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?