Given a vector bundle, its characteristic classes are cohomology classes of the base space which measure how 'twisted' the vector bundle is. In other words, they are obstructions to the vector bundle being trivial. One such class is the Euler class, which is Poincare dual to the zero set of a section which is transverse to the zero section of the bundle. We will discuss Thom isomorphism and discuss the relation between Euler class and Thom class. Lastly, we will discuss equivariant Euler class (which will be used in localization formula).

Meeting ID: 997 4840 9440 Passcode: 911104

At the 1900 International Congress of Mathematicians, Hilbert claimed that the Riemann zeta function is not the solution of any algebraic ordinary differential equation its region of analyticity. In 2015, Van Gorder addresses the question of whether the Riemann zeta function satisfies a non-algebraic differential equation and constructs a differential equation of infinite order which zeta satisfies. However, as he notes in the paper, this representation is formal and Van Gorder does not attempt to claim a region or type of convergence. In this talk, we show that Van Gorder's operator applied to the zeta function does not converge pointwise at any point in the complex plane. We also investigate the accuracy of truncations of Van Gorder's operator applied to the zeta function and show that a similar operator applied to zeta and other $L$-functions does converge.