I graduated last year and my advisor passed away unexpectedly soon after. What should I do now for subsequent academic job searches? I received a PhD in mathematics last May. Last December, my advisor passed away quite unexpectedly. 

My advisor was an extraordinary person; in addition to being a fantastic advisor mathematically, he was a wonderful mentor who I always thought of as a friend – this is very much a personal loss for me. My initial emotional response was that I should clearly leave mathematics forever. Rationally, that seems uncalled for. Before all of this, I had been aiming for academic jobs in mathematics. However, I expect that a recommendation (as well as other help) from one's advisor is a significant part of an academic job application (for example, this recent answer states that the letter from one's advisor is read first even when applying for a tenure-track job after a postdoc). 

How does one go about an academic job search when one's advisor has passed away? 

For example, my advisor wrote a letter for me for my job search out of grad school. Does one try to use that letter (via perhaps my graduate department) for subsequent job applications? Do I mention in every cover letter from now on end that my advisor has died (that seems horrifying, since I like my vague sense of denial)? 

I am in the first year of a (3-year) postdoc, so I won't be on the job market for a while (in fact, it seems simultaneously very soon and too far away). Of course I can tell that my productivity and motivation have slumped in the last few months, and I know that I should try to get back into gear as soon as possible. 

PS: It’s entirely possibly that there’s no ‘practical’ difference between this situation and other situations where one doesn't have a letter from an advisor (as commentor Sumyrda has said), in which case that would be a perfectly reasonable answer that I would happily accept. 



Related questions: 

While in more of a 'denial' stage, I asked this question to try to get at this. 

The analogous question for current graduate students was asked here and here.

There are several questions about dealing with not having a letter from one's PhD advisor on this site, such as this one, or this one, or this one.
This is a very difficult situation. As per the comment by Andy Putman, let your other letter writers explain the situation. Considering you got the Postdoc job, your current supervisor must have been impressed with what your advisor said in his letter. Your current supervisor can include a personal story about something your advisor said about you or even include a few of his favorite verbatim quotes about you from the letter of recommendation for your current postdoc. Anyone on a hiring committee will understand, and will possibly even be personally touched. This will not count against you. 

I wouldn't send a copy of your postdoc letter in your application materials If your postdoc supervisor highlighted the strongest quotes from the advisor's letter. If he chooses not to do this or if your advisor's letter in its entirety is much stronger than the best few quotes from it and your postdoc supervisor still has the letter he could send it along with his letter.(note that in other fields this letter may be a bit inappropriate because it could contain a lot of detailed information about how you are a great fit for the particular postdoc position but as Pete points out this usually is not true in pure math).

Unfortunately this does not heal any wounds caused by the unfortunate passing away of your advisor and life long friend, but you should not worry about the letter writing situation. I can't say that I even begin to imagine the emotional pain of losing an advisor so close to graduation, but as someone who lost a parent while in college I greatly sympathize with you and I can think of no greater honor to your advisor than to continue in academia and continue doing great work.