Mathematics

Dr. Gautam Bharali
IISc Bangalore

$H^\\infty$ control theory is a branch of the theory of robust control of systems comprising interconnected devices each of whose outputs depend linearly on the inputs. Classical diagnostics for deciding whether such a system can be stabilised, given uncertainty in its parameters, are now recognised to be overly conservative. One approach to developing better diagnostics rigorously involves some interesting geometric invariant theory (GIT) in the Stein category. Both the classical and the contemporary quests for diagnostics reduce to certain Pick--Nevanlinna-type interpolation problems: into the ``unit ball'' -- relative to a homogeneous functional called the structured singular value -- in the modern view. Such a ``unit ball'' is an unbounded, (usually) non-hyperbolic Stein domain. From the work of Agler--Young in the early-2000s (who make no mention of GIT) one is led to suspect that the modern interpolation problem -- for interpolation data in general position -- is equivalent to an interpolation problem on a hyperbolic domain of much lower dimension. It turns out that the latter domain is a GIT quotient of the above-mentioned ``unit ball'' under the action of a reductive complex Lie group. That such a quotient exists follows from the work of Snow and collaborators on Stein GIT quotients. In this talk, we shall first elaborate upon this and describe a family of these GIT quotients. Not every set of generators of the invariant ring -- whose prime spectrum is the GIT quotient of interest -- is useful to the question of interpolation that we have in mind. We shall describe a special set of generators using which, together with analytical methods, we shall establish the conjectured equivalence between the relevant interpolation problems.