Mathematics

An application of cointegration techniques to cause-specific mortality in USA, Japan, England & Wales and Australia

Severine Arnold (University of Lausanne (HEC Lausanne), Actuarial Science)

Abstract: Non-stationary time series have been widely studied by economists over the past decades. One interesting feature of non-stationary vari- ables is that we can distinguish between long-run relations, that are stationary, and short-run adjustments towards these. These long-run relations are known as cointegrating relations and represent long-run equilibriums. Since mortality rates are non-stationary variables, the theory developed for economic variables can also be used to gain in- sights into the long-run relations that may exist between mortality rates for different causes of death. In this work, cointegration is de- veloped for cause-of-death mortality. We analyze the five main causes of death across four major countries, including USA, Japan, England & Wales and Australia. Our analysis identifies long-run equilibrium relationships between the five main causes of death, providing new insights into the dependence structure that exists between competing risks. We show how male and female mortality trends by cause have been similar, although these differ across countries.

Wave decay and resolvent estimates on conic manifolds

Dean Baskin (Northwestern)

Abstract: We consider manifolds with conic singularities that are isometric to R^n outside a compact set. Under natural geometric assumptions on the cone points, we prove the existence of a logarithmic resonance-free region for the cut-off resolvent. The estimate also applies to the exterior domains of non-trapping polygons via doubling process. As applications of the estimate, we obtain local energy decay for the wave equation in odd dimensions and Strichartz estimates for the Schrödinger equation exterior to non-trapping polygons. This is joint work with Jared Wunsch. The Strichartz estimates are also joint with Jeremy Marzuola.

Gap distribution of Farey fractions determined by subgroups of SL(2,$\mathbb{Z}$)

Byron Heersink (UIUC Math)

Abstract: For a given finite index subgroup $H\subseteq$ SL(2,$\mathbb{Z}$), we use a process developed by Fisher and Schmidt to lift a Poincaré section of the horocycle flow on SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$) found by Athreya and Cheung to the finite cover SL(2,$\mathbb{R}$)/H of SL(2,$\mathbb{R}$)/SL(2,$\mathbb{Z}$). We then relate the properties of this section to the gaps in Farey fractions and describe how the ergodic properties of the horocycle flow can be used to obtain the limiting gap distribution of various subsets of Farey fractions.