The best way to prepare an undergraduate student before entering scientific research

Question

I am a student in the second year of a university majoring in mathematics. I am trying to publish scientific research. Of course I know that it will be very difficult to do it at this level of study. At this level I try to prepare myself to do it. I try to study anything in detail with the proofs, and to solve some diverse and difficult problems that are often from IMO.

Since I started doing this I have noticed that I find it very difficult to combine rigorous self-study with solving very hard problems. It seems that it is better to do one thing. By this I mean either I try to study without going too deep and without knowing the details, with solving difficult problems, or I try to study in depth without trying to solve very difficult problems.

As a note, by hard exercises I mean Olympic exercises, not medium-difficult exercises like the one in "calculus" Spivak's book. My question is this:

• Do you need to be good at solving difficult problems or just enough to understand everything you have studied, in order to be creative in mathematics?

• From your experience in scientific research, what is your advise for a student who is still trying to hone their skills to enter the field of research?

Answer 1

Focusing on some terms in your two questions ...

• Problem solving is honed through disciplined practice. While some folks may have an inherent ability, most if not all folks benefit by focusing time on improving the practice itself independent from the need to solve a problem.

• The chances of solving a "difficult" problem in a reasonable time by randomly applying everything you understand at any given moment are far smaller than the chances of solving the same problem by disciplined practice in problem solving.

• "Difficult" problems do not continue to exist because we have too few smart folks who understand enough about everything that we can know to connect their knowledge with successful answers. Difficult problems continue to exist because the steps to connect one field of knowledge with another require both the fullest mastery in the fields and the ability to apply careful, rigorous problem solving to make the connections between the fields in a robustly acceptable manner

• At any point in life, everything you may have studied to that point could be only a subset of everything you will need to comprehend to solve a particular problem. Hence the phrase behind the practice of life-long learning.

• Creativity without discipline can fall / fail to anarchy.

• Understanding has various levels of meaning in theory and practice. Find references on the Bloom taxonomy for but one case (know, comprehend, apply, analyze, synthesize, evaluate).

So, to your questions directly ...

• Creativity in any discipline is near the top in the taxonomy of mastery. Before you get to that level, you must build your confidence in your knowledge and your ability to analyze problems on hand your knowledge. Creativity has nothing to do with just solving (difficult) problems or just understanding (comprehending) everything you have studied as an either-or choice. It is the application of both to analyze wide ranges of problems.

• Hone your ability to analyze problems that test your comprehension of the knowledge you currently have. Hone your ability to report on the results of your analysis in a way that others can also comprehend what you have done and apply your method to their own needs.

In summary, difficult problems will still exist a few years from now, and they will not be the same ones as now. As second year undergraduate student, this may not be the best time to try to find fields that have "difficult" problems. Rather, it can be the best time to explore to find your best skills, greatest abilities, and deepest passions.

Answer 2

Frankly, you are just at the beginning of your career in Mathematics and should give yourself more time. Hopefully, your program of studies will teach you how to make and formulate a valid proof. Once you get into these classes, your professors will take note of you if you have the exceptional capabilities that you ascribe to yourself by wanting to solve difficult problems. Hopefully, you can find a mentor this way. If you can, take the Linear Algebra or Algebra classes early as they will give you an introduction to how modern Mathematics works and will also give you some idea how big Mathematical knowledge now is.

Also, notice that you need mathematical creativity in addition to problem solving skills. When you do a Math Olympiad problem, you know that there is a solution and you just need to find it. Finding a handle on an unknown field requires more creativity.

Finally, if you want to do research NOW and independently of others, you need to find an area that has not been mathematized for centuries like algebraic number theory, algebraic geometry, or complex analysis of several variables. In this areas, even a genius will need a few years to understand what has already been done. It used to be relatively easy for amateurs to make contributions to Graph Theory, but I am not sure that this is the case any more. Some other topic in Combinatorics might still be fresh enough. If you are good at programming, you might want to look at some aspects in applied mathematics. Modeling biological systems probably still has some low-hanging fruits. You might also look into adjacent areas. Algorithms in Computer Science is a rather Mathematical topic, but there is more to discover there.

Answer 3

While problems from mathematical olympiads can be fun and rewarding, I would claim that at your stage, they do not really prepare you for research.¹ These problems are very specific in their structure. By their very nature, they generally have an intended solution that is less than a page and uses nothing but high-school knowledge and some tricks that in principle can be taught to any gifted student in less than an hour or two.

Contrast this to mathematical research, which often builds upon material that takes years of university lectures to assemble all the required concepts and often involves proofs that in total extend along many pages, where the difficulty rarely consists of finding some clever trick, but mostly of splitting the problem into the right steps and sub-problems that then can each be solved using some standard methods.

This splitting into sub-problems is something you can best learn from an advisor, e.g. when writing a thesis. It is something that can rarely be put into the form of a problem, as there often is no clear right or wrong way to do it, just a better or worse one. So to learn this, it is best to have a guide, who at each step can tell you the problems with your specific approach and point you in a better direction.

Now, if you are not at the level of having an advisor yet, the thing you can study on your own is the other prerequisite mentioned, application of the standard methods. And by this I mean methods from university topics, so you will not find those in the IMO-problems but in the same medium level textbook exercises you mentioned.

¹Just to make sure that I am not misunderstood, there are indeed many brilliant mathematicians who did well in the IMO. But apart from the problem of reversing causality, they studied IMO-problems when they were in high-school, which helped them by introducing them to mathematical thinking earlier than most. But at a late undergraduate level, you should already know mathematical thinking well enough.

Answer 4

Math can be very useful in many fields of scientific endeavor, statistical analysis, and engineering. It can be useful in some fields of computer programming, but not as much as some would think.

So my answer would be: It depends on what 'science' you want to do, and whether you envision graduate studies in pure math, applied math, physics, engineering.

I would advise you to study 'Numerical Analysis' on a computer, as most of these fields (except possibly pure mathematics) will use that extensively.

I myself have a Ph.D. in electrical engineering(EE), and a solid grounding in 'Fourier Analysis' can get you far in EE.

Also, you are very young yet, so over your lifetime you may switch course many times. I got my degrees in EE but now am exclusively a software engineer!

One thing is to try to appreciate general concepts such as design, documentation, research, and how to analyze a problem (e.g. convert a story problem to a math equation).

Hope this helps, best wishes for a bright future.