A primary concern of algorithmic topology, as its name suggests, is to develop efficient algorithms for solving topological problems. For example, an open problem is to find a polynomial time algorithm to see if a knot is unknotted.
Experimental mathematics continues to grow in importance within mathematics, and computation and simulation are playing an increasing role in both the sciences and mathematics, weakening the objection that mathematics does not utilize the Scientific Method.
Although arithmetic computation is crucial to accountants, their main concern is to verify that computations are correct through a system of doublechecks.
In computer science, computational geometry is the study of algorithms to solve problems stated in terms of geometry.
The main impetus for the development of computational geometry as a discipline was progress in computer graphics, computer-aided design and manufacturing (CAD/CAM), but many problems in computational geometry are classical in nature.
The primary goal of research in combinatorial computational geometry is to develop efficient algorithms and data structures for solving problems stated in terms of basic geometrical objects: points, line segments, polygons, polyhedra, etc.
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