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- 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.
- Alternatively, one can define the braid group purely algebraically via the braid relations, keeping the pictures in mind only to guide the intuition.
- 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.
- One of the earliest application of a non-commutative algebraic structure for cryptographic purposes was the use of braid groups to develop cryptographic protocols.
- This assignment is onto, compatible with composition, and therefore becomes a surjective group homomorphism from the braid group into the symmetric group.
- 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.
- In mathematics, an affine braid group is a braid group associated to an affine Coxeter system.
- In mathematics the Lawrence-Krammer representation is a representation of the braid groups.
- 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.
- 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.
- Her thesis title was Homology representations of braid groups and her thesis adviser was Sir Michael Atiyah.
- The Yang-Baxter equation also shows up when discussing knot theory and the braid groups where R corresponds to swapping two strands.
- 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 braid group B 3 in turn is isomorphic to the knot group of the trefoil knot.
- It is also related to integrable models, knot theory and the braid group, quantum groups and subfactors of von Neumann 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.
- Representative topics are the structure theory of 3-manifolds and 4-manifolds, knot theory, and braid groups.
- He has written highly cited papers on domino tiling of Aztec diamonds, topological quantum computing, and on the representation theory of braid groups.