In the year since they posted their proof, Calegari, Dimitrov and Tang have continued their collaboration. They\u2019ve now returned to the types of problems in transcendental number theory that originally sparked their interest in the conjecture. \u201cWe\u2019re trying to finish what we started,\u201d Tang said. In fact, they\u2019ve already used their techniques to prove that several numbers of interest are irrational.<\/p>\n

\u201cThey\u2019re really pushing the [method] to the limit,\u201d Fres\u00e1n said. \u201cI\u2019m really very excited about this.\u201d<\/p>\n

These methods might also be applicable to other problems in number theory.<\/p>\n

Techniques aside, the resolution of the unbounded denominators conjecture marks one of the first big milestones in the effort to gain a better understanding of noncongruence modular forms. \u201cThis is an amazing achievement, that we can make some progress on noncongruence forms in this way,\u201d Franc said. \u201cI\u2019m excited for the next 10, 20 years, to see what happens.\u201d<\/p>\n

Li, Voight and others are already starting to look for patterns in the kinds of numbers that show up in the denominators of these mysterious modular forms. They hope that in doing so, they can find hints of deeper structure.<\/p>\n

\u201cThis unbounded denominators conjecture was just the beginning,\u201d Li said.<\/p>\n<\/div>\n

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