Differential Geometry and Binary Operations

Natural Operations in Differential Geometry

Differential geometry: a natural tool for describing symmetry operations.

Differential geometry provides a useful mathematical framework for describing the fundamental concepts in crystallography. The notions of point and associated vector spaces correspond to those of manifold and tangent space at a given point. A space-group operation is a one-to-one map acting on the manifold, whereas a point-group operation is a linear map acting between two tangent spaces of the...

Binary operations and groups

Example 1.3. The examples are almost too numerous to mention. For example, using +, we have (N,+), (Z,+), (Q,+), (R,+), (C,+), as well as vector space and matrix examples such as (Rn,+) or (Mn,m(R),+). Using subtraction, we have (Z,−), (Q,−), (R,−), (C,−), (Rn,−), (Mn,m(R),−), but not (N,−). For multiplication, we have (N, ·), (Z, ·), (Q, ·), (R, ·), (C, ·). If we define Q∗ = {a ∈ Q : a 6= 0}, ...

Binary Operations

Let A be a set. A unary operation on A is a function from A into A. A binary operation on A is a function from [:A, A :] into A. We adopt the following convention: u is a unary operation on A, o, o′ are binary operations on A, and a, b, c, e, e1, e2 are elements of A. In this article we present several logical schemes. The scheme BinOpEx deals with a non empty set A and a ternary predicate P , ...

Cohomology operations and algebraic geometry

This manuscript is based on a ten hours series of seminars I delivered in August of 2003 at the Nagoya Institute of Technology as part of the workshop on homotopy theory organized by Norihiko Minami and following the Kinosaki conference in honor of Goro Nishida. One of the most striking applications of homotopy theory in “exotic” contexes is Voevodsky’s proof of the Milnor Conjecture. This conj...