Research topics

Kurusch's research aims at developing mathematical methods useful to science and engineering. Topics of his work range from geometric integration methods, to nonlinear control theory, to Voiculescu’s free probability theory, to perturbative quantum field theory. Algebraic and combinatorial structures on graphs, partitions and other combinatorial objects are central in this respect. Currently, he is mainly interested in problems in nonlinear control theory, numerical analysis of ordinary and stochastic differential equations, and free probability theory. He also works on various aspects of multiple zeta values and related topics in number theory.

Stochastic Calculus:

  • matrix valued SDEs
  • stochastic exponentials
  • shuffle and quasi-shuffle algebras in stochastic integration
  • Hopf algebraic aspects
  • rough paths on manifolds
  • regularity structures for SPDEs

Nonlinear Control Theory:

  • non-linear systems interconnections
  • Faà di Bruno-type Hopf algebras for feedback
  • Poincaré's center problem
  • Abel equations
  • combinatorial tools

Free Probability:

  • shuffle Hopf algebra approach
  • Lie algebra and group theoretical aspects
  • moment-cumulant relations
  • cumulant-cumulant relations
  • non-commutative Wick polynomials
  • planar quantum field theory
  • Dyson-Schwinger equations
  • operads and operator-valued expectations

Numerical Integration Methods:

  • pre- and post-Lie algebras
  • Magnus- and Fer-type expansions
  • Butcher and Lie-Butcher series
  • order theory
2021-12-01, Kurusch Ebrahimi-Fard