Instanton SlicesMPIM
by
MPIM, Vivatsgasse, 7 - Lecture Hall
Max Planck Institute for Mathematics
Acquiring geometric information about algebraic varieties via counting Fq-points and employing various p-adic techniques
has always been a challenge, classically and neoclassically (..., perfectoid spaces). Surprisingly, this can be done for
S3∖K for certain(!) hyperbolic knots, including the famous K12n242, which approach requires
instanton slices. These are Quot-stacks of torsion-free modules over isolated singularities with prescribed conductors,
in general.
I will begin with major theories of superpolynomials and the connection conjectures, focusing on the motivic superpolynomials of
plane curve singularities, counterparts of the corresponding L-functions. They fully capture topological types of such singularities.
Generally, L-functions are insufficient to recover the underlining objects, which seems different in topology of isolated singularities.
The main theorem will state that the superpolynomials of instanton slices for Fq[[x,y]] (non-singular) and proper conductors
are sufficient to obtain motivic superpolynomials of plane curves singularities in arbitrary rank, those due to Ch. Philipp.
We will then explain how hyperbolic knots enter this theory.
In general, instanton slices are not related to "curves", which dominate modern geometric representation theory and its applications.
Moreover, the core constructions are fully applicable to char=0, the case of mixed characteristics.
The ref. is arxiv.org/abs/2607.25666, which also contains the DAHA direction and applications to Nekrasov's instanton sums.