Should I ask for permission to write an expository post about someone else's research? 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 The presentation was about some of his current (unpublished) methods and results to attack a problem in combinatorics.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.