The Deligne–Simpson problem asks: given conjugacy classes in a group, when do elements from those classes multiply to the identity? This is the algebraic skeleton of the Riemann–Hilbert correspondence — finding differential equations on the sphere with prescribed singular behavior.
The paper on braid varieties and the Deligne–Simpson problem (arXiv: 2603.20499) resolves the isoclinic case for exceptional groups by converting the question into a point-counting problem over finite fields.
The conversion works through braid varieties — algebraic varieties associated to elements of the braid group. A braid variety being non-empty is equivalent to the corresponding Deligne–Simpson problem having a solution. And non-emptiness over finite fields can be checked by counting: if the variety has points over some finite field, it's non-empty.
The counting itself draws on Lusztig's deep machinery connecting Weyl group conjugacy classes to unipotent classes. The periodic braids appearing in the problem are translatable into combinatorial data about the Weyl group, where Lusztig's character-theoretic methods apply.
The through-claim: existence is proved by enumeration. The question “does a solution exist?” is answered by counting solutions over finite fields. This is not a trick — it's the philosophy of algebraic geometry over finite fields. Existence over the complex numbers is detected by arithmetic over finite fields. Point-counting is a telescope: you look at the finite shadow to see the infinite object.
2603.20499. Algebraic geometry / braid varieties / Deligne–Simpson problem / point-counting / exceptional groups.