I will first introduce the idea of integer relations and discuss practical concerns for their computation by numeric techniques (i.e., using floating point arithmetic). To this end I will discuss the PSLQ and LLL algorithms (and will mention, in passing, some others). I will then extend the idea of integer relations into the relations consisting of algebraic integers. I will discuss, in particular, the case of algebraic integers from quadratic extension fields. As with the first part of the talk, I will discuss practical concerns for calculation by numeric techniques of these quadratic integer relations.

I introduce and demonstrate the Coq assisted theorem prover.

The use of GPUs for scientific computation has undergone phenomenal growth over the past decade, as hardware originally designed with limited instruction sets for image generation and processing has become fully programmable and massively parallel. This talk discusses the classes of problem that can be attacked with such tools, as well as some practical aspects of implementation. A direction for future research by the speaker is also discussed.

Our most recent computations tell us that any counterexample to Giuga’s 1950 primality conjecture must have at least 19,907 digits. Equivalently, any number which is both a Giuga and a Carmichael number must have at least 19,907 digits. This bound has not been achieved through exhaustive testing of all numbers with up to 19,907 digits, but rather through exploitation of the properties of Giuga and Carmichael numbers. We introduce the conjecture and an algorithm for finding lower bounds to a counterexample, then present our recent results and discuss challenges to further computation.

Giuga's conjecture will be introduced, and we will discuss what's changed in the computation of a counterexample in the last 17 years.

We shall continue from last week, performing more hyperbolic constructions, as well as some elliptic constructions for comparison. In particular, some alternating projection algorithms will be explored in hyperbolic space. With some luck we shall confound some more of our euclidean intuitions. Audience participation is actively encouraged.

Hyperbolic geometry will be introduced, and visualised via the Cinderella software suite. Simple constructions will be performed and compared and contrasted to with Euclidean geometry. Constructions and examples will be quite elementary. Audience participation, specifically suggestions for constructions to attempt, during the demonstration is actively encouraged.The speaker apologises in advance for not being nearly as knowledgeable of the subject as he probably ought to be.