Kommende Vorträge

I will carefully review the various equivalent definitions of principal $G$-bundles (free and proper $G$-action, fiber bundle with free and transitive $G$-action on fibers, local transition functions on cover satisfying cocycle condition, classifying map to $BG$). Then I show how these notions generalize to Lie $\infty$-groupoids. The main result is that we can still move between principal groupoid bibundles, anafunctors, and classifying maps in the $\infty$-categorical setting.

In 1990, R. C. Baker and H. L. Montgomery conjectured that for almost all fundamental discriminants d, the derivative of the Dirichlet L-function associated to the quadratic character modulo d has around $\log\log |d|$ real zeros on the interval $[1/2, 1]$. Baker and Montgomery's motivation in studying these zeros stems from their connection to real zeros of Fekete polynomials and to sign changes of real character sums. In this talk I will present recent work that settles this conjecture (up to a small factor of $\log\log\log |d|$).

In this talk we shall discuss the status of local-global principles for semi-integral points on orbifold pairs of Markoff type. If time permits, we will discuss a way to count Markoff orbifold pairs which satisfy the semi-integral Hasse principle while the corresponding Markoff surface lacks integral points. This talk is based on a joint work with Justin Uhlemann.