Algebra and Discrete Mathematics
The area of algebra and discrete mathematics encompasses both theoretical and applied aspects
of mathematics that are foundational for matrix analysis, modern algebra, number theory, combinatorics,
and graph theory. This area of study has a significant impact on applications arising in
statistics (linear models, experimental designs), probability (random models), operations research (mathematical programming,
network analysis), communication engineering (coding theory, cryptography), and computer science (analysis of algorithms,
nonnumerical computing). Students interested in the underlying theory of algebraic and discrete structures will
also gain insights into how these concepts are fundamental to a wide array of practical problems.
Faculty
- J. V. Brawley »
finite field theory and applications, linear algebra, combinatorics
- N. J. Calkin »
combinatorics, number theory, probabilistic methods
- S. Gao »
finite fields, cryptography, coding theory, computer algebra, combinatorial designs,
algorithmic number theory
- K. James »
number theory, modular forms, elliptic curves
- R. E. Jamison »
graph theory, finite and discrete geometry, closure systems
- J. D. Key »
finite geometries, combinatorial designs, error-correcting codes, groups
- R. C. Laskar »
graph theory, combinatorics
- H. Maharaj »
algebraic function fields, algebraic-geometric codes
- G. Matthews »
algebraic coding theory
Curriculum
The algebra and discrete mathematics concentration is structured around the following courses:
abstract algebra, matrix analysis, applicable
algebra, combinatorial analysis, and graph theory.
All of these courses emphasize the algebraic, combinatorial, and graph-theoretic structures used
to model problems arising in engineering, the life sciences, economics,
statistics, operations research, and computer science.
Abstract algebra surveys groups, rings, fields, and lattices.
Matrix analysis treats a variety of topics in matrix theory which support a modern applied curriculum.
Combinatorial analysis emphasizes applied topics from enumeration, graph theory, optimization, and
block designs.
Graph theory is the study of paths and networks, connectivity, trees,
coverings, and coloring problems.
The applicable algebra course and associated selected topics courses (cryptography, coding theory,
finite fields, computational algebra)
cover material that is of great interest in computer design and in ensuring the accuracy and security
of digital information.
Additional courses are available that integrate concepts from algebra and discrete mathematics
with the areas of analysis, computational mathematics, operations research, and
probability/statistics.
Courses
Sample Curricula
- Sample Program for M.S. Concentration in Algebra
FALL: 805, 810, 853
SPRING: 860, 821, 851
SUMMER: 803
FALL: 822, 852, 855
SPRING: 854, 856, 985, 892
- Sample Program for M.S. Concentration in Discrete Mathematics
FALL: 800, 810, 853
SPRING: 805, 821, 854
SUMMER: 860
FALL: 814, 855, 865
SPRING: 851, 856, 985, 892
Recent M.S. Graduates (master's project title)
- Nicole Bannister ("Tetrominoes Attack!")
- Mary Ann Coleman ("Semigroups and Minimum Distances of Codes from a
Quotient of the Hermitian Curve")
- Jill Cuneaz ("Discrete Logarithms in Finite Fields")
- Sean Daugherty ("The Influence and Total Influence Numbers of a Graph")
- Nate Drake ("Exact Minimum Distances of Some Two-Point Hermitian Codes")
- Katie Durham ("Some Weierstrauss Semigroups on Certain Maximal Curves")
- Jeffrey Farr ("Gowers, Roth, and the Proof of Szemeredi's Theorem")
- Tim Flowers ("Ultimately Periodic and Aperiodic Sum-Free Sets")
- Travieso Gonzalez ("Exploring the Partition Function")
- Sarah Graham ("Decoding Arrays for Two-Point Codes")
- Ray Heindl ("Fourier Transform, Polynomial GCD, and Image Restoration")
- Jason Howell ("The Index Calculus Algorithm for Discrete Logarithms")
- Kevin Hutson ("Signed Graph Clutters and the Set Covering Problem")
- Joe Johnson ("Algebraic and Graph Theoretic Properties of Polynomial Digraphs")
- Travis Kidd ("Naming the Trees in the Forest")
- Jira Limbupasiriporn ("Complexity of Arithmetic Operations")
- Mark Liu ("The Linear Diophantine Problem of Frobenius")
- Natalie Lochner ("Tiling Fringed Chessboards with Dominoes")
- Caroline Lucheta ("On a Class of Permutation Polynomials")
- Todd Mateer ("Video Poker in South Carolina: A Mathematical Study")
- Andrea McMakin ("Characterization, Enumeration, and Products of Involutory Matrices")
- Todd Michel ("One-Point Codes Using Places of Higher Degree")
- Shannon Purvis ("The Kings Problem: A Matrix Problem in Statistical Mechanics")
- Christopher Seawright ("Generalizations of Eisenstein's Criteria for Irreducibility of Polynomials")
- Padmapani Seneviratne ("Permutation Decoding for Binary Codes of Lattice Graphs")
- Ethan Smith ("On Easily Computable Subspaces of Modular Forms of Weight 3/2")
- Andreas Tsolakis ("Points of Finite Fields of High Order")
- John Villalpando ("Some Computing Aspects of Broadcasting and Gossiping")
- Zach Voller ("A Construction of a Nonlinear Code")
Recent Ph.D. Graduates (dissertation title)
- Laura L. Carpenter ("Designs and Codes from Hyperovals")
- Kelle L. Clark ("Bounds for the Minimum Weight of the Dual Codes of Some Classes of Designs")
- Peng Ding ("Minimum-Weight Generators for Generalized Reed-Muller Codes")
- Jeffrey Farr ("Computing Grobner Bases with Applications to Multivariate Pade Approximation
and Algebraic Coding Theory")
- Mark Fitch ("Slope Critical Configurations Generated from Regular Polygons")
- Jira Limbupasiriporn ("Partial Permutation Decoding for Codes from Designs and Finite Geometries")
- Prasit Limbupasiriporn ("Hidden Subgroup Problem in Quantum Computing")
- Donald Mills ("Root-Based Polynomial Compositions over Finite Fields")
- Daniel Pillone ("Rankings and Minimal Rankings of Graphs")
- Virginia Rodrigues ("Multivariate Polynomials: Irreducibility and Grobner Bases")
- Franklin Shobe ("On a Class of Steiner Systems and Their Codes")
- Tim Teitloff ("Permutation Polynomials on Unions of Algebras with Applications to Symmetric Matrices")
- John Villalpando ("Graph Parameters: Channel Assignment as Related to L(2,1)-Colorings, and Domination Parameters")
- Charles Wallis ("Domination Parameters of Graphs of Designs and Chessboard Graphs")
Current Ph.D. Students (dissertation advisor)
Additional Algebra/Discrete Math Links
Last Updated: June 30, 2005
Send comments to:
shierd@clemson.edu