Students are taught to solve problems in a systematic, analytic manner. They learn about numbers, polynomials, inequalities, sequences and sums, many types of functions, and how to solve them. With these skills students are able to put together mathematical models so they can find good quality solutions to real world situations.
Students usually focus on abstract algebra, which studies algebraic structures, such as groups, rings, fields, modules, and vector spaces. Students learn about the systems of equations and matrices, vectors, linear transformations, determinants, eigenvalues, and eigenvectors. Students also study the geometry of algebraic curves with applications to elliptic curves and computational algebraic geometry. They learn about lane curves, affine varieties, the group law on the cubic, and various applications.
Algebraic number theory is also taught in conjunction with abstract algebra. Topics include unique factorization, Dedekind domains, class numbers, Dirichlet's unit theorem, solutions of Diophantine equations, and Fermat's "last theorem".