Comparison inequalities

For a lot of my career I compared myself to this guy named Max Bemis.  He's not a mathematician and you probably haven't heard of him.  He's roughly my age and sings in a mediocre band called Say Anything that was almost popular in the early 2000s.  (I don't actually recommend listening unless you like bad emo music.  He and his music are also all kinds of problematic - I can't say I endorse him anymore.)  When he was young, some people said he would be the next Bob Dylan.  In interviews and his music he talks about this pressure to be the next great thing and it ultimately hospitalized him.  Earlier in my career I envied his intensity in search of some nebulous notion of greatness, and I would always be frustrated why I couldn't be as dedicated, or a visionary, or have a JAMS paper as a postdoc.  This has weighed on me throughout my career. 

As a result, along the way, I have also compared myself to a lot of other mathematicians.  As an undergrad, it seemed like everyone had so much more mathematical maturity than I did.  As a graduate student, it seemed like my advisor knew everything and the postdocs had this tremendous knowledge in areas I knew nothing about.  Many of my peers were getting papers in better journals, getting the jobs I had applied for, winning awards, etc.  As a postdoc at UT, I was really intimidated by how good a lot of the faculty were, but even the grad students.  Was I doing something wrong?   

In general, I think making comparisons with other people can be both good and bad.  The expectations in math are so nebulous that sometimes comparisons at least give you a loose sense of what progress can look like, what are career milestones, where to try to publish your papers, what types of service/research/advising are expected at this career stage, etc.  However, I think all of that can be misleading as well.  It took me a long time to realize a few things about making comparisons, so I wanted to mention some things that I find helpful when I get stuck feeling some kind of way about all of this.  (Some of this has been mentioned in previous posts, but it's easier to just rewrite things than reference.)

- This first one is the hardest one for me.  Math isn't the most important thing in my life, so it is ok if I am not the best mathematician in the world, or more realistically, the best mathematician that I could possibly be.  (The more I let go of this, the more fulfilled I tend to be because I find more balance and make space for the scores of other things in life that are also important and enjoyable to me.)  I think this can be especially difficult if your job search is impending.

- A lot of things in your career are necessarily governed by success in the traditional sense (getting a job, writing papers, getting grants, etc), but try to keep stock of your accomplishments in the other aspects of the career that are meaningful to you: maybe it's getting to travel the world for conferences, making friends through collaborations, helping to get someone integrated into the math community, inspiring students through teaching, outreach, etc.  There are a lot of aspects of this job which have value (and maybe the person you're comparing yourself to has completely abdicated other responsibilities you feel are important, like service or mentorship).

- Don't compare yourself to people that are further along in career stage.  Everybody continues to improve throughout their career: you know more, you get more efficient at doing things, you have more collaborators so can write more papers, etc.  (For example, when I was a grad student I would spend like an entire week writing a seminar talk.  Now I can sit down and write a talk in an hour, but it took me the practice of writing 100 talks to learn how to do it quickly.)  

- One doesn't need to be great at every aspect of being a mathematician to be a great mathematician.  I have come to realize that for every mathematician I come across, I know things and have skills they don't and vice versa.  Some can do hypertechnical computations; some have visions for solving huge problems; some know about every field; some can talk a big game but don't know actually know what they are talking about; some think very quickly while others are slow but very deep.  Even inside of low-dimensional topology, there's such a wide range: I know people who are terrible at topology but great at algebra, analysis, or representation theory and hence can do amazing work.  It's important to identify your strengths and weaknesses and work on them, but having different strengths/weaknesses than other mathematicians isn't actually a bad thing. 

- With anyone, there's always a lot going on behind the scenes.  Some people have partners who take care of their life full time so they can put infinite time into math; others maybe are undergoing a major health crisis and the fact that they wrote anything last year is actually astounding.  

- Math is heavily trend-dependent.  If you work in a hot field, there is a huge boost to your profile, job prospects, grant fundability, etc.  Even if one could order "best" or "most difficult" mathematics, I don't think it would line up with the reward system we have in place.  (For example, I think the bar for getting Heegaard Floer homology papers into good journals is quite different than Khovanov homology.)  So, it is not necessarily the case that these external measures necessarily indicate you're not a good mathematician.

- Math is also very much nonlinear.  Someone who does very well in courses in undergrad/grad school may not write a great thesis.  Soemone's hyperspecific skill may suddenly be the key to a huge problem.  It sometimes just takes a few clever thoughts or having just the right toolkit to get a great result, so maybe your time to shine will come soon.  It's also not clear that working extremely hard is the key to success - everyone has a very different mathematical process.  

- What you think about yourself isn't always reality!  You may think you're a mediocre mathematician, a fraud, don't belong at your department, etc, but usually we are harder on ourselves and self-criticize for things that aren't reasonable.  Someone might be comparing themselves to you and thinking they're not good enough!  Try surrounding yourself with mathematicians (or other people outside of math) that make you feel good about who you are and what you do.