Will Troiani (LIPN and Melbourne (Australia)), Computation in logic as the splitting of idempotents in algebraic geometry; two models of multiplicative linear logic



A hypersurface f = 0 in complex affine space is singular if and only if there exists a non-contractible matrix factorization (to be defined in the talk). Matrix factorisations are organised into a bicategory where composition is defined via a two-step process, first an infinite model of the composite is described, and then a terminating procedure is followed to extract a finite  presentation. Is this terminating procedure a semantics of cut-elimination? By considering simple cases, Daniel Murfet and myself have uncovered two models of multiplicative linear logic, one in the space of coordinate rings where cut-elimination corresponds to the celebrated Buchberger Algorithm, and the other in the space of Quantum Error Correction Codes, where cut-elimination corresponds to quantum error correction. The general picture has led Murfet and myself to postulate that the splitting of idempotents has fundamental relevance to the theory of computation. This talk defends this proposition.