Origin and Research Philosophy

From the beginning of my academic journey, I was fascinated by the beauty that complex systems embody. Gathering a deep understanding of such was my motivation to graduate in the field of dynamical Hamiltonian systems, symplectic geometry, and the topological invariants associated with such systems. My engagement in the Max Planck Institute of Molecular Cell Biology and Genetics (MPI-CBG) and the Center for Systems Biology (CSBD) allowed me to accumulate valuable insights into interdisciplinary research and real-world challenges.  

A fulfilling aspect of my work is the ability to construct a bridge between the abstract, theoretical world of pure mathematics and the practical, empirical world of the sciences. This connection is a source of immense satisfaction and a driving force behind my ambitions of scientific research.

Research Focus

In 2000 the SIAM Guest-Editors Jack Don-Garra and Francis Sullivan put together a list they call the “Top Ten Algorithms of the Century.” It is no surprise that two algorithms of that prominent list are the heart of fast (pseudo) spectral methods:

Fast Fourier Transform (FFT) 1965: James Cooley of the IBM T.J. Watson Research Center and John Tukey of Princeton University and AT&T Bell Laboratories unveil the fast Fourier transform. Easily the most far-reaching algorithm in applied mathematics, the FFT revolutionized signal processing. The underlying idea goes back to Gauss (who needed to calculate orbits of asteroids), but it was the Cooley-Tukey paper that made it clear how easily Fourier transforms can be computed. Like Quicksort, the FFT relies on a divide-and-conquer strategy to reduce an ostensibly O(N*N) chore to an O(NlogN) frolic. 

Fast Multipole Methods (FMMs) 1987: Leslie Greengard and Vladimir Rokhlin of Yale University invent the fast multipole algorithm. This algorithm overcomes one of the biggest headaches of N-body simulations: The fact that accurate calculations of the motions of N particles interacting via gravitational or electrostatic forces (think stars in a galaxy, or atoms in a protein) would seem to require O(N*N) computations—one for each pair of particles. The fast multipole algorithm gets by with O(N) computations. It does so by using multipole expansions (net charge or mass, dipole moment, quadrupole, and so forth) to approximate the effects of a distant group of particles on a local group. A hierarchical decomposition of space is used to define ever-larger groups as distances increase. One of the distinct advantages of the fast multipole algorithm is that it comes equipped with rigorous error estimates, a feature that many methods lack.

The article closes by: What new insights and algorithms will the 21st century bring? The complete answer obviously won’t be known for another hundred years. One thing seems certain, however."...The new century is not going to be very restful for us, but it is not going to be dull either!"

Our research goal is to contribute to the ongoing effort of finding the ”Top Algorithms of the 21st Century” by encountering the broadly common computational bottleneck that appears in challenges across disciplines. Herbey, we highlight our recent achievement: 

The Fast Newton Transform (FNT): The FNT is a novel algorithm for multivariate polynomial interpolation with a runtime of nearly Nlog(N), where N scales only sub-exponentially with spatial dimension, surpassing the runtime of the tensorial Fast Fourier Transform (FFT). We have proven and demonstrated the optimal geometric approximation rates for a class of analytic functions—termed Bos–Levenberg–Trefethen functions—to be reached by the FNT and to be maintained for the derivatives of the interpolants. This establishes the FNT as a new standard in spectral methods, particularly suitable for high-dimensional, non-periodic PDE problems, interpolation tasks, arising as the computational bottleneck in solving, e.g. 6D Boltzmann, Fokker-Planck, or Vlasov equations, multi-body Hamiltonian systems, and the inference of governing equations in complex self-organizing systems. Further applications can be reached by the Newton Neural Operator, realizing fast (de-)convolution in machine learning tasks.

A preprint and talk:

Teaching Philosophy

It is my great pleasure to currently teach a course on ”Approximation Theory and Approximation Practice” in my role of a visiting professor at the Mathematical Institute, University of Wrocław. In synchronicity with my research the course follows the equally titled book of Prof. L.N. Treftehen, Oxford University and bridges the theoretical mathematical world with the practical and applied aspects of numerical analysis.

As per my own experiences, I believe that excellent teaching is essential for providing the next generation of scientists with the opportunities they deserve. I’ve found that effective teaching occurs when the audience is encouraged to approach the subject matter flexibly based on their individual knowledge levels. I appreciate the continuous expansion of my own understanding of the subjects I teach, driven by the valuable feedback that I receive from my students.

On the other hand, I provide clear course structures to guide and ensure fair and transparent grading and support student-centered learning, promoting critical thinking, and providing opportunities for hands-on experience. These structures include defined learning objectives, self-test questions, and specific excercices designed for deepening the covered subjects.

Former teaching includes several basic and advanced courses in mathematics, both as tutor and lecturer, at University Leipzig & Max Planck Institute for Mathematics in the Sciences, Leipzig, and Max Planck Institute of Molecular Cell Biology and Genetics, Dresden & Technical University Dresden.

A short list of covered topics reads as: Classic Calculus, Linear Algebra, Algebra, Differential Geometry, Algebraic Topoplogy, Functional Analysis, Spatial Modeling and Simulation, Numerical Methods.