Succes in research, awful in solving in basic things [duplicate]

Question

I am a fourth-year Ph.D. student in pure mathematics. I am studying abroad. I need some advice. Before getting through to my question, let me say something about myself. I am studying at a good university in Europe. I have 3 papers, which two of them are my own papers(without co-authors). I am planning to defend my thesis in the following summer. Moreover, I am applying for a postdoc position.

My Ph.D. story is a sad story. After two years, my Ph.D. supervisor informed the university that I do not know any things about math, and he could not work with me. Then, the university chose a co-supervisor for me. After 1year, I and my co-supervisor wrote a paper. To be honest, my co-supervisor had a lot of nice ideas and I wrote his idea in papers. I did not fill good because that was most of his idea, but my colleges told me we always did the same things. There was a lot of pressure on me because all of the people found out my problems with my supervisor and they thought I am weak and stupid.

Last summer, I wrote two papers in the area which I have been working since my Ph.D. Then, everything changed. My supervisor said hello to me every day( before he did not say) and behavior everyone changes to me. I got some emails from great mathematicians about my papers.

I have two big problems. First, I can not solve basic things in math. That makes me feel upset and worried. For instance, when I got up today, I was thinking the following statements are equal: For all x, there is y such that x+y=4& There is x, for all y such that x+y=4, but somethings in my heart was saying I am wrong. So I start to read a logic book. That is not shaming for a Ph.D. student? Is it? or, Why I do not remember Linear Algebra? and other whys?

These kinds of questions make me feel I got lucky to write two papers. Anyone get lucky. My supervisor might be right, I do not know any things from math. How I can pass PhD exam?

Answer 1

I have seen my fair share of mathematicians failing basic tasks. Probably more iconic ones are related to arithmetics but that is more understandable.

For example I have seen a +40 years old working mathematician volunteering to teach differential equations because s/he didn't feel confident in his/her ability do solve them and wanted to force him/her-self to improve.

I don't think it is detrimental to fail basic tasks as long as you behave accordingly. As long as you discuss with others while doing your research and prepare before you teach, you can avoid most of the potential issues and fix them along the way.

Answer 2

It is hard to give you well-targeted advice, not knowing a lot more, but I can offer a few suggestions for thinking about the issues.

First, your advisor's past comments may just have been momentary frustration that should probably not have been expressed. Perhaps he just has a volatile personality. But you seem to have convinced him now, so no problems.

More important, you need to realize that since soon after the year 1900 it was no longer possible for anyone to know all of mathematics. The field is very broad and increasingly so. I knew (retired now) quite a lot about real analysis and general topology and had a lot of insight into both. But I had (and have) little actual insight into abstract algebra beyond the basics. But I was very good at what I did at the time.

Success in mathematics, today, is more about depth than breadth. The topics of research interest are very esoteric and not all topics give much insight into other topics. It isn't something that working mathematicians worry a lot about.

But, if you know some field well enough to get published in it, you are doing fine. You can branch out a bit after the intensity of doing a dissertation subsides if you like, or you can keep fishing in the same pond until it is no longer productive or you want to move on.

As to messing up with simple things, none of us is perfect. When our mind works intensely on some topic, we focus on it to the exclusion of other things. Then, we are less likely to miss simple errors since our mind is deeply engaged elsewhere. I'm working on an elementary book about mathematical thinking. I produced a draft and in reviewing it, I noticed (thankfully) that I'd reversed set union and set intersection in an important place. Embarrassing, but no more.

Think how boring life would be if you were perfect.

Answer 3

Your story reminded me of the Monty Hall problem. Even great mathematicians didn't believe:

Many readers of vos Savant's column refused to believe switching is beneficial despite her explanation. After the problem appeared in Parade, approximately 10,000 readers, including nearly 1,000 with PhDs, wrote to the magazine, most of them claiming vos Savant was wrong (Tierney 1991). Even when given explanations, simulations, and formal mathematical proofs, many people still do not accept that switching is the best strategy (vos Savant 1991a). Paul Erdős, one of the most prolific mathematicians in history, remained unconvinced until he was shown a computer simulation demonstrating vos Savant’s predicted result (Vazsonyi 1999).

As a physicist, mathematics was never a strength of mine, but I'm a postdoc and sub-group leader now and the thing you have to see is: Academia not only needs the savants, wiz, Einsteins to work efficiently. Academia also needs organizers, problem solvers apart from researchers with very bright and big ideas.

I remember some stories on Stephen Hawking, how he picked his assistants, postdocs to work with him and help him as he can hardly move/speak. He would probably have written similar letters like your first advisor, but that doesn't mean academia is the wrong place for you. There is so much competition in academia on various jobs with various criteria that you will get a very good and objective picture of your self after PhD, which way to go. But don't base it on a experience with a single advisor. Maybe he is just very ambitious and smart and seeks to work only with such persons. Understandable, but not not the norm, in a PhD the student is still and mostly learning a lot.