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I am a first year Maths PhD student in Singapore. As I am going to give my first talk at a Mathematics conference soon, I look for some tips on how to give a good talk. While googling, I come across Terrence Tao's blog post entitled 'talks are not the same as paper'. In the post, he stated in third paragraph that,

Instead, a talk should complement a paper by providing a high-level and more informal overview of the same material, especially for the more standard or routine components of the argument; this allows one to channel more of the audience’s attention onto the most interesting or important components, which can be described in more detail.

I have difficulty understanding the meaning of 'high-level'.

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An illustrative example:
Let's say you have been working for a very long time on a great paper in which you finally manage to construct a widget which has some amazing properties, which have important consequences for some other area of math.

Now, most of the paper consists of the construction which is extremely deep and technical (this is, after all, why nobody else had been able to construct this widget before you).

To give a high level overview of your work, you need to talk mainly about what the widget does, rather than how it is constructed. Talk about these properties and the amazing consequences to other areas in generic terms that can be understood by everyone.

Maybe towards the end, you can talk a bit about the construction. But you should still keep it high-level, meaning that you can talk for example about the main ingredients ("we use a construction analogous to the one for constructing wadgets, but whack it with a hammer in certain places to make it fit our needs"). But leave out any actual technical details.

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    I love the idea of Maths students whacking wadgets with hammers to make them into widgets. – Wildcard Jun 14 '18 at 21:40
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    if you whack a widget's "i" hard enough it turns into a w_.dget – user253751 Jun 15 '18 at 4:44
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It may help to consider an analogy: flying over the topic at high altitude (=level), the details can't be seen, but the shape of the whole landscape can be appreciated.

This is "high-level" in the same sense as a high-level programming language, i.e. highly abstracted from the fiddly details.

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This means that you don’t derive each equation shown in the paper but focus on the, perhaps generic, results and conclusions.

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As referenced in Chris H's answer, this use of "high-level" is the same as when computer scientists speak of high level and low level: there is a spectrum of different ways to view something, with "more physical, more detailed, more concrete" at the "low" end, and "more abstract, more theoretical, more conceptual" at the "high" end.

We might describe an action at the lowest level in terms of electrical signals. A slightly "higher" viewpoint would be to talk about network packets. An even higher viewpoint would be to talk about email. At the highest level, we would talk about the user interactions of a little girl sending a birthday email greeting to her grandmother.

All of those are describing the same activity but at different "levels". The choice of levels depends on what topic you're trying to address and the familiarity of your audience with that topic.

And of course, those aren't truly the lowest nor highest "levels" of how to view that activity, but only the lower/upper bounds which typically involve computer scientists! :-) Lower ones would involve physicists, higher ones would involve sociologists.

Also important to keep in mind that "lower" and "higher" in this context are not value judgements. None of these viewpoints is better, but some might be more applicable than others to solving a particular problem.

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