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

A bstract When a quantum field theory is trivially gapped, its infrared fixed point an invertible theory. The partition function of the records response to various background fields in long-distance limit. set can include spacetime-dependent coupling constants, which case we call corresponding parameterized We study such theories arising from free Dirac fermions with mass parameters using Atiya...

We give a proof of the cobordism invariance of the index of elliptic pseudodifferential operators on σ-compact manifolds, where, in the non-compact case, the operators are assumed to be multiplication outside a compact set. We show that, if the principal symbol class of such an elliptic operator on the boundary of a manifold X has a suitable extension to K1(TX), then its index is zero. This con...

In this article, we survey the gluing formulae of the spectral invariants the ζ-regularized determinant of a Laplace type operator and the eta invariant of a Dirac type operator. After these spectral invariants had been originally introduced by Ray-Singer [30] and Atiyah-Patodi-Singer [1] respectively, these invariants have been studied by many people in many different parts of mathematics and ...

We consider the problem of automatic vocal melody transcription: translating an audio recording of a sung melody into a musical score. While previous work has focused on finding the closest notes to the singer’s tracked pitch, we instead seek to recover the melody the singer intended to sing. Often, the melody a singer intended to sing differs from what they actually sang; our hypothesis is tha...