Computational Mathematics: Dynamics, Geometry, and Algebra in Numerical Integration

Explore the intersection of computational dynamics, differential geometry, and combinatorial algebra in this 53-minute MUNI Seminar Series talk by Hans Munthe-Kaas. Delve into the evolution of numerical analysis, focusing on the importance of preserving geometric structures in simulations of dynamical systems. Learn about Geometric Numerical Integration and its significance in long-term simulations, where the quality of error becomes more crucial than its magnitude. Discover how this field has enriched computational engineering, with examples illustrating the interplay between dynamics, geometry, and algebra. Gain insights into topics such as Kepler's Solar System, the history of computers, vector spaces, symmetries, global analysis, object-oriented programming, finite element analysis, Euler's methods, symplectic integrators, and Taylor series. Suitable for scientists with a general background in computational engineering problems, this talk provides a comprehensive overview of recent advancements in structure-preserving discretization of differential equations.

Introduction
Keplers Solar System
History of Computers
First Experiments
Computational Mathematics
Pure and Applied Mathematics
Abstractions
Vector Spaces
Abstractions in Mathematics
Symmetries
Global Analysis
Classical View
ObjectOriented Programming
Finite Elements Analysis
Finite Element Exterior Calculus
Eulers Method
symplectic oiler
geometric integration
structure preservation
method
computators
classical theory
group action
thermostat
more efficient
Taylor series
Different flavors