Mathematical Research at FSU

Complex Analysis

Clifford analysis or Dirac analysis is an active area of research in our department. A growing population of graduate students are working in this area. This topic involves higher dimensional function theories which extend results from complex and harmonic analysis. For example Cauchy integral formulas, Fourier series and boundary value problems. At the same time new phenomena occur which are absent in low dimensions. The function theories are based on Clifford algebras in the same way that complex analysis is based on the algebra of complex numbers. Novel features include noncommutivity and in some cases zero divisors. Analytic functions are defined as null solutions to certain Dirac operators which generalize the Cauchy-Riemann operators. Dirac operators also factor the Laplacian and wave operators in space. The success of complex analysis in the plane underscores the importance of Clifford analysis. Moreover, the mathematics has applications in modern physical theories. For example relativity, particle physics and signal processing.

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