We will show that the famous, intractible 1959 Kadison-Singer problem in C∗-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. This gives all these areas common ground on which to interact as well as explaining why each of these areas has volumes of literature on their respective problems without a satis...

Abstract Relations between the Atiyah–Patodi–Singer rho invariant and signatures of links have been known for a long time, but they were only partially investigated. In order to explore them further, we develop versatile cut-and-paste formula invariant, which allows us manipulate manifolds in convenient way. With help this tool, give description multivariable signature link $L$ as some closed t...

In this paper, we are interested in new versions of Farkas lemmas for systems involving convex and DC-inequalities. These versions extend well-known Farkas-type results published recently, which were used as main tools in the study of convex optimization problems. The results are used to derive several strong duality results such as: Lagrange, Fenchel-Lagrange or Toland-Fenchel-Lagrange duality...

These notes outline some ideas for approaching the KadisonSinger problem.

These maps correspond respectively to the families index theorem of Atiyah and Singer for fibrewise pseudodifferential operators for a smooth fibration of a compact manifold M −→ Y, to the equivariant index for invariant and transversally elliptic operators for the smooth action of a compact Lie group on a compact manifold Z and to the pseudodifferential extension of the numerical index map of ...

This is a primer on frame theory geared towards the parts of the theory needed for people who want to understand the relationship between the Kadison-Singer Problem and frame theory.

Teaching a new concept with gestures – hand movements that accompany speech – facilitates learning above-and-beyond instruction through speech alone (e.g., Singer & GoldinMeadow, 2005). However, the mechanisms underlying this phenomenon are still being explored. Here, we use eye tracking to explore one mechanism – gesture’s ability to direct visual attention. We examine how children allocate th...

In this paper we construct Riemannian metrics and weight functions over Casson handles. We show that the corresponding Atiyah–Hitchin–Singer complexes are Fredholm for some class of Casson handles of bounded type. Using these, the Yang–Mills moduli spaces are constructed as finite dimensional smooth manifolds over Casson handles in the class. AMS Classification numbers Primary: 57M30, 57R57