Speaker: Piper Harron (UH Manoa)

Title: Equidistribution of shapes of number fields of degree 3, 4, and 5.

Abstract: In her talk, Piper Harron will introduce the ideas that there are number fields, that number fields have shapes, and that these shapes are everywhere you want them to be. This result is joint work with Manjul Bhargava and uses his counting methods which currently we only have for cubic, quartic, and quintic fields. She will sketch the proof of this result and leave the rest as an exercise for the audience. (Check your work by downloading her thesis!).

Speaker: Gideon Zamba (U. Iowa)

Title: Applied Mathematics in Action through Biostatistics

PBRC Seminar
Title: Hydrodynamic reception and predator avoidance by free-swimming organisms

Dr. Daisuke Takagi
Dept. of Mathematics and PBRC

where: AgSc 219
When: Thurs, Jan 26, 4-5pm

Speaker: Gideon Zamba (U. Iowa)

Title: A Semi-parametric Random-cell type of goodness-of-fit Test when Observations are Recurrent

Abstract: Goodness-of-fit of the distribution function governing time to occurrence of recurrent events is considered. We develop a chi-square type of test based on the nonparametric maximum likelihood estimator (NPMLE) of the inter-event time distribution. The test is based on the minimum chi-squared estimator of a parametric family and compares the estimated parametric null to the NPMLE on k partitions of a calendar time over a study monitoring period. Small sample and asymptotic properties of the proposed statistic are investigated. Simulation results for Weibull lifetime models are discussed and large sample properties of the test statistic are established using empirical process tools. The approach is then applied to jet planes air conditioning system failures.

Title: Multiscale inverse problems for partial differential equations and applications to sonar imaging

Abstract
A common objective in many data-driven sectors is to accurately describe intrinsic features of a complex process. This is a typical inverse problem for finding parameters in a model from given data, for example determining coefficients in partial differential equations (PDEs) from solution data. Inverse problems for PDEs pose daunting theoretical and computational challenges. For example, the classical inverse conductivity problem posed by Calderon is severely ill-posed, even in the case of smooth, isotropic coefficients. The situation is worse when modeling heterogeneous materials such as composites, lung airways and vasculature, and sedimentary layers in the Earth’s crust. For a variety of reasons, including the high cost of simulations and uncertainty in the measurements, the models are often simplified by a smoothing or homogenization process. Using the analysis of inverse conductivity problems, our results identify key parameters in highly oscillatory coefficients that withstand the loss of information due to homogenization. Multiscale methods for numerical homogenization are then used to efficiently predict the forward model while recovering microscale parameters. Ideas presented in this strategy can also be applied to solving inverse problems in ocean acoustics that aim to characterize properties of the ocean floor using sonar data. Here, forward solvers incorporate simulations of Helmholtz equations on a wide range of spatial scales, allowing for detailed recovery of seafloor parameters including the material type and roughness. In order to lower the computational cost of large-scale simulations, we take advantage of a library of representative acoustic responses from various seafloor configurations.

Title: Mapping class group action on character varieties and the ergodicity

Abstract: Character varieties of a surface are central objects in several beaches of math-
ematics, such as low dimensional topology, algebraic geometry, differential geom-
etry and mathematical physics. On the character varieties, there is a tautological
action of the mapping class group – the group of symmetries of the surface, which is expected to be ergodic in certain cases. In this talk, I will review related results
toward proving the ergodicity and introduce two long standing and related conjectures: Goldman’s Conjecture and Bowditch’s Conjecture. It is shown by Marche and Wolff that the two conjectures are equivalent for closed surfaces. For punctured surfaces, we disprove Bowditch’s Conjecture by giving counterexamples, yet prove that Goldman’s Conjecture is still true in this case.