# Algebra, Number Theory, and Combinatorics

Algebra is one of the fundamental areas of mathematics. Like most of modern mathematics, it is no exaggeration to say that Algebra is very abstract.  The many abstract structures and constructions that exist in Algebra can be difﬁcult to grasp upon ﬁrst encounter. For this reason, it is sometimes helpful to have a “handle” to lend support.  In its essence, Algebra is the study of polynomial equations.  While not intending to be an oversimpliﬁcation of the matter, keeping this in mind can be of help to a student trying to make sense of the many abstract notions that arise.

For instance, Number Theory can be considered as that subset of Algebra that is concerned with polynomial equations for which the coefﬁcients involve only natural numbers.  Likewise, the origins of Group Theory lie in the study of solutions to polynomial equations in one variable. It was Galois who stressed the importance of looking at the permutations of the set of roots of a polynomial in one indeterminate. This led to what is now called Galois Theory, as well as to the notion of a group acting on a set, hence to what is now called Group Theory.

The set of solutions to a system of polynomials in several variables is called an algebraic variety. Algebraic Geometry arose as the study of algebraic varieties. Linear Algebra is the study of systems of linear equations.  Arising out of this study are what we now call vector spaces, and more generally, modules.  Matrices turn out to have both practical and theoretical importance in Linear Algebra. Ring Theory can be thought of as the natural abstraction of the addition and multiplication operations possessed by the set of square matrices.  Commutative Algebra naturally developed out of the study of properties of rings of functions on algebraic varieties.

A polynomial equation in one indeterminate that has no repeated root is said to be separable. The recently published book “Separable Algebras” by Professor T. J. Ford, is concerned with algebraic structures that arise as abstractions of the notion of separable polynomials. Included, for example, are the study of Azumaya algebras, the henselization of local rings,and Galois theory. Interwoven throughout these applications is the important notion of étale algebras. Essential connections are drawn between the theory of separable algebras and Morita theory, the theory of faithfully ﬂat descent, cohomology, derivations, differentials, reﬂexive lattices, maximal orders and class groups.

Mathematical Sciences Ph.D. Candidate Alex Milbrand is Leading the Way

Robert Lubarsky speaks in a Hausdorff Trimester Program Types, Sets and Constructions

## Faculty

 Dr. Dr. Papiya Bhattacharjee Dr. Kevin Drees Dr. Timothy J. Ford Dr. Frederick Hoffman Dr. Koray Karabina Dr. Stephen Locke Dr. Robert Lubarsky Dr. Aaron Meyerowitz Dr. Daniela Nikolova Dr. Markus Schmidmeier Dr. Richard Voss Dr. Paul Yiu

## Grants and Current Projects Books

 51st & 52nd Southeastern International Conference on Combinatorics, Graph Theory and Computing2/10/20-2/9/21Co-Director: Fredrick HoffmanFunding Agency: NSF A Degeneration Problem Invariant Subspaces 9/1/12 - 8/31/18 PI: Marcus Schmidmeier Funding Agency: Simons FoundationAmount Awarded: \$35,000 50th Southeastern International Conference on Combinatorics, Graph Theory and Computing2/20/18 - 2/27/19PI: Frederick Hoffman and Stephen LockeFunding Agency: National Security AgencyAmount Awarded: \$23,994

 Ford, Timothy, J., Separable Algebras, Graduate Studies in Mathematics, vol. 183, The American Mathematical Society, Providence, R.I. 2017. Klingler, Lee, Levy, Lawrence S., Representation Type of Commutative Noetherian Rings III: Global Wildness and Tameness. Memoirs of the American Mathematical Society, American Mathematical Society, Number 832, Vol. 176, July 2005. Dimitry Khavinson and Erik Lundberg:  Linear Holomorphic Partial Differential Equations and Classical Potential Theory.Mathematical Surveys and Monographs Volume: 232; 2018. American Mathematical Society. M. Burmester, G. Tsudik, S.S. Magliveras, and I. Ilic (eds.): Information Security,  13th International Conference, ISC 2010, Springer, Lecture Notes in Computer Science, vol. 6531, 2011. Fan Chung, Ron Graham, Frederick Hoffman, Leslie Hogben, Ronald C. Mullin, and Douglas B. West, (editors), 50   Years of Combinatorics, Graph Theory, and Computing, CRC Press, Boca Raton, 2020.