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

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.

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The Convenient Setting of Global Analysis

The diamond lemma for ring theory

IN this paper, we present a framework for studying moduli spaces of finite dimensional representations of an arbitrary finite dimensional algebra A over an algebraically closed field k. (The abelian category of such representations is denoted by mod-A.) Our motivation is twofold. Firstly, such moduli spaces should play an important role in organising the representation theory of wild algebras. Secondly, such moduli spaces can be identified with moduli spaces of vector bundles on special projective varieties. This identification is somewhat hidden in earner work ([6], [7]) but has become more explicit recently ([4], [12]). It can now be seen to arise from a 'tilting equivalence' between the derived category of mod-A and the derived category of coherent sheaves on the variety. It is well-established that mod-A is equivalent to the category of representations of an arrow diagram, or 'quiver', Q by linear maps satisfying certain 'admissible' relations. Thus, the problem of classifying A -modules with a fixed class in the Grothendieck group K0(mod-A), represented by a 'dimension vector' a, is converted into one of classifying orbits for the action of a reductive algebraic group GL(a) on a subvariety VA(a) of the representation space 9t{Q, a) of the quiver. Now, the moduli spaces provided by classical invariant theory ([1], [18]) are not interesting in this context. This is because the classical theory only picks out the closed GL(a)-orbits in VA{a), which correspond to semisimple /4-modules, and the quiver Q is chosen so that there is only one semisimple A -module of each dimension vector. On the other hand, we can apply Mumford's geometric invariant theory, with the trivial linearisation twisted by a character x of GL(a), which restricts our attention to an open subset of VA(a), consisting of semistable representations. Within this open set there are more closed orbits and the corresponding algebraic quotient is then a more interesting moduli space. In fact, this approach also has a classical flavour, since it involves the relative (or semi-) invariants of the GL(a) action. The main purpose of this paper is to show that the notions of semistability and stability, that arise from the geometric invariant theory, coincide with more algebraic notions, expressed in the language of mod-A Indeed, the definition is formulated for an arbitrary abelian category as follows:

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Moduli of representations of finite dimensional algebras

The invariant theory of n × n matrices

Complete symmetric varieties

General Methods and Ideas.- Symmetric Functions.- Theory of Algebraic Forms.- Lie Algebras and lie Groups.- Tensor Algebra.- Semisimple Algebras.- Algebraic Groups.- Group Representations.- Tensor Symmetry.- Semisimple Lie Groups and Algebras.- Invariants.- Tableaux.- Standard Monomials.- Hilbert Theory.- Binary Forms.

Lie Groups: An Approach through Invariants and Representations

The motivation stems from our attempt to understand Drinfeld's construction (el. [Dr2]) of special solutions of the Khniznik-Zamolodchikov equation (of. [K-Z]) with some prescribed asymptotic behavior and its consequences for some universal constructions associated to braiding: universal unipotent monodromy representations of braid groups, the construction of a universal Vassiliev invariant for knots, braided categories etc. The K-Z connection is a special flat meromorphic connection on C ~ with simple poles on a family of hyperplanes. It turns out that the prescription of the asymptotic behavior for such connections is controlled by the geometry of a suitable modification of C ~ in which the union of the polar hyperplanes is replaced by a divisor with normal crossings. In the process of developing this geometry we realized that our constructions could be developed more generally for subspace arrangements and became aware of the paper of Fulton-MaePherson [F-M] in which a Hironaka model is described for the complement of the big diagonal in the power of a smooth variety X. It became clear to us that our techniques were quite similar to theirs and so we adopted their notation of nested set in the appropriate general form. Although we work in a linear subspaces setting it is clear that the methods are essentially local and one can recover their results from our analysis applied to certain special configurations of subspaces. In fact the theory can be applied whenever we have a subvariety of a smooth variety which locally (in the gtale topology) appears as a union of subspaces.

Claudio Procesi

Papers

The invariant theory of n × n matrices

Complete symmetric varieties

Lie Groups: An Approach through Invariants and Representations

Wonderful models of subspace arrangements

A characteristic free approach to invariant theory