A classical result is the Picard Lefschetz formula, detailing how the monodromy round the singular fiber acts on the vanishing cycles, by a shear mapping.
32.
The flag variety is a quotient of a Lie group by a parabolic subgroup, and the monodromy group is an arithmetic subgroup of the Lie group.
33.
For generic monodromy data, the answer to Hilbert's twenty-first problem is'yes'- as was first proved by Josip Plemelj.
34.
This is in contrast to transcendental and logarithmic branch points, that is, points at which a multiple-valued function has nontrivial monodromy and an essential singularity.
35.
In terms of the inverse global analytic function " ? " & minus; 1, branch points are those points around which there is nontrivial monodromy.
36.
Another place in which tannakian categories have been used is in connection with the Grothendieck Katz p-curvature conjecture; in other words, in bounding monodromy groups.
37.
In complex analysis, the "'monodromy theorem "'is an important result about analytic continuation of a complex-analytic function to a larger set.
38.
Recently important connections between this theory and complex dynamics, particularly the notion of iterated monodromy groups, have been uncovered in the work of Volodymyr Nekrashevych . and others.
39.
Since the functions " ce z " are defined on the whole affine line " A " 1, the monodromy of this flat connection is trivial.
40.
He also worked with hypergeometric differential equations in 1857 using complex analytical methods and presented the solutions through the behavior of closed paths about singularities ( described by the monodromy matrix ).
