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This research aims to extend the Atiyah-Singer index theorem for a broader range of integral operators using a constructive approach.

January 17, 2026Open Access

Integral Operator Index Theory: A Constructive Framework with Certified Computations

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This research aims to extend the Atiyah-Singer index theorem for a broader range of integral operators using a constructive approach.
Developed a novel filtered differential algebra Kgenint for integral operator representation
Classified integral operators based on kernel properties and symbols
Derived explicit index formulas for various types of integral operators
Created certified criteria for determining Fredholm properties
Developed algorithms with complexity analysis and guaranteed error control
Provided explicit index formulas for elliptic, weakly elliptic, and non-Fredholm operators
Established certified symbolic calculus and error propagation rules
Demonstrated applications in fields such as image inpainting and quantum field theory

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This work establishes a constructive framework to compute indices of integral operators, enhancing index theory with certified numerical methods.

shifa liu (2025) studied this question.

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This work establishes a constructive framework to compute indices of integral operators, enhancing index theory with certified numerical methods.