The study of integers and prime numbers has intrigued people since ancient times and continues to be an active field of research today. Recent breakthroughs in this area have been significant. For instance, Wiles' solution to Fermat's Last Theorem in 1995 has stimulated much related research activity that persists to the present, such as the recent solution by Khare and Wintenberger of Serre's conjecture. Analytic number theory, which employs calculus and complex analysis to study integers, is concerned with the Riemann hypothesis, a Clay Millennium Problem. In this area, there have been recent strides, including the Green-Tao proof that prime numbers appear in arbitrarily long arithmetic progressions. The Langlands Program is a broad set of conjectures that connects number theory with representation theory, and this field also has connections with cryptography, which has applications in computer science.”
Combinatorics is a field of study concerned with discrete objects, and it has broad applications in mathematics and science. For instance, combinatorial reasoning plays a crucial role in solving biological challenges such as deciphering genomes and constructing phylogenetic trees. Researchers in quantum gravity also rely on combinatorial techniques to evaluate integrals, while many problems in statistical mechanics are formulated as combinatorial problems. In recognition of the significance of combinatorics, three of the four Fields Medals awarded in 2006 were related to combinatorial work. Specifically, Okounkov's research on random matrices and Kontsevich's conjecture, Tao's work on primes in arithmetic progression, and Werner's work on percolation were all awarded Fields Medals.
Graph theory is a branch of mathematics that deals with the study of graphs, which are abstract representations of objects and their relationships. Combinatorics is closely related to graph theory, as it provides the mathematical framework for analyzing the structures of graphs. In fact, many problems in graph theory are formulated as combinatorial problems, and combinatorial methods are often used to solve these problems.”
Department Members in This Field
Faculty
- Dr Benedict Vasco Normenyo Diophantine Equations, Lucas Sequences and their Generalizations
- Dr Kenneth Dadedzi Spectral Graph Theory
Graduate Students
- TBA