The Bass trace conjecture and the characteristic-zero Kaplansky idempotent conjecture
We prove the complex group-ring Bass trace conjecture for every discrete group: the Hattori–Stallings trace of a finitely generated projective module over its complex group ring is supported on conjugacy classes of finite-order elements. As a consequence, for every torsion-free group G and every commutative unital domain R of characteristic zero, the only idempotents in $RG$ are 0 and 1. This proves Kaplansky's idempotent conjecture in characteristic zero.
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One of the papers is a solution to the Bass trace conjecture, a beautiful conjecture made by Hyman Bass in 1976, implying the older idempotent conjecture (from the 50ies maybe?) that there should be non non-trivial idempotent in the group ring of a torsion-free group. I was privileged enough to…
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100+ reactions to 100+ solutionsProofs and PromptsOct 8, 2026
This trick and these two papers now appear in OpenAI’s deeply creative solutions of problems 207 and 335 – problems so different, they could not be farther apart! I am stunned. What do I take away from this night? We cannot defensively narrow our focus to what we currently regard as human…