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Within the research focus group Analysis, there are diverse research areas pertaining to dynamical systems and harmonic analysis. In the broadest sense dynamical systems means the investigation of mathematical structures with a time-like variable, and harmonic analysis the investigation of objects by decomposition into specific components. Within dynamical systems, a distinction is made between continuous time, which leads to differential equations, and discrete time, which occurs during the iteration procedure. The theory of dynamical systems can be applied within the mathematical fields of number theory, measure and probability theory, and differential equations. Applications outside of mathematics can be found in climate research, geophysics, ecology, neurobiology, and fluid mechanics. Harmonic analysis combines methods in which an object is analyzed with a linear transformation and finally processed in the transformation domain. One application is JPEG compression, which many people use daily when saving photographs on their mobile phones; this method uses discrete cosine basis vectors instead of pixel values as building blocks.

The topics of the groups within the research focus group Analysis include the following:

*Dynamical Systems and Geometry*

  * Ergodic theory (Pohl, Keßeböhmer)
  * Hyperbolic geometry and homogeneous spaces (Keßeböhmer, Pohl)
  * Fractal geometry (Keßeböhmer)
  * Quantum chaos (Pohl)
  * Ordinary and partial differential equations (Rademacher, Vogt)
  * Branches, pattern formation, non-linear waves and stability (Rademacher)
  * Applications in geophysics, ecology, fluid mechanics, control engineering (Rademacher)

*Harmonic Analysis*

  * Harmonic analysis on locally compact abelian groups (King)
  * Finite and infinite frame theory (King)
  * Time-frequency analysis and time-scale analysis (King)
  * Applications in image and signal processing, neural networks (King)
  * Harmonic analysis on fractals (Keßeböhmer)
  * Spectral theory of symmetric spaces (Pohl)

*Functional Analysis*

  * Operator theory (Vogt)
  * Non-linear partial differential equations, differential equations (Rademacher, Vogt)
  * Spectral theory (Keßeböhmer, Pohl, Rademacher, Vogt)
  * Transfer and Laplace operators (Keßeböhmer, Pohl)

*Number Theory*

  * Analytic number theory (Pohl)
  * Metric number theory (Keßeböhmer)



Involved Working Groups

Other working groups are also dealing with analysis.