How is algebra used to solve problems in various fields— in mathematics and otherwise?
Abstract Algebra
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Algebra has been used extensively to solve a myriad of mathematical and scientific problems in recent years. For example, a problem that has plagued two mathematical areas—topology and algebra—was recently solved: the relationship between different mathematical structures. First formulated by German mathematician Friedrich Hirzebruch in the mid20th century, the problem was to determine which numbers were topological invariants of complexalgebraic varieties (or the zerosets of polynomials). The researchers found that no such numbers were topologically invariant, and thus, the numbers depend on the algebraic structure of the variety, not the topological properties. Still another study used algebra in the field of DNA hybridization—a process central to most biotechnological devices that monitor changes in cells’ gene expressions. Based on their calculations and applications of algebra within their model, the researchers were able to monitor cellular changes.