that there would then be less need for moderation.

One pattern I've found successful for conversation starting is this: (1) work on an exercise in a textbook or blog post, (2) get stuck or intrigued, (3) ask a question based on that exercise. I wonder if having a short list of "conversation starter" exercises/questions somewhere might be helpful to some.

But anyways, having a bank of "conversation starters" is just one idea. More generally, I wonder if it could be productive to consider strategies by which we can help make it easier to start and continue good conversations on this zulip. Again, the hope is to help make it easier for newcomers to engage in ways that are less likely to require moderator actions.

John Baez (Mar 12 2026 at 19:55):

You became quite an expert on starting conversations here on this Zulip! Then you left. :wink:

Are you suggesting that we write up a little thing about "how to start conversations here"?

David Egolf (he/him) (Mar 12 2026 at 20:07):

John Baez said:

You became quite an expert on starting conversations here on this Zulip! Then you left. :wink:

Are you suggesting that we write up a little thing about "how to start conversations here"?

I still hope to return at some point, health allowing! We shall see.

I'm mostly inviting people to consider what can be done to make it easier for newcomers to start and continue conversations. Some resource on "how to start conversations here" could be one approach. I'm also intrigued by the idea of providing some specific examples of possible conversation starters. But perhaps other people have some ideas in this direction, too!

Kevin Carlson (Mar 12 2026 at 20:15):

I think providing specific conversation starters seems tough, because if they're technically precise, they might not lead to conversations that are very interesting to have repeatedly, whereas if they lead to repeated engaging conversations, they may tend to be more philosophical or political, in a less valuable direction. I think having a "welcome" channel with some pinned message suggesting how you might like to engage with this server in general, including advice like what you provided, David, could be very useful, though.

David Egolf (he/him) (Mar 12 2026 at 20:19):

Kevin Carlson said:

I think providing specific conversation starters seems tough, because if they're technically precise, they might not lead to conversations that are very interesting to have repeatedly ...

I agree that repeatedly discussing the same things might not be very interesting. To promote variation, one could imagine some kind of inclusive event, perhaps a "question of the month" thread, or something in that spirit. One could also imagine someone periodically changing items in a list of conversation starters. However, ideas like these require significant energy and some organization, and so I hesitate to suggest them.

John Baez (Mar 12 2026 at 20:21):

I keep suggesting that simple yes-or-no math questions are great, but I'm starting to suspect this comes across as "intimidating". Maybe people fear that questions of that sort are more likely to be condemned as dumb. Even if one drops the goal of writing an "impressive" question, which is a bad goal for beginners, it takes some real work to write a yes-or-no question that makes sense, e.g. uses math words correctly. But even an ill-formed basic math question like "is division by 2 a functor?" is a lot more productive, in my opinion, than something like "How you would you apply topos theory to molecular biology?"

Graham Manuell (Mar 13 2026 at 10:43):

@John Baez Fwiw, I don't think I agree with this push for simple yes-no questions. I understand that these are easier to answer, but for this very reason, these are precisely the questions which would generally be very easy for the questioner to find the answer to themselves. (Not that I am now trying to discourage people from asking them, just that I would not be so likely to ask them myself.)

When you brought this up before you mentioned an example of asking about Noetherian rings. You said that asking something like "Why is the Noetherian condition important?" or "How is it used?" or "What is the intuition?" were too vague and something like "Are finitely generated rings Noetherian?" is a good question, but the latter question is trivial to find answered in textbooks and/or follows very easily from famous results, so I don't know why I'd ever ask it here. But the other 'vague' questions are ones that really get to the heart of the matter, but are often infuriatingly absent from formal sources and are also heavily discouraged on mathoverflow, so of course these are the questions people would be more likely to ask here.

Jean-Baptiste Vienney (Mar 13 2026 at 12:08):

It makes think to schemes: they are difficult to learn about because most sources don’t provide the intuition which is not given by answers to yes or no questions. When I found some sources actually explaining the intuition (Algebraic Number Theory by Neukirch and The Geometry of Schemes by Eisenbud and Harris) I was very happy and under the impression that I discovered the hidden gems which could finally make me understand algebraic geometry. Then I realized that the book “The Geometry of Schemes” was absolutely loved by those knowing it, as can be seen on Amazon where the rating is 4.9 stars with something like 20 ratings.

Jean-Baptiste Vienney (Mar 13 2026 at 12:13):

On the other hand, for beginners being able to ask yes-or-no questions is still very important in the process of learning how to do mathematics seriously

Matteo Capucci (he/him) (Mar 13 2026 at 14:44):

Graham Manuell said:

When you brought this up before you mentioned an example of asking about Noetherian rings. You said that asking something like "Why is the Noetherian condition important?" or "How is it used?" or "What is the intuition?" were too vague and something like "Are finitely generated rings Noetherian?" is a good question, but the latter question is trivial to find answered in textbooks and/or follows very easily from famous results, so I don't know why I'd ever ask it here. But the other 'vague' questions are ones that really get to the heart of the matter, but are often infuriatingly absent from formal sources and are also heavily discouraged on mathoverflow, so of course these are the questions people would be more likely to ask here.

I kind of agree with this, but I also see the point John was making. I can imagine someone asking those questions in some context, like "I'm studying algebraic geometry and everyone keeps assuming rings are commutative, what breaks in the non-commutative case?" would be an open-ended but great conversation starter, but "Why is commutativity important?" would be awkward.

Matteo Capucci (he/him) (Mar 13 2026 at 14:45):

In fact I think those are the best questions!

Matteo Capucci (he/him) (Mar 13 2026 at 14:45):

Another kind of interesting question sis "I noticed X looks really similar to Y, can you help me understand why?"

David Egolf (he/him) (Mar 13 2026 at 16:45):

One pattern I find especially satisfying is when a broader/vaguer question can be answered (at least in part or in some particular cases) in a precise/specific way. It can be challenging to bridge from big-picture thinking to relevant particulars, though.

David Egolf (he/him) (Mar 13 2026 at 16:48):

I love it when I ask a relatively vague question, and someone responds in a way that helps me think about some specific aspect of the question in a precise way.

Jean-Baptiste Vienney (Mar 13 2026 at 17:09):

There is an example of this that I love: :upside_down:

For every polynomial $f(x) \in k[x]$ for $k$ a field of characteristic $0$, we have the Taylor formula:
$f(x)=\underset{n \ge 0}{\sum}\frac{f^{(n)}(0)}{n!}x^n.$

Question: We can't divide by $n!$ in a field of characteristic $p>0$, but can we still make sense of the Taylor formula?

Answer: Yes, in a precise way. Let $k$ be any field (of any characteristic: $0$ or a prime number $p>0$).

Define the $n^{th}$ Hasse-Schmidt derivative of a polynomial in $k[x]$ as follows:

Define the map $D^n:k[x] \rightarrow k[x]$ as the unique $k$-linear map such that $D^n(x^r)=\binom{n}{r}x^{r-n}$ if $r \ge n$ and $D^n(x^r)=0$ if $n>r$.

For $f(x) \in k[x]$, $D^n(f)$ is called the $n$-th Hasse-Schmidt derivative of $f(x)$. Note that $D^n(f)=\frac{f^{(n)}}{n!}$ if $k$ is of characteristic $0$.

We have the following formula:

$f(x)=\underset{n \ge 0}{\sum}D^n(f)(0)x^n.$

Jean-Baptiste Vienney (Mar 13 2026 at 17:12):

Now, I don't claim that it is that easy to think about this question if you're not very curious about differentiating polynomials over any field :)

John Baez (Mar 13 2026 at 17:23):

My point about vague questions was not that they're unimportant. They are in fact the questions that we are most desperately concerned with when learning mathematics. My point, which I tried to explain a few weeks ago, was that vague question impose a heavy burden on the person trying to answer them. To answer them well can require a carefully written essay. Thus, someone who shows up on a forum and starts asking a large number of vague questions will often get a poor reception. They may not understand how much they're demanding. They may not understand how much the experts enjoy precise questions.

So I keep reminding new mathematicians that yes-or-no questions are greatly cherished by the people answering the questions. This is one reason it's good to practice the art of asking powerful yes-or-no questions. The other is that it coming up with a good yes-or-no question requires a lot of thought. It's not easy... and in fact, maybe I should quite telling new arrivals about this: maybe it's a skill to learn later.

But still, just for fun, here's my philosophy. It has to do with h-levels:

If you ask a yes-or-no question you are asking for a single bit. This is a small ask: someone can provide the bit and then decide how much explanation they want to give. If you ask for the solution of an equation you are asking for an element of a set. If you are asking for an object in a category, this is an even bigger ask. And so on. Property, structure, stuff....

Jean-Baptiste Vienney (Mar 13 2026 at 17:35):

Just for fun, what are the levels above the third one for the solution of an equation? I would be interested in getting some concrete understanding of this. If a linear combination of solutions is a solution, then you have: 1. yes/no (the equation has a solution or it hasn't), 2. the set of solutions, 3. the vector space of solutions, 4. ?

Jean-Baptiste Vienney (Mar 13 2026 at 17:35):

Or do you have another example where this phenomenon can be understood somewhat concretely?

Jean-Baptiste Vienney (Mar 13 2026 at 17:36):

I know we can define homotopies between homotopies between homotopies etc... but that's not what I call concrete :sweat_smile:

Jean-Baptiste Vienney (Mar 13 2026 at 17:39):

For instance, saying that each level gives us a better way to compute the solutions would be an amazing answer if it was true. That's the kind of understanding I'm looking for.

Jean-Baptiste Vienney (Mar 13 2026 at 17:47):

I’ve just realized it could be a not yes-or-no question placing a lot of burden to be answered :sweat_smile:

John Baez (Mar 13 2026 at 18:19):

By the way, if we treat the kernel you're talking about as a subspace of an existing vector space $V$, it's an element of a set (the set of all subspaces of $V$). But if we treat it as an object equipped with a mono to $V$, it's an object of a category. Category theorists know how to reconcile these two viewpoints.

When I ask for a category with some property, I'm asking for an object in a 2-category.

John Onstead (Mar 13 2026 at 18:24):

Matteo Capucci (he/him) said:

Another kind of interesting question sis "I noticed X looks really similar to Y, can you help me understand why?"

These are my favorite questions, but as they are hard to put into yes-or-no format, I don't often ask them anymore.
A good example of something along these lines I encountered just yesterday:

John Onstead (Mar 13 2026 at 18:25):

This is the Chapman-Kolmogorov equation for composition of Markov kernels $k: X \to Y$ and $h: Y \to Z$:

$$(h\circ k)(C|x) = \int_Y h(C|y)\,k(dy|x)$$

And this is the usual profunctor composition formula of profunctors (using the same variable names) $k: X \to Y$ and $h: Y \to Z$:

$$(h\circ k)(C,x)=\int^{y\colon Y} h(C,y) \times k(y,x)$$

I don't see a difference! I know the answer probably has to do with matrix multiplication or something, but I still find this similarity interesting.

Kevin Carlson (Mar 13 2026 at 18:27):

I think this would be a perfectly reasonable thing to ask about. I wouldn't over-index on the "yes or no" aspect. "Here are two really concrete things that look similar. What's up with that?" is pretty easy to answer with either a "they're both blah blah blah" or "no idea." If you really want you could yes-no-ify it by asking "Are they really the same in some sense?" But I think the yes-no question advice is one way of concretizing the request to ask concrete questions for which it's unambiguous what constitutes a sufficiently complete answer, and also ideally that answer could plausibly be short.

Kevin Carlson (Mar 13 2026 at 18:28):

You also personally have a lot of credit on this server, I think, John, as someone who asks thoughtful questions and engages rigorously with responses, so you'd probably succeed with questions based more on your own judgment than on general rules of thumb extended to newbies.

John Onstead (Mar 13 2026 at 21:42):

Kevin Carlson said:

You also personally have a lot of credit on this server, I think, John, as someone who asks thoughtful questions and engages rigorously with responses, so you'd probably succeed with questions based more on your own judgment than on general rules of thumb extended to newbies.

Thank you! I'll keep that in mind.

John Onstead (Mar 17 2026 at 10:06):

I'm starting a discussion on a new topic I don't see discussed often enough, and I think it's got a good potential for a long discussion! If any of you here are interested, drop by the thread on "connected components" and let me know what you think!

JR Learnstomath (Mar 17 2026 at 16:46):

John Baez said:

You became quite an expert on starting conversations here on this Zulip! Then you left. :wink:

I'm also a seasonal engager here! When I started asking questions, it was a combo of seeing what kinds of questions people were asking and being so stuck and confused that I had to ask. The "no dumb questions" slogan in learning:questions encouraged me to try.

I asked questions about "I'm interested in X: who is working on it now? Can someone point me to their favorite resources?", and "I'm confused between the concepts A and B ". People gave great (and kind!(*)) answers and I went off to look things up/continue parsing resources. For that, though, I never expected sustained conversations.

I think that a pinned message to the intro would be really helpful, with maybe 3 examples of a range of questions, or a link to a few "favorite questions".

(*) I didn't know very much about Alexander Grothendieck when I asked a question a few months ago, and the response - I think it was Morgan's - was a master class in informing me without making me feel bad. I even had a good laugh at myself!

John Baez (Mar 17 2026 at 16:49):

You didn't put an end quote symbol in your last comment, @JR Learnstomath. I didn't say most of that stuff.

JR Learnstomath (Mar 18 2026 at 05:20):

John Baez said:

You didn't put an end quote symbol in your last comment, JR Learnstomath. I didn't say most of that stuff.

Woah! My apologies. Thank you, I have fixed it now!