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Anton Betten, Associate Professor, Department of Mathematics, Colorado State University
The Power of Algebraic Computing
Richard Hamming, a pioneer of coding theory, once said “The purpose of computing is insight, not numbers.” The statement could not be more true today.
New theorems can be proved based on observed data from computations. These theorems would be very difficult to obtain otherwise. The symmetry of objects plays an important role. This is because symmetry of a space defines equivalence of objects and creates the need for classification in the first place. Secondly, observed objects which admit non trivial symmetries are often much easier to understand and generalize.
What are some examples of this? In geometry, we can classify finite field versions of classical objects such as algebraic surfaces, curves, etc. The finite field version may be able to tell us about objects over the real numbers where classification up to isomorphism is impossible. Other objects of interest are tensors and semifields.
Most of these algebraic algorithms require high powered tools from Computer Science to perform the heavy lifting. Clique finding, exact cover, diophantine systems are some of the problems that arise (all of them NP-hard). In order to facilitate all this, the interplay between combinatorics, group theory and algorithms is essential.
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