How deep do you have to go before you can contribute to the research frontier

Question

When you do university level math, you take a sequence of courses:

E.g.

1. Analysis on $R$

2. Analysis on $R^n$

3. Functional analysis

4. ....

At no point during these courses do we have to do research frontier level math.

My question is: how deep does one have to go before one can contribute to the research frontier?

Obviously there is no clear-cut answer to this question, because it both differs per topic and it depends on exactly what you mean by research frontier.

Nevertheless I'm wondering if people who are actually at various points of the research frontier (unlike myself) could give me a sense of "how big the map is".

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To clarify my question, here are two pictures.The first one seems very unrealistic to me. But how unrealistic is it? Is the second one exxagerating?

I am intending these pictures as kind of capturing the "work involved" in learning a topic. i.e. one unit area captures one unit of "time of focused work" required to master the level. I don't mean by these pictures that Analysis 1 is "a part of" analysis 2, but rather that before one can understand analysis 2, one has to have studied analysis 1. In other words, by "getting to the research frontier" I mean conquereing an area of the space, so that one gets to the "unknown"

or Which one is closer to the truth? What does the actual map look like?

Answer 1

Both pictures don't really show how it really is. I am not good at drawing, so I'll try to explain how I see it:
Topics are related not by inclusion, as in your images, but in a directed graph. If $T_1$ and $T_2$ are two topics, then an edge $(T_1,T_2)$ means that $T_2$ depends on $T_1$. It might be an extension of the topic, it might be something that uses techniques and results from $T_1$, there are many ways how one topic can depend on another.
During your studies, you learn a lot of topics that are assumed to have the most outgoing paths, i.e. they are important for many other topics. Your analysis chain would be one such example, although the relation is not that clear, depending on what you do there might be dependencies in both directions.

Now where can we find new topics, unsolved problems and frontiers in this graph? The answer is: everywhere. In every last topic, there are open questions, and there is an infinite number of new topics out there. If you want to do research, you should keep this graph in mind. If your topic is important, in the sense that there are already many outgoing paths known, then chances are high that many people already worked on that, so you have a lot of sources that might help you. However, for the problem to be still unsolved, chances are that it will be really hard to solve it. On the other hand, a topic with few (or no known) outgoing paths might allow you to find results easier, but less people might be interested in them. Thus, it is also part of research to generate outgoing edges from ones topic, to show that it is interesting for questions where it wasn't previously considered.

I myself had a hard time understanding that as a student, I had to live it first. If you want to find the "closest" frontier, I would suggest number theory. Problems like e.g. the Goldbach conjecture can be explained in a single talk, including all the needed information on what a prime number is, but are still unsolved.

Answer 2

You can get to the frontier quite sooner than you probably expect. For example, with one course in analysis and one course in linear algebra you can reach the frontiers (reaching meaning you understand the statement of the conjectures) of several fields in functional analysis, e.g. frame theory. However, making meaningful contributions to the frontier is still very far away. Just because one can read the statement of the HRT conjecture and understand what it is asking on a superficial level does mean that they can say anything meaningful about it.

Answer 3

Assume that this is my functional analysis map:

If I were living in the year 1976, the Uniform Boundedness Principle would be unknown for me and it would be in the frontier. Then, after reading the first proof, my map could be upgraded to something like this:

Then, I would think "yeah, the frontier was beyond my previous knowledge":

However, this would not be true. As shown in this paper, here is where the Uniform Boundedness Principle really lies:

Thus, if I were smart enough, I could have attained the frontier without expand my map of knowledge. In other words: there was a piece of frontier very close to me, but I could not see it. This happens because math it is not only about knowledge. It is also about creativity:

Good, he did not have enough imagination to become a mathematician.
— David Hilbert, upon hearing that one of his students had dropped out to study poetry.

In a mathematical examination ... the examiner is not allowed to content himself with testing the competence and the knowledge of the candidate; his instructions are to provide a test of more than that, of initiative, imagination, and even of some sort of originality.
— G. H. Hardy

In order to know the distance between your knowledge and the unknown results, you necessarily have to know the path. But if you know that path, the result is not unknown anymore (of course, I'm talking about proofs - which is the only way to contribute). Thus your question "how big the map is?" cannot be satisfactorily answered.

And for the other question "What does the actual map look like?", I would say none of them. I believe that, in a more realistic map, the frontier should intersect even the elementary knowledge. Otherwise, elementary proofs would never be found.

Remark 1. As explained above, there is more than one path to the Uniform Boundedness Principle (and thus it could be reached by people with different levels of knowledge). In fact, as listed here and here, there are many results with this characteristic.

Remark 2. Although (in my opinion) it is not possible to see the distance to the frontier (as I explained above), I'm not saying that a map (of knowledge) does not exist. In fact, some high level mathematicians have maps. Here is how Laurent Schwartz described his map (which his called "internal castle"):

... all the mathematical knowledge which I accumulated, live inside my brain in a well-structured manner. Each part is connected to other parts, each part is preceded and followed by other parts. The whole forms a beautifully ordered set. This structure is as beautiful to me as a palace. It has a rigid structure.

... I can't see how I could do new mathematics if my internal mathematics were not so weIl-organized.

... When I receive an impression from outside, I have to put a whole series of phenomena back into their places and include the new idea into my own scheme of things. My castle is then even more perfect than before. Maybe some parts of it have been rejected as useless from time to time, but it's necessary to clean up every now and then. Other elements have been incorporated into it. And the castle slowly becomes modified, without actually seeming to get larger.

... I know many mathematicians who feel about new ideas the way I described above, and others who have an incredibly rapid mind. The latter kind must have an interior castle with a different kind of structure.

... Of course, there are often many ways to get from one point to another, and a well-constructed castle contains them all. !t's more or less the same system as neurons.

... My mathematical memory is deserting me in giant steps. This book shows that my memories of the events of my life have not left me, but that isn't sufficient to create mathematics. I remember the moments and circumstances of my past creations, but much less of the creations themselves .... My beautiful inner castle is deteriorating, the connections are disappearing and sometimes I get lost.

— A Mathematician Grappling with His Century.