PDE Seminar Spring 2022

Fridays, 3:30 — 4:20pm, Stevenson Center 1312 (in-person) or via Zoom (online).

Date: Friday, April 15, 2022.

  • Speaker: Wenrui Hao, Penn State University.
  • Title: Data-driven modeling on Alzheimer’s disease.
  • Abstract: Alzheimer’s disease (AD) affects more than 5 million people in the US. Recently, personalized treatment of AD provides a new way to manage AD patients’ treatment plans. Such treatment requires a new approach to analyze the growing electronic AD brain data. In this talk, I will introduce a mathematical modeling approach to describe the progression of AD clinical biomarkers and also incorporate patient data for personalized prediction and optimal treatment. More specifically, an AD personalized prediction is provided via validating the mathematical model on a multi-institutional dataset of AD biomarkers. Personalized therapeutic simulation studies for AD patients are performed via adding optimal controls to this model.
  • Zoom link: https://vanderbilt.zoom.us/j/92847654394

Date: Friday, April 22, 2022.

  • Speaker: Alex Pandya, Princeton University.
  • Title: Initial investigations of causal, stable first-order relativistic hydrodynamics.
  • Abstract: Fluid mechanics has proven to be remarkably successful in describing a wide variety of substances. Of these, the quark-gluon plasma is of particular interest, as experiments are now sensitive to dissipative effects typically neglected in relativistic models. This fact, combined with indications that similar effects might also impact the dynamics of neutron star mergers, provides strong motivation for the development of relativistic fluid theories with dissipation. In this talk, I will review the modern interpretation of fluid mechanics as a gradient expansion about equilibrium, and define two fluid theories which include dissipation: Muller-Israel-Stewart (MIS) theory, the standard in the field, and the promising alternative known as BDNK theory. The main difference between the two is that BDNK arises at lower order in the gradient expansion, rendering it simpler in many respects than MIS and allowing for easier control over the so-called hydrodynamic frame, a property crucial to ensuring the theory be free of pathologies. To capitalize on these advantages, one must solve a series of technical problems in designing a numerical scheme for the BDNK equations. I will review these challenges and their solutions, and evaluate our BDNK solver on a suite of standard tests. To conclude, I will provide a roadmap of steps to generalize our approach for applications in nuclear physics and astrophysics. This is joint work with Elias Most and Frans Pretorius.

Date: Friday, April 29, 2022.

  • Speaker: Delilah Gates, Princeton University.
  • Title: Observable Emission from Circular Equatorial Kerr Orbiters.
  • Time & Location (note: this differs from our usual schedule): 3 pm at 4309 Stevenson Center
  • Abstract: We characterize observational emission from emitters on stable circular equatorial orbits around Kerr black holes. Our analytic characterization of the emission from these orbiters arises from understanding two critical phenomena related to the Kerr geometry: the presence of the photon shell (a region of critical null geodesics), and the emergence of a throat of divergent proper length in the near horizon of near maximally rotating black holes.

    First, we discuss monochromatic and isotropic photon emission from circular equatorial Kerr orbiters. We derive analytic expressions for the photon escape probability and the redshift-dependent total flux collected on the celestial sphere as a function of emission radius and black hole spin. Next, we consider the observable emission from collisions between circular equatorial orbiters at the innermost stable circular orbiters and infalling particles (Banados-Silk-West type collisions) that have arbitrarily high center of mass energy when the black hole is near maximally rotating. Lastly, we analyze the maximum observable blueshift of emission from an equatorial disk of emitters on stable circular equatorial orbits around a Kerr black hole to an observer at fixed position on the celestial sphere which provides constraints on the spin or inclination in terms of the other.