Course Identification

*First degree in mathematics.

*Familiarity with basic algebraic topology (the concepts of homology and cohomology) and with basics on manifolds.

-In the first class I will introduce the Witten KdV conjecture.

-The first lectures will be devoted for studying moduli spaces of stable sphere, elliptic curves, Teichmuller spaces and psi classes.

-In parallel the TA will concentrate on the basics of vector bundles, characteristic classes, modular forms and hyperbolic geometry.

-The next lectures will be devoted to Kontsevich's proof of Witten's conjecture.

-The parallel TA sessions will be devoted to background on integrable hierarchies, Virasoro representations and matrix integrals.

-Then, depending on our progress we will discuss generalizations of the theory.

We will get to know with Witten's KdV conjecture and Kontsevich's proof.

Along the way we'll become familiar with moduli spaces of curves, vector bundles and their characteristic classes, hyperbolic geometry, differential forms, Feynman integrals, matrix integrals and integrable hierarchies.

If time permits we will discuss generalizations of Witten's-Kontsevich's theory.