CECM: Research

Research

Symbolic Computation

These projects use symbolic computation in an essential way both in the process of discovery and proof. Each aims at producing robust software.

Computationally Assisted Inequality Validation
Identity Checking
Polynomial Greatest Common Divisors

Complexity Issues And Computational Phenomena

These concern the theoretical behaviour of analytic algorithms and the exhibition of unusual related computational phenomena.

Complexity of Analytic Computations
Interesting and Unusual Computational Phenomena

Numerical Computation

The following projects involve differing mixtures of symbolic and numerical computation. The mathematics involved suggests the following classification.

Fast Algorithms in Classical Analysis.
Hypergeometric Functions, Modular functions and q-Series.
Special Functions.
Complexity of Approximations.
Geometry of Polynomials and Computational Complex Analysis.
Analytic and Polynomial Inequalities.
Orthogonal and Markov Systems.
Functional Equations

Computational Modern and Applied Analysis

Convex Programming and Maximum Entropy Optimization.
Moment Problems.
Projection and Relaxation Methods.
Fixed Points and Iterative Methods for Solving Inverse Problems.
Nonsmooth Analysis and Existence of Best Approximations.

Computational Number Theory

Special Expansions.
Computational Diophantine Number Theory.
Integer Chebyshev Problems.
Irrationality Questions.
Partitions.

Scientific Computation

Computation of invariant and inertial manifolds
Spectral methods for PDEs
Multigrid Methods
High Precision ODE Solvers
Automatic and Symbolic Differentiation

Visualization of Mathematics

Closely connected to the philosophy of experimental mathematics, these projects represent explorations into visualizing a largely abstract domain of science which strongly constrains the bounds of rigorous knowledge.

n-Traces: Visualizing a Problem in Philosophical Logic
Zeros of Random Polynomials with Coefficients 0 and 1