Topics in Mathematics 2016/2017

Partendo dall'esempio delle curve ellittiche, darò una breve introduzione ad alcune varietà complesse Ricci piatte. Nel caso delle superfici, queste sono esclusivamente dei tori complessi o delle superfici semplicemente connesse dette K3. Dopo aver analizzato le loro proprietà in bassa dimensione, passeremo ai casi di dimensione più alta, guardando analogie e differenze.

Partendo dall'esempio delle curve ellittiche, darò una breve introduzione ad alcune varietà complesse Ricci piatte. Nel caso delle superfici, queste sono esclusivamente dei tori complessi o delle superfici semplicemente connesse dette K3. Dopo aver analizzato le loro proprietà in bassa dimensione, passeremo ai casi di dimensione più alta, guardando analogie e differenze.

Partendo dall'esempio delle curve ellittiche, darò una breve introduzione ad alcune varietà complesse Ricci piatte. Nel caso delle superfici, queste sono esclusivamente dei tori complessi o delle superfici semplicemente connesse dette K3. Dopo aver analizzato le loro proprietà in bassa dimensione, passeremo ai casi di dimensione più alta, guardando analogie e differenze.

Partendo dall'esempio delle curve ellittiche, darò una breve introduzione ad alcune varietà complesse Ricci piatte. Nel caso delle superfici, queste sono esclusivamente dei tori complessi o delle superfici semplicemente connesse dette K3. Dopo aver analizzato le loro proprietà in bassa dimensione, passeremo ai casi di dimensione più alta, guardando analogie e differenze.

In 1976 Paul Malliavin proposed in the paper "Stochastic calculus of variation and hypoelliptic operators" a probabilistic proof of Hormander's "sum of square" theorem. His proof was based on a new infinite dimensional differential calculus on the Wiener space. The theory was further developed by Stroock, Bismut and Watanabe, among others, to become what is nowadays known as the Malliavin calculus. This calculus has become a fundamental tool in the theory of stochastic (partial) differential equations and has found important applications in mathematical finance. This short course aims to provide a coincise introduction to the subject together with a sketch of Malliavin's proof of Hormander's theorem. Few remarks on the applications in mathematical finance will also be provided.

This is a collection of three lectures, at a graduate school level, devoted to the main problems in CR and pseudohermitian geometry and focusing on the impact of subelliptic theory, as a tool similar to elliptic theory in Riemannian geometry, and on the interrelations between pseudohermitian geometry and general relativity theory. The classical themes in CR geometry are the CR embedding problem and the CR extension problem, to which one may add the relationship between hyperbolic and subelliptic theories, as stemming from the existence of Fefferman’s metric, a Lorentzian metric associated to a positively ori- ented contact form on a strictly pseudoconvex CR manifold. Topics to be dealt with lie at the cross road of three main mathematical disciplines i.e. complex analysis in several complex variables, partial differential equations, and differential geometry.

A major theme in Lie algebra theory is that algebraic structures of complex simple Lie algebras are fully revealed by their Dynkin diagrams. Therefore, additional algebraic structures on Lie algebras can be studied by combinatorial information on Dynkin diagrams. We shall discuss the resulting interplay between algebra and combinatorics, including the real forms and automorphisms on Lie algebras.

A major theme in Lie algebra theory is that algebraic structures of complex simple Lie algebras are fully revealed by their Dynkin diagrams. Therefore, additional algebraic structures on Lie algebras can be studied by combinatorial information on Dynkin diagrams. We shall discuss the resulting interplay between algebra and combinatorics, including the real forms and automorphisms on Lie algebras.