SEO Research, how do you track your progress?

Hello everyone!

We are two marketing students from Denmark, researching the U.S. market for keyword rank tracking. If you focus on SEO in your company, it would mean the world to us if you could take 5 minutes of your time to answer this survey.

Any responses are much appreciated, but we’re especially interested in American companies :)

Thanks a lot!

Research on formalization of what constitutes a good theorem

This question is about automating theorem proving. Now automated theorem proving, as currently understood in CS literature, has little to do with pure mathematics as practiced on this site (I believe the former tends to be about statements of low level of abstraction such as some theorems of Euclidean geometry; the question is not about such statements).

So let me clarify what I mean by automating theorem proving: this means (a) to be able to formalize the proofs of deep results in pure mathematics (e.g. V. Lafforgue’s results on Langlands) (b) to have a machine that is capable of producing “interesting” results of equal/higher depth on a consistent basis. The progress on (a) tends to be by current or former pure mathematicians (e.g. Kevin Buzzard) which, I believe, makes this question appropriate to MO.

Of course, if you have sufficient computational power, you can just try all possible ways to adjoin axioms and syllogisms, and you would produce a lot of theorems. If you are persistent enough, you will produce some number of interesting theorems but the percentage they would make up would be vanishingly small, I believe. Therefore, to do (b) you need your machine to know what is an “interesting” theorem.

Different people tend to find different things interesting but I believe that if you fix a particular area of mathematics, there do exist theorems which 90% of mathematicians working in this area find interesting, so the set of “interesting theorems” is not empty. The question is: how do we formally define it? Are there any papers on this subject? If not, maybe some people who have tried hard to do this could share their insights?

One concern I have is that if you tried to explain what humans mean by “interesting” to a machine, it might look pretty arbitrary (and reflect certain pieces of intuition hard-coded into our brains by nature/evolution). As one homotopy theorist puts it: “studying something like mathematics is essentially understanding the way the human mind thinks.” Maybe that homotopy theorist is right (i.e. you could learn something about an alien civilization just by looking at the pure math section of their arXiv).

I do not know but for some reason, whenever I look at the definition of etale fundamental group, I feel like it was always a part of the primate geometric intuition, it just takes some time to write it out explicitly. But let us see what research says about this.

Research topics in Microlocal Analysis

Before asking this question here I did some research on web but I would like to get the opinion of those directly interested if there are any , (as I did in this thread Research topics in distribution theory since a good connection exists). I have seen that the applications and/or research topics are varied. I also tried to see more about mathematical analysis, but every time I am more passionate about these topics.

I also mention this thread for applications: Applications of Microlocal Analysis?

For example, even if the question I’m doing is quite generic, these theories that are mostly applied to the study of linear PDEs, as can be applied in non-linear PDE contexts (I read that they are very important for this case too very difficult). The non-linear case techniques on what are based? Clearly I refer for example to the existence and uniqueness of the solution or regularity.

Another thing to add, I have read of applications in various classification of PDEs, ellipticals obviously (fairly well known), but also hyperbolic or parabolic. I would be grateful for every answer.

Clearly any reference is welcome

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