MSI Weekly Bulletin - Week starting Monday 31 July, 2006

Let $X$ be a projective variety. For many years now, we have known how to form an infinite completion for the derived category $D^b(Vect/X)$, whose objects are bounded complexes of vector bundles on $X$. This allows us to prove theorems about finite complexes of vector bundles, using infinite techniques. Only very recently have we understood how to handle the derived category $D^b(Coh/X)$, whose objects are bounded complexes of coherent sheaves on $X$. Krause published a paper, in 2005, showing that $K(Inj/X)$ is an infinite completion for $D^b(Coh/X)$. Jorgensen, also in 2005, showed that, as long as $X$ is affine, $K(Proj/X)$ is an infinite completion for the dual category $D^b(Coh/X)^{op}$. Grothendieck's local duality studies conditions under which the categories $D^b(Coh/X)$ and $D^b(Coh/X)^{op}$ are equivalent; it suffices for there to exist a "dualizing complex", and dualizing complexes exist for many interesting $X$. One can ask what happens to the infinite completions; it turns out that, when a dualizing complex exists, then $K(Inj/X)$ and $K(Proj/X)$ are also equivalent. We will discuss this, as well as related results.

This talk will be concerned with inequalities relating to the eigenvalues of the Laplacian, that is, solutions to Helmholtz's Equation: $\triangle\phi+\lambda\phi=0$ on a bounded domain in ${\bf R}^n$. This equation is a result of performing separation of variables on both the wave and heat equation, and so has many physical applications. The study of these eigenvalues forms a large part of the field of Isoperimetric Inequalities, which relates physical properties not readily obtainable to geometric properties that are. I will discuss some classic results and compare the methods used to prove them.

RSA cryptography is attributed to Rivet, Shamir and Adleman, and the method was published by them in 1977. Following an article in a Martin Gardner column in the same year, the relevant 5 lines of computer code was banned from export by the US government. It was printed on T-shirts which formally became prohibited munitions exports. But in fact RSA cryptography was discovered by Clifford Cocks in 1973 working in the British secret service. In the best tradition of spook agencies this information was not made publicly known* until 20 years later in 1997. RSA cryptography is the basis of all eBay, amazon and internet banking transactions, EFTPOS and ATM, digital signatures and authenticity verification. It has the amazing property that everyone knows how to encode a secret message meant for me, they all use the same encoding map, but only I (not the CIA, not Mossad, not the KGB) can decode the message. If we regard a secret message as a number in the set for some large (say 300 digits in length), and the encoding function as a bijective function then anyone can compute for given On the other hand, if is the inverse (i.e. decoding) function then only I know how to compute for, within any reasonable time frame (e.g. the life time of our solar system) using current technology and methodology. I will prove the RSA theorem (it only requires a Year 10 school mathematics background), demonstrate a "real life" example or two with Maple, and discuss the difficulties involved in cracking the code (prime factorisation issues and the official RSA challenge). This is a non standard "maths education" MSI colloquium, not current research, for those who do not know how RSA works but are curious. *"Secret Intelligence gives the Government a vital edge", Tony Blair, Prime Minister, see http://www.gchq.gov.uk/