Applications of the symplectic cutting construction will be discussed.

We use the degree formula for connective K-theory to study rational contractions of algebraic varieties. As an application we obtain a condition of rational incompressibility of algebraic varieties and a version of the index reduction formula. Examples include complete intersection, rationally connected varieties, twisted forms of abelian varieties and Calabi-Yau varieties.

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.

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 $\tilde g$.

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.

I will try to cover the following topics: 1. Equivariant cohomology: Definition, motivation, 2. Two constructions of E, 3. Cartan Model and some examples.

Two points x,y in a spacetime X are said to be causally related if there is a nonspacelike curve between them, i.e. if one can get from one point to the other moving not faster than the light speed. For globally hyperbolic spacetimes X the light rays through a point x form a Legendrian sphere $S_x$ in the contact manifold N of all light rays in X. We show that if the universal cover of a level set of a timelike function is an open manifold, then x and y are causally related exactly when the Legendrian link $(S_x, S_y)$ is nontrivial in N. In particular this proves the Low conejcture and the Legendrian Low conjecture formulated by Natario and Tod.

This talk will give a historical motivation for stacks from the nonexistence of moduli spaces, give a non-technical definition of a stack, and describe some examples.

Let $G$ be a reductive group. A $G\times G$-variety $X$ is called an equivariant compactification of $G$ if $X$ is normal, projective, and contains $G$ as an open and dense orbit. Regular compactifications and reductive embeddings are the main source of examples. In the first case, the equivariant cohomology ring has been explicitely described by Bifet, de Concini, Procesi and Brion. Loosely speaking, it depends mostly on the torus embedding part and the structure of the $G\times G$-orbits. As for the second class, Renner has found that they have a canonical cell decomposition based on underlying monoid data. My goal in this talk is to give an overview of the theory of group embeddings, putting more emphasis on the monoid approach, and to describe the structure of the so called rational cells. Finally, I will explain how such cellular decompositions could lead to a further application of GKM theory to the study of reductive embeddings.