인공고관절의 세라믹 볼 헤드의 기계적 안정성 평가를 위한 3 차원 유한요소 해석

These lecture notes are devoted to the theory of equations of associativity describing geometry of moduli spaces of 2D topological field theories. Introduction. Lecture 1. WDVV equations and Frobenius manifolds. {Appendix A.} Polynomial solutions of WDVV. {Appendix B.} Symmetriies of WDVV. Twisted Frobenius manifolds. {Appendix C.} WDVV and Chazy equation. Affine connections on curves with projective structure. Lecture 2. Topological conformal field theories and their moduli. Lecture 3. Spaces of isomonodromy deformations as Frobenius manifolds. {Appendix D.} Geometry of flat pencils of metrics. {Appendix E.} WDVV and Painlev\'e-VI. {Appendix F.} Branching of solutions of the equations of isomonodromic deformations and braid group. {Appendix G.} Monodromy group of a Frobenius manifold. {Appendix H.} Generalized hypergeometric equation associated to a Frobenius manifold and its monodromy. {Appendix I.} Determination of a superpotential of a Frobenius manifold. Lecture 4. Frobenius structure on the space of orbits of a Coxeter group. {Appendix J.} Extended complex crystallographic groups and twisted Frobenius manifolds. Lecture 5. Differential geometry of Hurwitz spaces. Lecture 6. Frobenius manifolds and integrable hierarchies. Coupling to topological gravity.

http://cds.cern.ch/record/265648/files/9407018.pdf

Geometry of 2D topological field theories

Theories with Gauge-Mediated Supersymmetry Breaking

To Professor Shoshichi Kobayashi on his 60th birthday 1. Introduction. In this paper we shall introduce a new family of varieties, which we call quiver varieties, and study their geometric structures. They have close relation to the singularity theory and the representation theory of the Kac-Moody algebras. Our original motivation was to study solutions of the anti-self-dual Yang-Mills equations on a particular class of 4-dimensional noncompact complete manifolds, the so-called ALE spaces (or the ALE gravitational instantons), which were constructed by Kronheimer [Krl]. In [KN] we gave a description of the framed moduli space of all solutions in terms of solutions of a system of quadratic equations (called the ADHM equations) for representations of a quiver on an affine, simply laced Dynkin graph. It is an analogue of the description, given by Atiyah, Drinfeld, Hitchin, and Manin [ADHM], of the moduli space for IR 4 (or S4) in terms of solutions of a quadratic equation for certain finite-dimensional matrices. Once we set aside their gauge-theoretic origin, there is no longer reason to restrict ourselves to affine Dynkin graphs. Definitions can be generalized to arbitrary finite graphs. We get what we call quiver varieties. We study geometric structures of quiver varieties in this paper. In [Nal] it was noticed that the moduli space of anti-self-dual connections on ALE spaces has a hyper-K/ihler structure, namely a Riemannian metric equipped with three endo-morphisms I, J, K of the tangent bundle which satisfy the relations of quaternion algebra and are covariant constant with respect to the Levi-Civita connection: The same holds for general quiver varieties. In particular, quiver varieties have holomorphic symplectic forms. We study further properties of the quiver variety, such as a natural *-action, symplectic geometry, topology, and so on. As ALE spaces closely related to simple singularities, quiver varieties have very special kinds of singularities that enjoy very nice properties. Surprisingly, the ADHM equation appears in a very different context. In [L3] Lusztig used it to construct \"canonical bases\" of the part U-of the quantized enveloping algebra U associated by Drinfeld and Jimbo to the graph. Motivated by his results, we give a geometric construction of irreducible highest-weight integrable representations of the Kac-Moody algebra associated to the graph (Theorem 10.14). The weight space of the representation space will be given as a vector space consisting of constructible functions on a Lagrangian subvariety of a quiver variety. The action of the Kac-Moody …

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Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

Groupes et algébres de lie

Smooth functions invariant under the action of a compact lie group

© Publications mathématiques de l’I.H.É.S., 1980, tous droits réservés. L’accès aux archives de la revue « Publications mathématiques de l’I.H.É.S. » (http:// www.ihes.fr/IHES/Publications/Publications.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

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Lifting smooth homotopies of orbit spaces

Representations of simple lie groups with regular rings of invariants

The orbit space of a representation of a compact Lie group has a natural semialgebraic structure. We describe explicit ways of finding the inequalities defining this structure, and we give some applications.

Inequalities defining orbit spaces

(0.0) This is an expanded version of my talk of the above title at the Rutgers Conference on Topological Methods in Algebraic Transformation Groups, April 4–8, 1988. I report on recent work which shows there is a strong interconnection between the theory of quotients in the algebraic category (reductive groups) and in the topological category (compact groups). It turns out that one can represent quotient spaces of reductive groups as quotient spaces of compact groups. The precise method of doing this has important topological consequences for quotients of reductive groups and also provides some new insight into quotients of compact groups. I hope that my exposition will make these results more accessible to all.

Gerald W. Schwarz

Papers

Smooth functions invariant under the action of a compact lie group

Lifting smooth homotopies of orbit spaces

Representations of simple lie groups with regular rings of invariants

Inequalities defining orbit spaces

The Topology of Algebraic Quotients