Morse theory of stratified spaces has uses everywhere from pure mathematics topics such as braid groups and representations to robot motion planning and potential theory.A popular application in pure mathematics is Morse theory on manifolds with boundary, and manifolds with corners.…
This is outlined in the article on braid theory.Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.Braid groups were introduced explicitly by Emil Artin in 1925, although (as Wilhelm Magnus pointed out in 1974) they were already implicit in Adolf Hurwitz's work on monodromy (1891).…
Patrick Dehornoy (born September 11, 1952 in Rouen) is a mathematician at the University of Caen who works on set theory and algebra.He found one of the first applications of large cardinals to algebra by constructing a certain left-invariant total order, called the Dehornoy order, on the braid group.…
In mathematics, a double affine braid group is a group containing the braid group of an affine Weyl group.Their group rings have quotients called double affine Hecke algebras in the same way that the group rings of affine braid groups have quotients that are affine Hecke algebras.…
Non-commutative cryptography is the area of cryptology where the cryptographic primitives, methods and systems are based on algebraic structures like semigroups, groups and rings which are non-commutative.One of the earliest application of a non-commutative algebraic structure for cryptographic purposes was the use of braid groups to develop cryptographic protocols.Later several other non-commutative structures like Thompson groups, polycyclic groups, Grigorchuk groups, and matrix groups have been identified as potential candidates for cryptographic applications.…
By forgetting how the strands twist and cross, every braid on strands determines a permutation on elements.This assignment is onto, compatible with composition, and therefore becomes a surjective group homomorphism from the braid group into the symmetric group.…
Their group rings have quotients called affine Hecke algebras.They are subgroups of double affine braid groups.…
The braid group can be shown to be isomorphic to the mapping class group of a punctured disk with punctures.This is most easily visualized by imagining each puncture as being connected by a string to the boundary of the disk; each mapping homomorphism that permutes two of the punctures can then be seen to be a homotopy of the strings, that is, a braiding of these strings.…
In mathematics, an affine braid group is a braid group associated to an affine Coxeter system.Their group rings have quotients called affine Hecke algebras.…
In mathematics the Lawrence-Krammer representation is a representation of the braid groups.It fits into a family of representations called the Lawrence representations.…
One classical such representation is Burau representation, where the matrix entries are single variable Laurent polynomials.It had been a long-standing question whether Burau representation was faithful, but the answer turned out to be negative for, More generally, it was a major open problem whether braid groups were linear.In 1990, Ruth Lawrence described a family of more general "Lawrence representations" depending on several parameters.…
The loop braid group is a mathematical group structure that is used in some models of theoretical physics to model the exchange of particles with loop-like topologies within three dimensions of space and time.The basic operations which generate a loop braid group for n loops are exchanges of two adjacent loops, and passing one adjacent loop through another.…
Lawrence followed her first degree with a second degree in physics in 1986 and a D.Phil in mathematics at Oxford in June 1989, at the age of 17.Her thesis title was Homology representations of braid groups and her thesis adviser was Sir Michael Atiyah.Lawrence's first academic post was at Harvard University, where she became a Junior Fellow in 1990 at the age of 19.…
In one dimensional quantum systems, R is the scattering matrix and if it satisfies the Yang-Baxter equation then the system is integrable.The Yang-Baxter equation also shows up when discussing knot theory and the braid groups where R corresponds to swapping two strands.Since one can swap three strands two different ways, the Yang-Baxter equation enforces that both paths are the same.…
Under the dimension condition Y will be connected.With this definition, then, we can call the braid group of X with n strings the fundamental group of Y (for any choice of base point - this is well-defined up to isomorphism).The case where X is the Euclidean plane is the original one of Artin.…
The braid group B 3 in turn is isomorphic to the knot group of the trefoil knot.The quotients by congruence subgroups are of significant interest.…
In statistical mechanics, the Temperley-Lieb algebra is an algebra from which are built certain transfer matrices, invented by Neville Temperley and Elliott Lieb.It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann algebras.…
Their group rings have quotients called double affine Hecke algebras in the same way that the group rings of affine braid groups have quotients that are affine Hecke algebras.For affine A n groups, the double affine braid group is the fundamental group of the space of n distinct points on a 2-dimensional torus.…
In mathematics, low-dimensional topology is the branch of topology that studies manifolds of four or fewer dimensions.Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups.It can be regarded as a part of geometric topology.…
Larsen is known for his research in arithmetic algebraic geometry, combinatorial group theory, combinatorics, and number theory.He has written highly cited papers on domino tiling of Aztec diamonds, topological quantum computing, and on the representation theory of braid groups.…
