An interesting question was posed to me recently. If you were told you were going to die tomorrow, which 5 math topics/questions would you be most sad you never got to learn about/have answered? First of all, I must admit I freeze any time people ask me to rank my top five anything. It feels so final, and I really want to think about it carefully before I answer. Also, honestly, if I were told I had 24 hours to live I would be sad and upset but probably not about the math I was going to miss. But that is not the point of the question, I guess. In this post I will attempt to answer this question, with full awareness that I may change my mind in a few days. But I will also pose a few other questions and then leave it to you, my readers, to ponder them.
1. My immediate response to the question was the Riemann hypothesis. Not that given 100 more years to live I would have any hope of solving this problem, but I would like to see it proved in my lifetime. Especially because we are all pretty certain that it’s true.
Of course, then one can go through the list of Millenium Problems and I would add two more things:
2. the Birch and Swinnerton-Dyer Conjecture, and
3. the P vs NP problem.
Again, I am not saying I have any chance of solving them, just that I would like to see the solutions to these problems. But this is where it gets tricky. I basically have a list of three things that probably anyone could have made (these are some of the most famous problems in math!). So how do I add two more things to it? Nothing will seem as important (nothing else I can think of would make you a millionaire!). OK, there are three other millenium problems, but I’m just not as interested in them. So then I started thinking about the math topics I would be sad not to have learned if I were to die tomorrow.
4. I have gotten interested in mirror symmetry and its relation to physics and number theory, so I guess I would be sad if I died tomorrow without learning more about it.
5. Arithmetic dynamics, since I am very interested but kind of new to it.
But doesn’t the list become weak after I add these two things? Anyway, please share your Top 5 in the comments below.
The original question got me thinking about other fun questions on might ask:
– Which 5 math books would you take to a desert island? The funny thing is that I can’t think of a top 5 but I can always think of at least one or two things. For example, I would bring Serre’s A Course in Arithmetic. But of course, if you asked me to bring just one I would be stuck.
– Who are your Top 5 mathematicians of all time? Gauss? Ramanujan?
-Slight variation: which 5 mathematicians would you take to a desert island? See, here I would probably pick some fun/handy mathematicians. I don’t know if Gauss would be very good at building a hut.
– What are the best 5 math formulas? Euler’s formula is widely regarded as one of the most beautiful formulas in mathematics. Do you agree? Can you think of others?
– What are your 5 favorite functions? I know one: hypergeometric functions!
As a final comment, I wanted to say the first question was suggested by my friend Casey Douglas, who is an Assistant Professor at St. Mary’s College of Maryland. He thought of this question as he was preparing a talk for the SMCM math department’s annual “MATH WEEK OF AWESOME”, which sounds, indeed, awesome.
So now I open it to you. Do you have answers to these questions? Do you also find it slightly frustrating when these questions are posed (if so, I apologize)? Can you think of other questions like this?
Five math topics I would be sad to have not learned, or problems I would be sad over not seeing them resolved would be….
1) Riemann Hypothesis resolved- thought a lot about that problem… I’m hoping someone will alleviate my worries. 😉
2) Quantum Yangs-Mills theory developed- quantum field theory fascinates me, but it is a bit hard to swallow from a mathematician’s perspective…
3) I too would like learn more on arithmetic dynamics, but haven’t had much time to work through Silverman’s books… perhaps soon…
4) I would like to have a deeper understanding of category theory, higher categories, categorification, etc. The applications to theoretical physics intrigue me, but its a bit hard to get a grip on.
5) I would love to understand more about string theory and the various elements of the mathematical theory behind it, especially learning more about complex manifolds.
The top 5 mathematicians is a tough one…. there are so many fields, each with their pioneers, as well as saviors from stagnation, but… as of this moment I would go with…. Euler, Galois, Grothendieck, Gauss, and Riemann.
As for top 5 mathematicians to bring on a desert island… probably… Euler, Hilbert, Hardy, Galois (he had such a rough life- he could use the vacation! 😉 ) and…. not a mathematician, but can I bring Carl Sagan? I’d bring that man anywhere- never fails to enlighten, encourage, and excite me.
Top 5 formulas…. I would include Euler’s formula… as well as the functional equation for the Riemann zeta function, Stokes theorem (general version in terms of differential k-forms on R^n [or manifolds, I suppose], since it served as such a huge motivation for a vast spectrum of new fields), Cauchy’s integral formula, and the Euler product formula.
Top 5 functions: Riemann zeta function, Gamma function, exp(z), Euler phi function, Mobius function.
One more additional list (not sure I can list 5, but still)- most shocking/unexpected/surprising bits of mathematics I came across during my studies: different sizes of infinities, unit spheres of increasing dimension actually start shrinking in volume, and the Banach-Tarski paradox. 🙂
Since I’m trying to write up something and just hit a technical snag, I feel like procrastinating and answering the first question.
1. This seems silly, but I’d like to see a proof that there are no odd perfect numbers.
2. I’ve recently become interested in modularity, so I’d like to know in particular if all Calabi-Yau threefolds (not just the rigid ones!) are modular, and maybe more generally to what extent the Langlands program works out.
3. And now a pattern emerges, what are the necessary and sufficient conditions needed for a Calabi-Yau threefold in positive characteristic to lift to characteristic 0?
4. May as well throw in the Hodge conjecture.
5. I’ll copy you and say how mirror symmetry and arithmetic properties are related, since I only have weird little things left on the list that won’t be that interesting to other people.
Here are my top five math things to do before I die:
1) Prove something interesting (preferably in the next two years).
2) See BSD proved (not by me, of course).
3) Really, truly understand spectral sequences (I might try to do this later today).
4) Convince my advisor that schemes and representable functors are awesome.
5) Talk to Barry Mazur about some of his epic papers.
I’m going to limit mine to 3:
Formulas: Euler’s, Class Number formula, Euler product formula for the Riemann zeta function.
Mathematicians: Grothendieck, Gauss, Ramanujan.
Math books: A Mathematician’s Apology (Hardy), Grothendieck-Serre Correspondence (a book of letters that I’ve read probably 100 times), How to Solve It (Polya; changed my whole perspective the first time I read it).
I’d only need two math books: the Princeton Companion, and Grünbaum’s Convex Polytopes. After that I’d fill up my quota with a thick problem book — Putnam or Berkeley; and, as night reading, the collected works of Dunham and Ribenboim (if that’s cheating, then Euler: The Master of Us All, and My Numbers, My Friends).
Since I just finished my thesis (and thereby what hopefully leads to my first publication), I wouldnt be too upset to die tomorrow 😉 Still, it would be kinda sad that I never:
1) Learned algebra beyond a first course in Galois Theory. Specifically algebraic number theory
2) Learned sieve theory
3) Finished reading Ernie Croot’s thesis
4) Collaborated with anyone
5) (As a consequence of 4)) Obtained a finite Erdos number
To a deserted island I’d definitely bring Erdos, some number theory books, and Ernie Croot’s thesis, so I could still do all those things 🙂