Branwen Purdy prepares hands-on activities for kids at the OMSI Meet-A-Scientist Day in Portland, to share hands-on learning experiences about her research in topological data analysis.

Join us for these events hosted by the Department of Mathematics, including colloquia, seminars, graduate student defenses and outreach, or of interest to Mathematicians hosted by other groups on campus.

Given a polynomial dynamical system on real affine space, we show that the forward orbit of any point intersects any algebraic subvariety in at most finitely many infinite arithmetic progressions, with the remaining points of intersection lying in a set of (Banach) density zero. This may be viewed as a weak asymptotic version of the Dynamical Mordell-Lang Conjecture. The result actually holds for dynamical systems on any affine variety over any field in arbitrary characteristic. The proof uses methods of ergodic theory applied to compact Berkovich spaces, in particular a strong version of the Poincare recurrence theorem due to Furstenberg. Read more.

ABSTRACT:We consider a thin metal film on a thermally conductive substrate exposed to an external heat source in a setup where the heat absorption depends on the local film thickness. Our focus is on modeling film evolution while the film is molten. The film geometry modifies local heat flow, which in turn may influence the film surface evolution through thermal variation of material properties. We use asymptotic analysis to develop a thermal model that is accurate, computationally efficient, and that accounts for the heat flow in both the in-plane and out-of-plane directions. We apply this model to describe metal films of nanoscale thickness exposed to heating and melting by laser pulses, a setup commonly used for self and directed assembly of various metal geometries via dewetting while the films are in the liquid phase. We find that thermal effects play an important role, and in particular that the inclusion of temperature dependence in the metal viscosity modifies the time scale… Read more.

I will introduce the notion of halfspaces in group splittings and discuss the problem of when these halfspaces are one-ended. I will also discuss connections to JSJ splittings of groups, and to determining whether groups are simply connected at infinity. This is joint work with Michael Mihalik. Read more.

ABSTRACT: Finite element analysis requires a qualified analyst to generate the necessary input data, verify the output and post process the analysis results for a meaningful conclusion. The required expertise and labor efforts precluded the use of FEA in daily medical practice for example. Recent scientific advancements such as low dose CT scans, machine learning, and high order FEA which allows an inherent verification methodology of the numerical accuracy, make it possible to provide a fully autonomous process for assessing bone strength and fracture risk. This autonomous process, named autonomous finite element (AFE) analysis, introduces a paradigm shift in the use of FEA. This talk addresses a novel AFE for patient-specific analysis of human femurs used nowadays in clinical practice: it involves an automatic segmentation of femurs from CT-scans by U-Net, an automatic mesh generation and application of boundary conditions based on anatomical points, a high-order FE analysis with… Read more.

ABSTRACT: Earth system models solve exceedingly complicated multiphysics problems by breaking down the Earth system hierarchically into smaller sub-models (e.g. atmosphere, ocean, land, and sea ice), which are composed of smaller components themselves. This decomposition of an Earth system model (which may require millions of lines of code in its software implementation) into many small modules is a vital part of model development. However, naïve coupling of modular physics packages using first-order methods can significantly reduce model accuracy, or even produce numerical instability. This talk covers two examples from the Energy Exascale Earth System Model (E3SM). First, we will see that “sequential” (Lie-Trotter) splitting is a major source of error for E3SM’s cloud and precipitation physics. We will discuss our evaluation of several proposed alternatives, including Strang splitting and multirate methods. Second, we will see that E3SM is prone to spurious “oscillations” in winds… Read more.

ABSTRACT: Full waveform inversion to monitor changes in seismicity is a computationally expensive and challenging task. The latter is due to the fact that the discretization of the seismic wave equation can have millions of degrees of freedom. Moreover, aiming at estimating, for instance, the elastic structure at every grid point results in a large parameter space within the inverse problem. Model order reduction (MOR) techniques can help to speed up the computations, using low-dimensional models that capture the original system's important features. However, for large-scale wave propagation problems, constructing efficient reduced models is challenging as MOR methods can suffer from a slow decay of the Kolmogorov n-width for such problems, thus, requiring a large number of basis functions to reach the desired accuracy. In our work, we address the mentioned challenge as follows: We transform the problem to the Laplace domain, where we can exploit that the output of interest – the… Read more.

ABSTRACT:
An introduction to fractional derivatives and some of their properties will be presented. The regularity of solutions to Caputo fractional initial-value problems is then discussed; it is shown that typical solutions have a weak singularity at the initial time $t=0$. This singularity has to be taken into account when designing and analysing numerical methods for the solution of such problems. To address this difficulty we use graded meshes, which cluster mesh points near $t=0$, and answer the question: how exactly should the mesh grading be chosen?
Finally, initial-boundary value problems are considered, where the time derivative is a Caputo fractional derivative. (This is a fractional-derivative generalisation of the classical parabolic heat equation.) Once again a weak singularity appears at $t=0$, and the mesh in the time coordinate should be graded to compute satisfactory numerical solutions. This problem is the most widely studied fractional-derivative problem in the… Read more.

Math for all in Corvallis has the purpose of fostering inclusivity in mathematics by holding talks and discussions in both research and education. This conference will be targeted to undergraduate and graduate students, post-docs, and faculty members from all institutions in Oregon and provide a friendly, open environment to learn and discuss mathematics. Read more.