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Many mathematical problems have been stated but not yet solved. These problems come from many areas of mathematics, such as theoretical physics, computer science, algebra, analysis, combinatorics, algebraic, differential, discrete and Euclidean geometries, graph theory, group theory, model theory, number theory, set theory, Ramsey theory, dynamical systems, and partial differential equations.
Brauer laid out a list of problems that help to define representation theory. Number 23 is the Height Zero Conjecture, in which Brauer claims that abelian groups must have a particular quality ...
Representation theory is a useful method because it reduces problems in abstract algebra to problems in linear algebra, a subject that is well understood. [ 5 ] [ 6 ] Representations of more abstract objects in terms of familiar linear algebra can elucidate properties and simplify calculations within more abstract theories.
Open Problems in Mathematics. Springer International Publishing. ISBN 978-3-319-32162-2; Guy, R. (2013). Unsolved Problems in Number Theory. Problem Books in Mathematics. Springer New York. ISBN 978-0-387-26677-0
[2] [3] An important open mathematics problem solved in the early 21st century is the Poincaré conjecture. Open problems exist in all scientific fields. For example, one of the most important open problems in biochemistry is the protein structure prediction problem [4] [5] – how to predict a protein's structure from its sequence.
In mathematics, the representation theory of the symmetric group is a particular case of the representation theory of finite groups, for which a concrete and detailed theory can be obtained. This has a large area of potential applications, from symmetric function theory to quantum chemistry studies of atoms, molecules and solids.
Open problems on word-representable graphs can be found in, [3] [8] [9] [10] and they include: Characterise (non-)word-representable planar graphs. Characterise word-representable near-triangulations containing the complete graph K 4 (such a characterisation is known for K 4-free planar graphs [17]). Classify graphs with representation number 3.
This category is intended for all unsolved problems in mathematics, including conjectures. Conjectures are qualified by having a suggested or proposed hypothesis. Conjectures are qualified by having a suggested or proposed hypothesis.