T-theory is a branch of

discrete mathematics Discrete mathematics is the study of mathematical structures that can be considered "discrete" (in a way analogous to discrete variables, having a bijection with the set of natural numbers) rather than "continuous" (analogously to continu ...

dealing with analysis of

tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, including only woody plants with secondary growth, plants that are ...

s and discrete

metric spaces In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setti ...

General history

T-theory originated from a question raised by

Manfred Eigen Manfred Eigen (; 9 May 1927 – 6 February 2019) was a German biophysical chemist who won the 1967 Nobel Prize in Chemistry for work on measuring fast chemical reactions. Eigen's research helped solve major problems in physical chemistry and ...

in the late 1970s. He was trying to fit twenty distinct t-RNA

molecule A molecule is a group of two or more atoms held together by attractive forces known as chemical bonds; depending on context, the term may or may not include ions which satisfy this criterion. In quantum physics, organic chemistry, and bio ...

s of the ''

Escherichia coli ''Escherichia coli'' (),Wells, J. C. (2000) Longman Pronunciation Dictionary. Harlow ngland Pearson Education Ltd. also known as ''E. coli'' (), is a Gram-negative, facultative anaerobic, rod-shaped, coliform bacterium of the genus '' Esc ...

'' bacterium into a

. An important concept of T-theory is the

tight span In metric geometry, the metric envelope or tight span of a metric space ''M'' is an injective metric space into which ''M'' can be embedded. In some sense it consists of all points "between" the points of ''M'', analogous to the convex hull of a ...

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 In metric geometry, an injective metric space, or equivalently a hyperconvex metric space, is a metric space with certain properties generalizing those of the real line and of L∞ distances in higher- dimensional vector spaces. These properties ca ...

that contains ''X''. John Isbell was the first to discover the tight span in 1964, which he called the injective envelope. Andreas Dress independently constructed the same construct, which he called the tight span.

Application areas

* Phylogenetic analysis, which is used to create phylogenetic trees. *

Online algorithm In computer science, an online algorithm is one that can process its input piece-by-piece in a serial fashion, i.e., in the order that the input is fed to the algorithm, without having the entire input available from the start. In contrast, an o ...

s - ''k''-server problem

Recent developments

Bernd Sturmfels Bernd Sturmfels (born March 28, 1962 in Kassel, West Germany) is a Professor of Mathematics and Computer Science at the University of California, Berkeley and is a director of the Max Planck Institute for Mathematics in the Sciences in Leipzig si ...

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

References

* * * * Metric geometry Trees (data structures) {{combin-stub