CECM Home > 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.

- Inverse Symbolic Calculator (includes on-line demo)
- Computationally Assisted Inequality Validation
- Identity Checking
- Computational Convex Analysis
- 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.

- Mathematical Constants: Computing pi and related matters
- 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.

- Computational Classical Analysis
- EZ-Face (Multiple zeta values, Euler sums)
- 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.
- Rapid calculation of and new recurrences for Bernoulli numbers, Euler numbers and other Rational Poly-exponentials
- Functional Equations

##### Computational Modern and Applied Analysis

- Function Reconstruction, the MomEnt+ Project
- 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

##### Advanced Collaborative Network Technologies

These projects explore issues arising from the CECM's role in the development of network-based, environments for research and education in the mathematical sciences.

- M3Plexus: Multi-Modal Mathematical Document Delivery System
- Organic Mathematics Project: Phase I of M3Plexus (see
*Proceedings of OM*) - Inverse Symbolic Calculator (includes on-line demo)

##### 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.

- Math Constants Visualization Project
*n*-Traces: Visualizing a Problem in Philosophical Logic- Zeros of Random Polynomials with Coefficients 0 and 1
- Mathematical Visualization Resources