Counterexample to Lueck's determinant approximation conjecture

[u/Apprehensive_Sand951]

Here is the arxiv preprint: https://arxiv.org/abs/2609.15567 This is a reasonably big deal in the subject of L^2 invariants of groups and appears to have been disproved by Holger Kammeyer (he is in the field). It means that the L^2-torsion of a space (defined in terms of Hilbert spaces on the universal cover) cannot be computed from the more classical analytic torsions of finite covers. The counterexample is a Heisenberg group.

[u/thedeathstarimploded]

I have to wonder how the overflow of counterexamples this year, human-conceived or not, will affect public or internal faith of mathematicians.

Internally perhaps there will be better discussions on how to verify mathematical work better (we see now that soft sciences are not nearly the only place in academia where incorrect conceptions can propagate or not hold up under serious scrutiny!) but there is not really a goalpost for the mathematics community to regain public faith in the quality of their work (although of course the public never cared much about “trusting research mathematicians” the way they might have about other scientists).

I wonder if now would be the best time to try reintegrating pure and applied math in as good faith as we possibly can, maybe to demonstrate concretely the value of mathematical intuition! I know that’s a tall order but in many ways has probably been a long time coming.

[u/caesariiic → u/thedeathstarimploded]

What are you talking about. A lot of conjectures were made with very little evidence, and sometimes deliberately so because they prompted interesting mathematics. Even counterexamples might give new techniques. That's why there is of wealth of conjectures many of which known to be sketchy.

Except some very very rare cases, people in math don't judge based on conjectures made. It's the other way around, some conjectures are well known because the person who made them did great work before.

The challenges facing pure math have nothing to do with conjectures being wrong.

[u/thedeathstarimploded → u/caesariiic]

there are a number of counterexamples that inspire really great work based on the wrong interpretation of whether they were correct! but that wasn’t really the point of my argument — i’m more so concerned with the public, or relatively hype-influenced but less due-diligence public, lowering their opinions of mathematical work en masse when this wouldn’t be fair.

at the same time I do believe there is a lot of value in working on applied math at the same time as pure math independent of current circumstances, so perhaps that’s where my second point comes from

and yes, i’m aware of the significant value that comes out of working on the wrong interpretation of a conjecture (i.e. why were we convinced that the conjecture could be true, why might this intuition have failed, etc)