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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