[2] Starting at age 13, he took honors math courses at the University of Wisconsin–Madison and did research under the mentorship of Ken Ono while dual enrolled at Madison West High School.

[3][5] Prior to his 17th birthday, he resolved an open conjecture proposed years earlier by Andrews and Lewis; for this research, he was named Fellow Laureate of the Davidson Institute for Talent Development.

[6] He graduated Phi Beta Kappa from the Massachusetts Institute of Technology in 2007 with two bachelor's degrees, one in mathematics with computer science and the other in physics.

[8] He also won the Machtey Award as an undergraduate in 2005, with Tim Abbott and Paul Valiant, for the best student-authored paper at the Symposium on Foundations of Computer Science that year, on the complexity of two-player win-loss games.

[9] Kane received his doctorate in mathematics from Harvard University in 2011; his dissertation, on number theory, was supervised by Barry Mazur.