Algebraic K (1)

Algebraic K-theory is a branch of mathematics that deals with the abstract algebraic structures known as K-groups. These groups are used to encode detailed information about various geometric, algebraic, and arithmetic objects, but they are difficult to compute. One of the main challenges in the field is to find a way to compute the K-groups of the integers. The study of K-theory was initiated by Alexander Grothendieck in the 1950s as part of his research on intersection theory on algebraic varieties. The lower K-groups, such as K0 and K1, were discovered first, and have connections to concepts like vector space dimension, the Picard group of a ring, and the group of units in a ring. The higher K-groups, such as K2 and beyond, were more difficult to define and understand, but their study has led to important developments in areas like class field theory, the Hilbert symbol, and the solvability of quadratic equations.

Algebraic K-theory is a branch of mathematics that deals with the study of geometric, algebraic, and arithmetic objects known as K-groups. These groups contain detailed information about the original object, but they can be challenging to calculate. The study of K-theory has connections to fields such as geometry, topology, ring theory, and number theory. It was developed by Alexander Grothendieck in the 1950s through his work on intersection theory in algebraic varieties. K-theory has many applications, including the Grothendieck-Riemann-Roch theorem and the study of motivic cohomology and Chow groups. It also has connections to classical number theory topics like quadratic reciprocity and the embedding of number fields into the complex and real numbers. The higher K-groups, which describe the K-groups of rings, were defined by Daniel Quillen, and the basic properties of the higher K-groups of algebraic varieties were explored by Robert Thomason.

In the 19th century, Bernhard Riemann and Gustav Roch established the Riemann-Roch theorem, which states that if X is a Riemann surface, the sets of meromorphic functions and meromorphic differential forms on X form vector spaces. These vector spaces are finite dimensional if X is projective, and their difference in dimensions is equal to the degree of a line bundle on X plus one minus the genus of X. In the mid-20th century, Friedrich Hirzebruch extended the Riemann-Roch theorem to all algebraic varieties, resulting in the Hirzebruch-Riemann-Roch theorem. This theorem states that the Euler characteristic of a vector bundle on an algebraic variety (the alternating sum of the dimensions of its cohomology groups) is equal to the Euler characteristic of the trivial bundle plus a correction factor from the characteristic classes of the vector bundle.

In 1957, Alexander Grothendieck introduced the subject of K-theory in his generalization of the Hirzebruch-Riemann-Roch theorem, known as the Grothendieck-Riemann-Roch theorem. Grothendieck assigned to each vector bundle on a smooth algebraic variety X an invariant, called its class, and defined the group K(X) as the set of all classes on X. If a proper morphism f: X → Y to a smooth variety Y exists, then it determines a homomorphism f*: K(X) → K(Y), called the pushforward. The Grothendieck-Riemann-Roch theorem states that the pushforward in K-theory followed by the Chern character and Todd class of Y is equal to the Chern character and Todd class of X followed by the pushforward for Chow groups. The group K(X), now known as K0(X), can also be defined for non-commutative rings by replacing vector bundles with projective modules. K-theory was extended to topology by Atiyah and Hirzebruch, who defined topological K-theory. This associates a sequence of groups Kn(X) to each topological space X satisfying certain conditions, with the exception of the normalization axiom in the Eilenberg-Steenrod axioms. However, the definition of the higher Kn(X) was not clear in the setting of algebraic varieties, and technical issues often restricted the definition of Kn to rings rather than varieties.

The concept of algebraic K-theory, introduced by Alexander Grothendieck in the 1950s, assigns groups called K-groups to geometric, algebraic, and arithmetic objects. These groups contain detailed information about the original object, but are challenging to compute. The lower K-groups, including K0 and K1, were discovered first, with K0 related to vector space dimension and K1 related to the group of units. The higher K-groups, including K2 and beyond, were more difficult to define and were only able to be defined for rings, not varieties. In topology, K-theory is related to the Betti numbers of a manifold and the concept of simple homotopy equivalence. The Whitehead group, a predecessor to K-theory, was introduced by J.H.C. Whitehead and later found to be a quotient of K1. It was used to disprove the Hauptvermutung, a conjecture about the invariance of Betti numbers under triangulation.

Throughout the 19th century, the physics of Isaac Newton reigned supreme in the physical sciences. However, towards the end of the century, it was observed that matter and energy at the atomic and subatomic level behaved differently from matter and energy at a human-perceivable level. Newton's laws of motion, for example, were unable to accurately predict the location, path, and velocity of very small particles, even though these same laws were accurate in our everyday universe. This failure led to the development of a new theory, quantum theory, which was introduced in 1900 by Max Planck and further developed by other scientists. Quantum theory and its associated mathematics are able to accurately describe the behavior of small particles of energy and matter. In some cases, the math required for quantum physics involves the use of complex and imaginary numbers, which are not often used in other disciplines such as engineering. The advancements made possible through quantum theory include the creation of the laser, the electron microscope, and the transistor. It also played a crucial role in the development of modern computers and smart phones.