Written: Nov. 23, 2016
When you want to find out what's new in math (or physics) you don't go to the "official" journals, you go to arxiv.org . Then why do we still have "peer-reviewed" journals? We have the commercial ones like Elsevier's (that lead to Sir Timothy Gowers' famous boycott) charging an arm and a leg, "non-profit" ones, only charging an arm, and "free" ones, that do not suffer from the sin of greed, but do suffer from another deadly sin: to make some people feel superior to other people by rejecting submissions.
Nowadays the "importance" of a submission can wait for the future. Tim Gowers suggested an "overlay" journal, where the papers will stay in the arxiv, and will get a "like" by one of the "experts" of the "distinguished" editorial board, but anyone can create their own "overlay" journal. Ask a friend or colleague who is an expert to carefully go over your new article, and check it completely. Then you can thank him or her for checking it (and possibly improving it). Then the future will decide how "significant" a paper is.
But, I am being naive. Humans will be humans. They need their journals for sociological and psychological reasons. Some journals are more "prestigious" than others, but they all share the desire to improve their standing in the pecking order, and their editors, that by nature must have at least a small sadistic streak, (or they would refuse to do such a job), love to reject papers, because it makes them feel superior.
But accepting or rejecting papers is only one form of inclusion-exclusion. The (often below-average administrator type) editor-in-chief decides whom to include in the editorial board, and whom to exclude.
For example, Joseph Kung, the editor-in-chief of Advances in Applied Mathematics, a few days ago, wrote me email stating that he decided to kick out from the editorial board of his journal my esteemed Rutgers colleague, Vladimir Retakh, because Retakh accepts "not top tier" papers. When I asked him to give me the list of papers, he refused, but Retakh sent them to me, and these include two papers (this one and this one) by Dima Grigoriev, that are much better than (in my opinion) this bottom tier paper.
When I emailed Kung asking him to reconsider, saying that I am considering resigning from the editorial board if Retakh is not apologized and reinstated, he immediately told Elsevier that I am resigning. It is just as well, I do not wish to be on the editorial board of a person whose main joy in life is to exclude (and include his friends).
Joseph Kung has been doing this job for way too long (since ca. 1999). Under his mismanagement it turned into a second-rate combinatorics journal (that has very little to do with "Applied Mathematics"). I hope that Elsevier will "rotate" him soon, but it should also rotate his crony, Catherine Yan, since she listens to everything he says, even after he would finally "officially" retire.
Added Dec. 6, 2016: Read interesting feedback by Vince Vatter.
Added July 5, 2017: Read the appendix on a more efficient algorithm to take away greed from mathematics.