T-theory

T-theory is a branch of discrete mathematics dealing with analysis of trees and discrete metric spaces.

General history

As per Andreas Dress, T-theory originated from a question raised by Manfred Eigen, a recipient of the Nobel Prize in Chemistry, in the late seventies. He was trying to fit twenty distinct t-RNA molecules of the E. Coli bacterium into a tree.

One of the most important concepts of T-theory is the tight span of a metric space. If X is a metric space, the tight span T(X) of X is, up to isomorphism, the unique minimal injective metric space that contains X. John Isbell was the first to discover the tight span in 1964, which he called the injective envelope. Dress independently constructed the same construct, which he called the tight span.

Application areas

• Phylogenetic analysis, which is used to create phylogenetic trees.
• Online algorithms - k-server problem

Recent developments

• Bernd Sturmfels, Professor of Mathematics and Computer Science at Berkeley, and Josephine Yu classified six-point metrics using T-theory.

References

• Hans-Jurgen Bandelt and Andreas Dress (1992). "A canonical decomposition theory for metrics on a finite set". Advances in Mathematics 92: 47–105. doi:10.1016/0001-8708(92)90061-O. 
• A. Dress, V. Moulton and W. Terhalle (1996). "T-theory: An Overview". European Journal of Combinatorics 17 (2–3): 161–175. doi:10.1006/eujc.1996.0015. 
• John Isbell (1964). "Six theorems about metric spaces". Comment. Math. Helv. 39: 65–74. doi:10.1007/BF02566944. 
• Bernd Sturmfels and Josephine Yu (2004). "Classification of Six-Point Metrics". The Electronic Journal of Combinatorics 11.