# Brief Description of Algebraic Number Theory, Algebraic Geometry and Representation Theory

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Within Algebra, there are three main research areas at LSU: Algebraic Number Theory, Algebraic Geometry, and Representation Theory.An algebraic number field is a finite extension field of the field of rational numbers. Within an algebraic number field is a ring of algebraic integers, which plays a role similar to the usual integers in the rational numbers. The study of algebraic number theory goes back to the nineteenth century, and was initiated by mathematicians such as Kronecker, Kummer, Dedekind, and Dirichlet. Gauss called Algebraic Number Theory the ``Queen of Mathematics.'' One motivation for this study was an attempt to establish Fermat's Last Theorem (proved a few years ago by Andrew Wiles). It turns out that, while rings of algebraic integers bear many similarities to the usual integers, there can be significant differences. For example, one may lose the unique factorization of elements into products of powers of primes (the Fundamental Theorem of Arithmetic). However, there is still a unique factorization on the level of ideals in these rings of integers. The researchers in Algebraic Number Theory at LSU also are studying Quadratic Forms both from the algebraic point of view (function fields of quadrics, generic splitting, Milnor K-theory) and from the arithmetic point of view (integral properties, class numbers, zeta functions). The methods used in these studies include techniques from diverse areas of algebra.

Algebraic Geometry is the study of sets of common zeros of a family of polynomials (in several variables). Such a set is called an algebraic variety. Some geometers at LSU work mostly over the complex numbers, where there are connections with complex analysis and Riemann surfaces. Some work mostly over the real numbers, where one studies semi-algebraic sets whose points satisfy polynomial inequalities. Some work over finite fields, where there are connections with algebraic number theory and applications to areas such as error-correcting codes. Arithmetic algebraic geometry, the study of algebraic varieties over number fields, is also represented at LSU. The tools in this specialty include techniques from analysis (for example, theta functions) and computational number theory. In all these facets of algebraic geometry, the main focus is the interplay between the geometry and the algebra. For example, to each point of an algebraic variety one can associate a ring (actually a local ring, which is a ring with a unique maximal ideal) and the question of whether this point is a smooth point or a singular point (for example, a node or a cusp) on this variety can be answered by understanding the algebraic structure of this ring.

The influence of computers and software on both algebraic number theory and algebraic geometry has been considerable in recent years. Many of the researchers here make use of such software as PARI and Macaulay2 and use techniques such as Gröbner bases to help in their studies.