1. Here are the slides (Examples-Friends or Foes Part I and Examples-Friends or Foes Part II) for the talk, Examples–Friends or Foes that’s upcoming for Tuesday January 31, 2017 at 12:10 pm on the 2nd floor (Room 295) of the science building at Norwich University. While, the slides themselves are insufficient for getting the full experience of the talk, they may be useful after the talk, and they may help you decide whether or not you wish to attend the talk. In addition, the talk discusses a (probably) new proof of Viviani’s theorem, which is posted here. There is a also a brief reference to the use of a key example to understand the fundamental theorem of calculus, based on this paper. Finally, the talk references this video by 3 blue 1 brown that explains the circle division problem in about 9 minutes.
Here is the abstract for the talk:
While doing mathematics, examples can lead you astray, and they don’t prove much of anything! Yet a carefully chosen example can cut through all the difficulties, leading the way to a beautiful solution. This talk will feature selected examples of both situations from the history of mathematics—although some brand new mathematics will be introduced! As most of these examples will require little background, the talk will be accessible to a wide audience. Nevertheless it comes with the following—Warning: May inspire a deeper interest in geometry and discrete mathematics.
2. Former student Katy Smith (class of 2006) and I have a very nice result on vertex-magic total labelings of odd complete graphs. In fact we completely solved the spectrum problem for odd complete graphs, meaning that we described exactly the set of integers consisting of possible magic constants for vertex-magic total labelings of odd complete graphs. However, we also have a second much smaller paper with a similar sounding title, but on a different topic, with a much different result, on multiple complete graphs. Here is a preparing of this lesser known paper, Vertex magic total labelings of multiple complete graphs.
3. Another much older paper on vertex magic total labeling of cubic graphs, was a “major conceptual step forward,” according to the introduction of this paper. A preprint of the cubic graphs paper can be found here.
5. Our proof that every drawing of the complete graph in the plane must have at least 219 crossings is summarized on arxiv here. It is joint work with S. Pan and R. B. Richter. An improved version has since appeared and is available in JCT-B; the link to the improved version is here. In fact this new version was JCT-B’s hottest paper for July-September and October-December, 2015.
6. This preprint, Witt cancellation seen as a cancellation, is on the Witt Cancellation Theorem, with extra information about the Witt Ring. It is joint work with Sunil Chebolu and Ján Mináč. The illustrations are by Matthew Teigen. A different version of this paper has now been accepted for publication in Expositiones Mathematicae and it is available here.
7. Here are the slides for my talk, “Vertex-magic graphs,” from June 11, 2011, at the spring meeting of the Northeastern Section of the Mathematical Association of America.