Аннотация:

1) Conformal transformations. Definition. Infinitesimal conformaltransformations. Lie derivative of metric tensor with respect to agenerator of conformal transformation.

2) Mobius transformations of R^n. The Lie algebra of Mobius group.Theorem: For n>=3 the Lie algebra of local conformal vector fields in R^nis finite-dimensional and coincide with the Lie algebra of Mobius group.

3) Mobius transformations of R^n from isometries of R^{n+1,1}.

4) Weil, Schouten and Cotton tensors. The formula expressing the Riemanntensor in terms of metric and Ricci tensors for n=3. Transformationsproperties of the Weil and Cotton tensors under conformal changes ofRiemannian metric.

5) Necessary and sufficient conditions for conformal flatness (withoutproof).

6) Isothermal coordinate on two-dimensional surfaces. Beltrami equation.All 2-dimensional Riemannian manifolds are conformally flat. Localconformal maps are holomorphic or anti-holomorphic maps.

7) Genus of hyperelliptic Riemann surfaces. Vector fields anddifferentials on Riemann surfaces. Holomorphic differentials onhyperelliptic Riemann surfaces.

8) Tensor bundles on Riemann surfaces. The Riemann-Roch theorem (withoutproof).

9) Teichm\uller space and moduli space for tori. Fundamental domain in theTeichmuller space for tori.

10) Beltrami differentials on Riemann surfaces as generators of conformalstructures deformations. The tangent space to the moduli space.

11) Action of the Witt algebra (vector fields on the circle) at the modulispace. Variations of the Riemann period matrix under the action of vectorfields.

12) Virasoro algebra action on the Dirac space associated to a Riemannsurface. The origin of the central charge.

Список всех тем лекций

Лекция 1. Introduction to the conformal geometry.

Лекция 2. Conformal maps.

Лекция 3. Riemann tensor in dimension 3.

Лекция 4. Weyl and Cotton tensors.

Лекция 5. Weyl and Cotton tensors (continued).

Лекция 6. Cotton tensor.

Лекция 7. Beltrami equation.

Лекция 8. Introduction to the Riemann surfaces.

Лекция 9. Riemann bilinear relations I.

Лекция 10. Riemann bilinear relations II.

Лекция 11. Divisors and Riemann-Roch theorem.

Лекция 12. Moduli space.

Лекция 13. Moduli space II.