On isotropic and numerical equivalence of cycles

Isotropic motivic categories provide local versions of the Voevodsky category of motives. Considered over “flexible fields”, these categories are much handier than the global one and more reminiscent of the topological counterpart. The pure part of them, the category of “isotropic Chow motives” is hypothetically equivalent to the category of “numerical Chow motives” (with finite coefficients). This implies that isotropic realizations should provide a large supply of new points for the tensor-triangulated spectrum $\mathrm{Spc}(\mathrm{DM}^{\mathrm{c}}(\Bbbk))$ (in the sense of Balmer) of the Voevodsky category. I will discuss the proof of this Conjecture for a range of new cases.