Part I

Research Talks by Math Institutes

October 12, 2016 – Promenade Room 101A
12:00 – 1:00: Registration/Check-in (in front of Promenade Room 101A)
01:00 – 1:05: Welcome & Opening Remarks
01:05 – 1:35: SAMSI - David Stenning: “Bayesian analysis of multiple stellar populations in galactic globular clusters”
01:40 – 2:10: MSRI - Bobby Wilson: “Applications of $\ell^2$ decoupling”
02:15 – 2:45: MBI - Reginald McGee: “Uncovering Functional Relationships in Leukemia”
02:50 – 3:15: Break (Promenade Room 101B)
03:15 – 3:45: IMA - Shelvean Kapita: “Plane Wave Discontinuous Galerkin Methods for Scattering Problems”
03:50 – 4:20: NIMBioS - Nourridine Siewe: “Granuloma Formation and Immune Response to Infection by Leishmania: Mathematical Models”
04:30 – 5:30: Keynote Lecture by Joseph Teran (UCLA), “Scientific Computing in the Movies and Virtual Surgery” (Promenade Room 102B)
05:30 – 7:00: Poster Session & Reception (Promenade Room 101B)

SAMSI - David Stenning
“Bayesian analysis of multiple stellar populations in galactic globular clusters”
Computer models are becoming increasingly prevalent in a variety of scientific settings; these models pose challenges because the resulting likelihood function cannot be directly evaluated. For example, astrophysicists develop computer models that predict the photometric magnitudes of a star as a function of input parameters such as age and chemical composition. A goal is to use such models to derive the physical properties of globular clusters gravitationally bound collections of up to millions of stars. Recent observations from the Hubble Space Telescope provide evidence that globular clusters host multiple stellar populations, with stars belonging to the same population sharing certain physical properties. We embed physics-based computer models into a statistical likelihood function that assumes a hierarchical structuring of the parameters in which physical properties are common to (i) the cluster as a whole, or to (ii) individual populations within a cluster, or are unique to (iii) individual stars. A Bayesian approach is adopted for model fitting, and we devise an adaptive MCMC scheme that greatly improves convergence relative to its precursor, non-adaptive MCMC algorithm. Our method constitutes a major advance over standard practice, which involves fitting single computer models by comparing their predictions with one or more two-dimensional projections of the data.

MSRI - Bobby Wilson
“Applications of $\ell^2$ decoupling”
In this talk, we will discuss some applications of the pivotal work of Jean Bourgain and Ciprian Demeter on $\ell^2$-decoupling of functions whose Fourier supports lie near compact hypersurfaces. In particular, we will talk about work concerning the improvement of Strichartz estimates for the cubic nonlinear Schr\"odinger equation on the irrational torus. A surprisingly crucial aspect of the original problem, for the rational torus, is that we can reduce our estimates to counting lattice points contained in convex bodies. However, there is a sharp contrast, in what counting techniques are applicable, between the case of the irrational torus and the rational torus. We will discuss how $\ell^2$-decoupling allows us to bridge the gap between the two cases. This is joint work with G. Staffilani, C. Fan, and H. Wang.

MBI - Reginald McGee
“Uncovering Functional Relationships in Leukemia”
Mass cytometers can record tens of features for millions of cells in a sample, and in particular, for leukemic cells. Many methods consider how to cluster or identify populations of phenotypically similar cells within cytometry data, but there has yet to be a connection between cell activity and other features and these groups or clusters. We use differential geometric ideas to consider how cell cycle and signaling features vary as a function of the cell populations. This consideration leads to a better understanding of the nonlinear relationships that exist in the cytometry data.

IMA - Shelvean Kapita
“Plane Wave Discontinuous Galerkin Methods for Scattering Problems”
We apply the Plane Wave Discontinuous Galerkin (PWDG) method to study the direct scattering of acoustic waves from impenetrable obstacles. This problem is modeled by the Helmholtz equation in the unbounded domain exterior to the scatterer. To compute the scattered field, an artificial boundary is introduced to reduce the infinite domain to a finite computational domain. We then apply Dirichlet-to-Neumann (DtN) boundary conditions on a circular artificial boundary. Numerical experiments indicate that the accuracy of the PWDG method for the scattering problem is improved by the use of DtN boundary conditions.

NIMBioS - Nourridine Siewe
“Granuloma Formation and Immune Response to Infection by Leishmania: Mathematical Models”
Leishmaniasis is a disease caused by the Leishmania parasites. The two common forms of leishmaniasis are cutaneous leishmaniasis (CL) and visceral leishmaniasis (VL). VL is the more severe of the two and, if untreated, may become fatal. The hallmark of VL is the formation of granuloma in the liver or the spleen. In this paper, we develop a mathematical model of the evolution of granuloma in the liver. The model is represented by a system of partial differential equations and it includes immigration of cells from the adaptive immune system into the granuloma; the rate of the influx is determined by the strength of the immune response of the infected individual. It is shown that parasite load decreases as the strength of the immune system increases. Furthermore, the efficacy of a commonly used drug, which increases T cells proliferation, increases in a person with stronger immune response. The model also provides an explanation why, in contrast to humans, mice recover naturally from VL in the liver.

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