Abstract: Metric noncommutative geometry: In this series of talks, I am going to review some metric aspects of noncommutative geometry due to Alain Connes. To be more specific, I will discuss four formulas in Riemannian geometry and formulate them in algebraic forms, so that they can be considered in “noncommutative spaces” as well. These four formulas are concerned about geodesic distance, volume form, space of gauge potentials and Yang-Mills functional action. In the first talk last week, we looked at the spectral triple of a Riemannian manifold which in a sense captures our algebraic data. We also saw a formula for geodesic distance using just this piece of information. For the next talk, I'm planning to discuss volume forms and space of gauge potentials.

Abstract: A complete Riemannian manifold $(M, g)$ is a $Y^x_l$-manifold if every geodesic $\gamma(t)$ parametrized by arc length and originating at a point $\gamma(0)=x$ satisfies $\gamma(l)=x$ for $0\neq l\in \mathbb R$. Berard-Bergery proved that if $(M,g)$ is a $Y^x_l$-manifold of dimension $m>1$, then $M$ is a closed manifold with finite fundamental group, and the ring $H^*(M, \mathbb Q)$ is generated by one element. We say that $(M,g)$ is a $Y^x$-manifold if for every $\epsilon >0$ there exists $l$ with $|l|>\epsilon$ such that for every geodesic $\gamma(t)$ parametrized by arc length and originating at $x$, the point $\gamma(l)$ is $\epsilon$-close to $x$. We use Low's notion of refocussing Lorentzian manifolds to show that if $(M, g)$ is a $Y^x$-manifold of dimension $m>1$, then $M$ is a closed manifold with finite fundamental group. If $\dim M=2, 3$ and $(M,g)$ is a $Y^x$-manifold, then $(M, \tilde g)$ is a $Y^x_l$-manifold for some metric $\tilde g$.

Abstract: There is an old question in optics that has been called Alhazen's Problem. The name Alhazen honours an Arab scholar Ibn-al-Haytham who flourished 1000 years ago. The problem itself can be traced further back, at least to Ptolemy's Optics written around AD 150. The problem - while considered one of the 100 great problems of elementary mathematics - is very easy to state: Given two arbitrary balls on a circular billiard table, how does one aim the object ball so that it hits the target ball after one bounce off the rim. In this talk we introduce various methods of approach that has been studied, but mainly focus on the number of solutions and their distribution on the table.