How to listen math lectures?

Solution 1:

I'll interpret the word "listen" in a non-literal sense for this answer, i.e not just in the sense of "how to mentally process the new information during the lecture".

For your first question:

First of all, as mentioned by Swapnil Tripathi in the comments, reading about the topic beforehand is very helpful. While the main advantage of doing that is the obvious one - Being able to follow at least partially - it is also helpful on the psychological level: if in the beginning of the lecture you feel that you understand what's going on, it's easier to stay concentrated and motivated to try to follow during the rest of it as well.

Another method I find useful is to write down the basic definitions and notations given in the beginning of the talk, and review them every time you forget what they mean. It has often happened to me that 5 minutes into the talk, when the board is filled with expressions of the form $A_\alpha\xi = \int_{\Gamma}B\mathcal{I}$, I can't remember what $\xi$ stands for, and how in the world is makes sense to integrate $B\mathcal{I}$ over $\Gamma$ ("ah! I mistook $B$ for $b$. Now it all makes sense").

Another thing is to keep in mind one simple (but nontrivial) example of the object(s) discussed throughout the talk. Every time the speaker mentions some quality said object has, or some result it satisfies, apply it right away to your example. When the topic is hard for you and/or you don't have enough background, you can even think just about the example, i.e don't try to actually understand the results in the general case, just think why they are true for that small special case you found. In good lectures I've attended, the speaker provided such an example themselves, and told the audience that they may think of the object as that special case. For example, if the speaker talks about closed $3$-manifolds, think of the $3$-sphere. If it's about knots, think of the trefoil. If non-Archimedian fields are discussed, take the p-adic numbers, $\mathbb{Q}_p$ (technically, by Ostrowski's theorem, any number field is isomorphic to the above for some $p$, but it may not be clear if it's the first time you hear about the topic). And so on.

In order to see "the big picture" (which, I think, is the most important part - especially if the subject is unrelated to your field), try to translate everything mentioned to a (mathematical) language you know - Suppose the speaker talks about something like how (homotopy classes of) maps from the suspension $SX$ of a space $X$ to $Y$ are the same as (homotopy classes of) maps from $X$ the loop space $\Omega Y$. Well, you don't know what a suspension is, or what a loop space is, but you notice that there's an idea here: First we did something to the first object ($X$) and sent that thing into the second object ($Y$), and then we did something to the second object ($Y$) and sent $X$ to that thing. And the two processes are the same! You then recall that you've seen something like that before: If you take a set $M$ and and a vector space $V$, then you can do two things: You can build a vector space with $M$ as a basis, and then send that thing into $V$, or you can turn $V$ into just a set by forgetting its additional structure, and then send $M$ into that thing. And the two processes are the same! those are special cases of the concept of "adjunctions", although the speaker will probably omit that observation. Making such observations on your own (especially during a talk) might be the most difficult part, but in my opinion, also the most rewarding. It will also help you remember the general idea of the talk without actually knowing the details, and then you could come back to it later on and learn the details on your own if you wanted.

As for your second question, I think it happens for one of two reasons: Either the notion is so basic that it's not worth wasting time on (for example, in a lecture on Galois theory given to algebraists, one wouldn't give an example of a field extension), or the talk is bad (at least that part of it). When introducing a new notion, either there is a simple example which the speaker could mention in 5 seconds, or all nontrivial examples are complicated, in which case it doesn't make sense to expect the audience to come up with one in the 10 seconds left between the definition, and the theorem the speaker is about to prove. From this point on, a large part of the audience will only have a somewhat vague idea of what's going on (best case scenario), will only follow the technical computations (worst case scenario), or will take out their research-related drafts and start thinking about their latest project (really worst case scenario), And yes, I've been to a (specialized) seminar talk in which every single person who hasn't left 5 minutes into the talk was working on unrelated stuff, exactly because of a pool of definitions with no examples and no intuition.