These notes record the basic results about DVR’s (discrete valuation rings) and Dedekind rings, with at least sketches of the non-trivial proofs, none of which are hard. This is standard material that any educated mathematician with even a mild interest in number theory should know. It has often slipped through the cracks of Chicago’s first year graduate program, but then we would need at least...

This document is a record of my contribution to a panel discussion which took place on July 27, 2011 as part of the Infinity and Truth Workshop held at the Institute for Mathematical Sciences, National University of Singapore, July 25–29, 2011. In preparation for the panel discussion, Professor Woodin asked each panelist to formulate a yes/no question to be asked of a benevolent, omniscient mat...

1.1 Origin The traveling salesman problem (TSP) were studied in the 18th century by a mathematician from Ireland named Sir William Rowam Hamilton and by the British mathematician named Thomas Penyngton Kirkman. Detailed discussion about the work of Hamilton & Kirkman can be seen from the book titled Graph Theory (Biggs et al. 1976). It is believed that the general form of the TSP have been firs...

The master economist must possess a rare combination of gifts. ... He must be mathematician, historian, statesman, philosopher to some degree.