Abstract: Complex-valued neural networks (cv-NNs) have shown strong potential for tasks that rely on phase-based representations, short-term memory, and dynamic pattern selection — echoing ideas found in oscillator networks and wave patterns phenomena [3, 12]. Building on the previous work, Chapter 2 of this thesis demonstrates how a dynamic cv-NN can reconstruct partially occluded digit images by evolving toward one of several stable states, guided by the system’s oscillatory dynamics and eigenvalue structure [3]. This approach provides an interpretable mechanism for recovering missing information and highlights how complex eigenmodes can drive pattern completion in neural systems. This line of work naturally invites the question: how can such a framework be generalized to operate beyond specific matrix classes such as the HiPPO matrix [11] or other structured normal matrices? To address this, Chapter 3 establishes a new factorization result for diagonaliz-able complex orthogonal matrices [25], making it possible to extend the solvable cv-NN approach to arbitrary real matrices. This factorization bridges synchronization-inspired models with structured state-space representations, offering fresh insights into the spectral properties of HiPPO matrices and suggesting a unified view of oscillatory neural dynamics and modern sequential modeling.

Abstract: The central objects of algebraic geometry, schemes, are an intricate notion which takes some effort to define - in synthetic algebraic geometry, schemes are just sets with an additional property, but the rules for manipulating sets are different than what is assumed in the classical theory.

More precisely, the language of synthetic algebraic geometry is homotopy type theory together with three axioms. We will explain these axioms and some essential definitions needed to describe schemes and the example of projective space. To show non-trivial facts about projective space, like the classification of line bundles, we will make use of the higher types available in homotopy type theory.