Given a closed, connected, oriented 3-manifold with positive first Betti number, one can define an instanton Floer group as well as a quilted Lagrangian Floer group. The quilted Atiyah-Floer conjecture states that these cohomology groups are isomorphic. We initiate a program for proving this conjecture.This paper is the first in a series [11] [13] that prove various aspects of the quilted Atiyah-Floer conjecture. The present paper is dedicated entirely to the underlying Floer theory and differential geometry of the project.Section 2 begins with a brief history and overview of the conjecture. We also introduce our notation and conventions. The section ends with a description of the 1 quilted Floer group associated to a 3-manifold. This group was first introduced by Wehrheim and Woodward in [29], and is defined via a 2+1 field theoretic scheme that relates symplectic and gauge theoretic 3-manifold invariants. We point out that in [29] the authors define their quilted invariants for 3-manifolds via gauge theory on suitable PU(r)-bundles (one usually considers the case PU(2) = SO(3)). We adopt their approach by working with the higher rank PU(r)-bundles throughout this paper.In Section 3 we discuss Floer's instanton homology [14] [15]. We note that, as in the symplectic theory, the standard references (e.g., Donaldson's book [5], as well as Floer's original papers) focus almost entirely on the Lie groups SU(2) and SO(3). However, it is possible to define instanton homology for other Lie groups as well.To maintain the analogy with the Woodward-Wehrheim theory, we focus attention on PU(r) for r ≥ 2. A full treatment of higher-rank instanton Floer theory does not seem to appear in the literature, so we have spent some time developing the theory in this case, proving several folklore results that are well-known in the case r = 2. We note also that Kronheimer has defined higher-rank Donaldson invariants [23]. These can be viewed as the 4-dimensional version of the higher-rank instanton Floer theory discussed here.In Section 4 we make a precise statement of the quilted Atiyah-Floer conjecture, as well as a chain level version of the conjecture. We then describe our overall approach, and prove several chain level statements. This includes a detailed discussion of how to obtain transversality simultaneously in both Floer theories. Acknowledgments:The author is grateful to his thesis advisor Chris Woodward for his insight and valuable suggestions. In addition, the author benefited greatly from discussions with Katrin Wehrheim, via Chris Woodward, of her unpublished work on the Atiyah-Floer conjecture. This unpublished work analyzes various bubbles that appear in the limit of instantons with Lagrangian boundary conditions with degenerating metrics, and it outlines the remaining problems in such an approach. The present paper takes a different approach that avoids instantons with Lagrangian boundary conditions.The author would also like to thank Penny Smith, Dietmar Salamon, Fabian Ziltener, Peter Ozsváth, and Tom Parker for en...

The Gemini Remote Access to CFHT ESPaDONS Spectrograph has achieved first light of its experimental phase in May 2014. It successfully collected light from the Gemini North telescope and sent it through two 270 m optical fibers to the the ESPaDOnS spectrograph at CFHT to deliver high-resolution spectroscopy across the optical region. The fibers gave an average focal ratio degradation of 14% on sky, and a maximum transmittance of 85% at 800nm. GRACES achieved delivering spectra with a resolution power of R = 40,000 and R = 66,000 between 400 and 1,000 nm. It has a ~8% throughput and is sensitive to target fainter than 21 st mag in 1 hour. The average acquisition time of a target is around 10 min. This project is a great example of a productive collaboration between two observatories on Maunakea that was successful due to the reciprocal involvement of the Gemini, CFHT, and NRC Herzberg teams, and all the staff involved closely or indirectly.

A standardised protocol for radiographic skeletal survey protocol has been developed in New Zealand. We present it here for consideration by others.

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