Hi, I'm Ivan — a researcher working at the intersection of signal processing, geometry, and computation.
I treat problems as objects in functor categories between structured spaces. Solutions are morphisms. Ill-posed inverse problems are non-invertible maps whose cokernel retains residual geometry. I ask: what invariants (in the sense of natural transformations) survive under a given functor? When invertibility fails, what does the solution set's cohomology look like?
Core perspectives:
- Signal spaces as objects in Banach or Hilbert categories – filtering, sampling, reconstruction as natural transformations between functors.
- Invariants = fixed points under adjoint pairs. Solvability ⇔ existence of a section up to natural isomorphism.
- Ill-posed inverse problems – regularisation = choosing a left inverse in the bicategory of relations, i.e., restricting to the subobject where the forward map is monic.
- HPC pipelines – compositions of operators interpreted as commutative diagrams; optimisation = finding a Kan extension that factorises through hardware constraints.
- Hardware-aware computation – discretisation and bandwidth define a forgetful functor from the continuous category to the finite-dimensional. The loss of structure is not a bug; it's the object of study.
Key Research Questions:
- What category does this problem naturally live in? (Objects, morphisms, and a monoidal structure.)
- What adjunctions preserve its essential invariants? (Left adjoint = abstraction, right adjoint = refinement.)
- When the forward map is not invertible, what quotient or subobject characterises the solution set?
Open to opportunities — available upon request.