“Tony gave the best lecture I ever saw at Rice.” When Brendan Hassett introduces a speaker with such words, you know you are in for a treat. Tony Várilly-Alvarado gave an inviting AMS Invited Address Wednesday morning.

“Can you find three integers (you can use negative numbers) $x$,$y$, and $z$ such that $x^3+y^3+z^3=35$?” he playfully asked. What about $x^3+y^3+z^3=34$?

While $(3,2,0)$ and $(3,2,-1)$ provide solutions for these questions, if you consider $x^3+y^3+z^3=33$ the first solution appeared in 2019 and involves integers with 16 digits. And if you consider $x^3+y^3+z^3=32$, there are no solutions! What is going on? To explain this, Tony took us through the wonderful world of projective space and elliptic curves.

Let’s first look at an easier question by “only” considering tuples $(x,y,z)$ such that $x^2+y^2=z^2$, i.e., Pythagorean triples. Since once you have a solution, any multiple of it is also a solution, then we only need to consider $x,y,$ and $z$ to be co-primes. In other words, we can look at it with a projective lense.

To find solutions, rescale $z$ to be 1 and consider rational solutions to $(x/z)^2+(y/z)^2-1=0$. Setting $X=x/z$ and $Y=y/z$, we reduce the problem to finding rational solutions to $X^2+Y^2=1$ in projective space aka where you are allowed to rescale $X$ and $Y$. Using a bit of projective geometry, it turns out that the map

$$(X:Y)\to (2XY:X^2-Y^2:X^2+Y^2)$$

describe all such solutions, hence all Pythagorean triples are of the form $(2XY,X^2-Y^2,X^2+Y^2)$ or multiples of it.

By converting algebraic equations to geometric objects in projective geometry, Tony explained that the equation $x^3+y^3=z^3$ has actually no non-zero solutions (this is the $n=3$ version of Fermat’s Last Theorem) and that, going back to the initial question, $x^3+y^3+z^3=n$ seems to only have solutions when $n\neq \pm4 \mod 9$. To this date, this remains an open problem.

To be fair to Tony, he also talked about several other interesting objects: perfect cuboids, magic squares and how using similar methods, one can provide evidence for the veracity of conjectures involving these objects.

I left the room impressed by the talk. It was engaging, interesting, and quite understandable despite it touching technical mathematical tools. Well done Tony!