Undergraduate Research & Scholarships

Aaron Agulnick L&S Math & Physical Sciences

Higher-Order Invariants in Reconstruction of Crystalline Structures

The phase problem is the central problem of crystallography. It arises from the inconvenient fact that an x-ray image of a crystal does not produce data sufficient to completely determine the structure. The data that is obtained is instead the so-called moduli of the Fourier coefficients of the function describing the crystalline structure, placing the problem squarely in the mathematical realm of Fourier analysis. One path towards resolution is offered by higher-order correlations, which generalize the Fourier coefficients while still being physically measurable. It has been shown that, in general, correlations of all (infinitely many) orders are needed to uniquely determine a function; however, for a restricted class of functions which arise physically from the discrete Fourier transform, it has been shown that correlations of finitely many orders suffice. My project aims to turn the nonconstructive proof of this fact from Grünbaum and Moore (1994) into an algorithm, which would offer a step towards resolving the phase problem by allowing a crystalline structure to be reconstructed from its higher-order correlations.

Message To Sponsor

Thank you for sponsoring my project! I am excited to pursue a project of interest to mathematicians and chemists alike, while learning about the research process in an environment of my peers. I look forward to engaging with a topic that interests me and growing as a researcher. I am grateful for the opportunity to do so.
Major: Mathematics
Mentor: F. Alberto Grünbaum
Sponsor: Hellman Fund
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