Forthcoming events in this series
Aside from a few tangential problems, this seminar will include a proof of Ostrowski's Theorem. This states than any norm over the rationals is equivalent to either the Euclidean norm or the $p$-adic norm, for some prime $p$.
This talk will introduce Dirichlet's Theorem on the approximation of real numbers via rational numbers. Once this has been established, a stronger version of the result will be proved, viz Hurwitz's Theorem.
This second 'problem sheet' of the term includes a proof of Jensen's Theorem for the number of zeroes of an analytic function in a disc, the usefulness of which is highlighted by an application to the Riemann zeta-function.
A brief overview of consonance by way of continued fractions and modular arithmetic.
Defined in terms of $\zeta(\frac{1}{2} +it)$ are the Riemann-Siegel functions, $\theta(t)$ and $Z(t)$. A zero of $\zeta(s)$ on the critical line corresponds to a sign change in $Z(t)$, since $Z$ is a real function. Points where $\theta(t) = n\pi$ are called Gram points, and the so called Gram's Law states between each Gram point there is a zero of $Z(t)$, and hence of $\zeta(\frac{1}{2} +it)$. This is known to be false in general and work will be presented to attempt to quantify how frequently this fails.
I will review the construction of algebraic de Rham cohomology, relative de Rham cohomology, and the Gauss-Manin connection. I will then show how we can find a basis for the cohomology and the matrix for the connection with respect to this basis for certain families of curves sitting in weighted projective spaces.