Structural studies are responsible for much of our current understanding of protein science and enzymology. In spite of significant advances in other techniques, X-ray crystallography remains one of the most effective and commonly used methods of structure determination. The method consists of placing a protein crystal in an x-ray
beam and measuring the intensity of the diffracted rays.
Assuming that the density of electrons in a crystal can be reasonably approximated by a periodic function in three dimensions, the square of the magnitudes of the Fourier
coefficients of the electron-density function are
proportional to the intensity of diffracted x-rays. Unfortunately, the argument or "phase" of the Fourier coefficient is not easily captured.
Mathematically speaking the crystallographic "phase problem" consists of determining suitable arguments to accompany the
measured magnitudes of the Fourier coefficients of the electron-density In this limited context, without any further information regarding the nature of the function,
any set of phases for the Fourier coefficients is as valid as any other set. Physically, of course, the reconstructed function corresponds to the electron density of some real
molecular system; therefore, certain chemical constraints must be satisfied. Associating a phase set with a particular molecular model provides a means to evaluate different potential phase sets. Phase sets derived from molecular models that are chemically valid and produce
Fourier coefficient magnitudes coincident with the measured data are superior to those than either lack chemical validity or would produce significantly different diffraction data.
In this talk we will discuss the mathematics of diffraction and structure determination paying special attention to applications of the representation theory of locally compact
abelian groups and compact nonabelian groups.
Note from speaker regarding the mathematics in this talk:
"I will keep it simple, discussing three locally compact abelian groups: the integers, the real numbers, and the integers mod n; and a single compact non-abelian group: the groups of rotatations in three real dimensions.
"I will describe how Fourier transforms, Fourier series, discrete Fourier transforms and fast Fourier transforms are based on these groups. Possibly, these topics will likely be of greater interest to researchers outside of pure mathematics, for example physics and engineering." Refreshments will precede the talk at 3:30 in Kerchof 314
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