10 research outputs found
A unified Witten-Reshetikhin-Turaev invariant for integral homology spheres
We construct an invariant J_M of integral homology spheres M with values in a
completion \hat{Z[q]} of the polynomial ring Z[q] such that the evaluation at
each root of unity \zeta gives the the SU(2) Witten-Reshetikhin-Turaev
invariant \tau_\zeta(M) of M at \zeta. Thus J_M unifies all the SU(2)
Witten-Reshetikhin-Turaev invariants of M. As a consequence, \tau_\zeta(M) is
an algebraic integer. Moreover, it follows that \tau_\zeta(M) as a function on
\zeta behaves like an ``analytic function'' defined on the set of roots of
unity. That is, the \tau_\zeta(M) for all roots of unity are determined by a
"Taylor expansion" at any root of unity, and also by the values at infinitely
many roots of unity of prime power orders. In particular, \tau_\zeta(M) for all
roots of unity are determined by the Ohtsuki series, which can be regarded as
the Taylor expansion at q=1.Comment: 66 pages, 8 figure
Primary decomposition and the fractal nature of knot concordance
For each sequence of polynomials, P=(p_1(t),p_2(t),...), we define a
characteristic series of groups, called the derived series localized at P.
Given a knot K in S^3, such a sequence of polynomials arises naturally as the
orders of certain submodules of the sequence of higher-order Alexander modules
of K. These group series yield new filtrations of the knot concordance group
that refine the (n)-solvable filtration of Cochran-Orr-Teichner. We show that
the quotients of successive terms of these refined filtrations have infinite
rank. These results also suggest higher-order analogues of the p(t)-primary
decomposition of the algebraic concordance group. We use these techniques to
give evidence that the set of smooth concordance classes of knots is a fractal
set. We also show that no Cochran-Orr-Teichner knot is concordant to any
Cochran-Harvey-Leidy knot.Comment: 60 pages, added 4 pages to introduction, minor corrections otherwise;
Math. Annalen 201
The Characteristics and Interpretability of Land Surface Change and Implications for Project Design
Front-end process modeling in silicon
Front-end processing mostly deals with technologies associated to junction formation in semiconductor devices. Ion implantation and thermal anneal models are key to predict active dopant placement and activation. We review the main models involved in process simulation, including ion implantation, evolution of point and extended defects, amorphization and regrowth mechanisms, and dopant-defect interactions. Hierarchical simulation schemes, going from fundamental calculations to simplified models, are emphasized in this Colloquium. Although continuum modeling is the mainstream in the semiconductor industry, atomistic techniques are starting to play an important role in process simulation for devices with nanometer size features. We illustrate in some examples the use of atomistic modeling techniques to gain insight and provide clues for process optimization