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I am a math PhD student in North America about to enter my second year. Last week I attended a conference and was interested in a research presentation by a professor at another school. The presentation was about some of his current (unpublished) methods and results to attack a problem in combinatorics. I am thinking of writing an expository post about his talk in my personal blog.

Question: Should I contact the professor to ask for permission to write about his research? My concern is that he hasn't published the results and so might not want to put them in public yet. I have seen people writing expository posts about talks or papers that they read or hear about (for example here), but I'm not sure if they need permission from the original author(s) to do so.

I don't think you strictly need permission, but I would recommend asking the speaker anyway, out of simple courtesy. People are aware when they give a talk that they are making the information public, and that the audience members may share what they have learned, but putting it on the Internet may speed up that process more than the speaker is expecting. Posting about the talk without the author's consent, or at least notifying them, isn't unethical, but it is sort of "bad manners".

If the speaker asks you to wait, I would suggest you honor that request: partly out of general professional courtesy, and partly because it isn't good politics as a PhD student to get on the bad side of senior people, especially if you work in this area. You may be applying for a postdoc with this person someday. I would guess that at most, they might say something like "We plan to post a preprint in a few weeks; could you hold off until then?"

As a possible side benefit, when you contact the speaker to ask if it's okay with them, they might also volunteer some other information for you to use in your writeup: copies of their slides, further details, a preprint, etc.

If they say it's okay, then as a further courtesy, you may want to send them a copy of your writeup before you post it, to see if they have any comments or notice any errors. Or at least send them a link when you post it, and be ready to respond to any feedback they may have.

The general way to proceed has already been addressed in Nate's answer.

However, you should note that there are also conferences which have policies which explicitly forbid public use of information communicated at the conference (with the goal of fostering open exchange of unpublished results). The examples I know are the Gordon Research Conferences/Workshops, see. their policy. Thus, make sure to check whether the conference has such a policy in place (in the cases I remember, this policy has also been clearly communicated at the beginning of the conference).

tl;dr: No, it's fine (but send him a link).

My concern is that he hasn't published the results

But he has published his approach - by giving a presentation about it. And you can only write about what he talked about in that presentation.

Question: Should I contact the professor to ask for permission to write about his research?

As long as you don't misrepresent what he said in his presentation - there's really no need.

As a courtesy, however, I'd email him a note about your blog post with a link to it. If anything bothers him, he can write you.