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

We define new invariants of 3d Calabi-Yau categories endowed with a stability structure. Intuitively, they count the number of semistable objects with fixed class in the K-theory of the category ('number of BPS states with given charge' in physics language). Formally, our motivic DT-invariants are elements of quantum tori over a version of the Grothendieck ring of varieties over the ground field. Via the quasi-classical limit 'as the motive of affine line approaches to 1' we obtain numerical DT-invariants which are closely related to those introduced by Behrend. We study some properties of both motivic and numerical DT-invariants including the wall-crossing formulas and integrality. We discuss the relationship with the mathematical works (in the non-triangulated case) of Joyce, Bridgeland and Toledano-Laredo, as well as with works of physicists on Seiberg-Witten model (string junctions), classification of N=2 supersymmetric theories (Cecotti-Vafa) and structure of the moduli space of vector multiplets. Relating the theory of 3d Calabi-Yau categories with distinguished set of generators (called cluster collection) with the theory of quivers with potential we found the connection with cluster transformations and cluster varieties (both classical and quantum).

Stability structures, motivic Donaldson-Thomas invariants and cluster transformations

Convex Analysis: (pms-28)

Groupes et algébres de lie

We set up the technique of discriminant-forms, which allows us to transfer many results for unimodular symmetric bilinear forms to the nonunimodular case and is convenient in calculations. Further, these results are applied to Milnor's quadratic forms for singularities of holomorphic functions and also to algebraic geometry over the reals. Bibliography: 57 titles.

https://iopscience.iop.org/article/10.1070/IM1980v014n01ABEH001060/pdf

Integral Symmetric Bilinear Forms and Some of Their Applications

This monograph gives a coherent account of the theory of isolated singularities of complete intersections. One encounters such singularities often as the central fibres of analytic map-germs; that is why such map-germs (deformations) receive here a great deal of attention. The work treats both the topological side--including vanishing cycles and monodromy--and the analytic side--including properties of the discriminants of deformations--and explores the connections between them. It ends with a proof that the discriminant of the semi-universal deformation of an A-D-E singularity is isomorphic to the discriminant of the associated Coxeter group.

Isolated Singular Points on Complete Intersections

Construction of canonical stratifications.- Stratifications and flows.- Unfoldings of smooth map-germs.- Proof of the topological stability theorem.

Topological stability of smooth mappings

This paper describes a new invariant theory for root systems. We briefly outline the main geometric applications; for precise results we refer the reader to the relevant theorems. Let R be a root system, R v its dual and QV the lattice generated by R v. If E is an elliptic curve over tr, then A : = QV | is an abelian variety on which the Weyl group W of R acts. It is easily shown that there is a W-invariant symmetric bilinear form I on QV such that 1 takes the value 2 on the roots of smallest length. It follows from the theorem of Appel l -Humbert that there exists a line bundle 5O over A of "Chern c lass"I such that the action of W on A lifts to an action of W on 5 ~ with the property that W leaves the fibre over 0cA fixed. Then 5 ~ 1 is ample and the algebra of W-invariant sections

Root systems and elliptic curves

Motivic measures

We prove that any product of tautological classes of ℳ
 g
 of degreed (in the Chow ring of ℳ
 g
 with rational coefficients) vanishes ford>g−2 and is proportional to the class of the hyperelliptic locus in degreeg−2.

Eduard Looijenga

Papers

Isolated Singular Points on Complete Intersections

Topological stability of smooth mappings

Root systems and elliptic curves

Motivic measures

On the tautological ring of Mg