We show that two natural approaches to quantum gravity coincide. This identity is nontrivial and relies on the equivalence of each approach to KdV equations. We also investigate related mathematical problems.

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Intersection theory on the moduli space of curves and the matrix Airy function

© Publications mathématiques de l’I.H.É.S., 1977, 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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Infinitesimal computations in topology

2D gravity and random matrices

Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

/pdf/the-convenient-setting-of-global-analysis-378u9vden9.pdf

The Convenient Setting of Global Analysis

Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An important special case The even-dimensional case The odd-dimensional case The bounded odd-dimensional case The bounded even-dimensional case Completion of the proof Patterns of application: Manifold structures on Poincare complexes Applications to submanifolds Submanifolds: Other techniques Separating submanifolds Two-sided submanifolds One-sided submanifolds Calculations and applications: Calculations: Surgery obstruction groups Calculations: The surgery obstructions Applications: Free actions on spheres General remarks An extension of the Atiyah-Singer $G$-signature theorem Free actions of $S^1$ Fake projective spaces (real) Fake lens spaces Applications: Free uniform actions on euclidean space Fake tori Polycyclic groups Applications to 4-manifolds Postscript: Further ideas and suggestions: Recent work Function space methods Topological manifolds Poincare embeddings Homotopy and simple homotopy Further calculations Sullivan's results Reformulations of the algebra Rational surgery References Index.

Surgery on compact manifolds

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Arithmetic subgroups of algebraic groups

Differential Operators on a Semisimple Lie Algebra

/pdf/discrete-series-for-semisimple-lie-groups-ii-explicit-4m6asp439s.pdf

Discrete series for semisimple Lie groups. II: Explicit determination of the characters

Then tr 4̂ is actually independent of the choice of this base. Let VT denote the set of a l l /G^ i (G) such that ir(f) is of the trace class. Then Vr is a linear subspace of Li(G). Put 0 . ( / ) = tr *•(ƒ) ( f £ F T ) . Then ®T is a linear function on VT which we may call the character of 7T. Of course this concept would be useful only when the space V* is fairly large. Let 8(G) denote the set of all equivalence classes of irreducible unitary representations of G. It is easy to see that for any representation x, VT and ©^ depend only on the class co of 7r. Hence we may de-

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Harmonic analysis on semisimple Lie groups

I . THEORY ON TIIE LIE ALGEBRA l~educ t io f f of T h e o r e m 1 to t h e s e m i s i m p l e case . . . . . . . . . . . . . . . . . 243 S e c o n d r e d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249 T h i r d r e d u c t i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 250 N e w e x p r e s s i o n s fo r J a n d J ' . . . . . . . . . . . . . . . . . . . . . . . . . . 252 P r o o f of L e m m a 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 A c o n s e q u e n c e of T h e o r e m 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 258 S o m e e l e m e n t a r y f ac t s a b o u t t e m p e r e d d i s t r i b u t i o n s . . . . . . . . . . . . . . 259 P r o o f of L e m m a 17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 A n a u x i l i a r y r e s u l t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265 P r o o f of L e m m a 21 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 R e c a p i t u l a t i o n of s o m e e l e m e n t a r y f ac t s . . . . . . . . . . . . . . . . . . . . . 268 P r o o f of L e m m a 26 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 T e m p e r e d a n d i n v a r i a n t e i g e n d i s t r i b u t i o n s . . . . . . . . . . . . . . . . . . . 271 P r o o f of L e m m a 30 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 T h e d i s t r i b u t i o n T~ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 A p p l i c a t i o n of T h e o r e m 1 t o T~ . . . . . . . . . . . . . . . . . . . . . . . . . 281 P r o o f of L e m m a 41 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283

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Harish-Chandra

Papers

Arithmetic subgroups of algebraic groups

Differential Operators on a Semisimple Lie Algebra

Discrete series for semisimple Lie groups. II: Explicit determination of the characters

Harmonic analysis on semisimple Lie groups

Discrete series for semisimple Lie groups I: Construction of invariant eigendistributions