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Orals Weekly Readings (09/02-09/23)
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Over the past few weeks, I have been reading about various results from the theory of functional analysis, harmonic analysis, and (non)linear wave equations (as mentioned in the first blog post). I figured that it would be a valuable exercise to summarize what I have read each week in a blog post. This post is the first in such a series. Since the gap between my first two meetings with my advisor was three weeks apart, this blog post covers three weeks of readings, rather than the usual one.
Notes on Hyperbolic Partial Differential Equations
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As part of my orals exam for PhD candidacy in the math department at Stony Brook, I’m currently reading a compilation of lecture notes and texts on nonlinear wave equations, harmonic analysis, geometry, and fluids (the first three forming the bulk of my major topic, and the latter forming my minor topic). The following notes I’ve written are based on my latest understanding of these topics. The content itself is not unique, having been based strongly upon my readings (which include lecture notes on nonlinear wave equations by Profs. Gustav Holzegel, Andrew Lawrie, and Jonathan Luk), but perhaps the organization is. Over this semester and the next, I will gradually be adding more material to these notes, and revising already existent notes.
portfolio
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publications
A New Framework for Understanding Systematic Errors in Cluster Lens Modeling. III. Deflection from Large-Scale Structure
Published in The Astrophysical Journal, 2024
Interpreting and reconstructing distant sources that are gravitationally lensed by galaxy clusters requires accurate and precise lens models. While high-quality data sets have reduced statistical errors in such models, systematic errors remain important. We examine systematic lensing effects caused by density fluctuations due to large-scale structure along the line of sight. We use a multiplane ray-tracing algorithm with the IllustrisTNG 100-3 cosmological simulation of matter distribution and compute the statistical distributions of shear, convergence, and higher-order deflections using two Hubble Frontier Field clusters as examples (A2744 and MACS J0416.1−2403). The cosmic shear distribution is Gaussian in each component, while the cosmic convergence distribution is skewed such that 1 +κ is consistent with a log-normal distribution; the standard deviations for these quantities are at the level of a few to 10%, depending on the redshift of the source. The deflection from higher-order terms beyond convergence and shear has significant scatter: the rms deflection is ∼15″, considerably larger than the image position residuals for current lens models. These results indicate that line-of-sight deflection effects due to largescale structure can significantly impact lens models and should not be neglected. We present results in forms that can be incorporated into future cluster lens models. Abstract:
Recommended citation: Madhava A. & Keeton C.R. (2024). "A New Framework for Understanding Systematic Errors in Cluster Lens Modeling. III. Deflection from Large-Scale Structure." The Astrophysical Journal. 975(2), e287.
Download Paper | Download Bibtex
talks
The Isoperimetric Inequality, Calculus of Variations, and Lagrangian Mechanics
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Talk delivered to the First and Second Year Honors Seminar run by the Department of Mathematics at Rutgers. I presented the theory behind calculus of variations and some of its applications in physics and differential geometry.
Quantifying Systematic Errors in Cluster Lens Models due to Cosmological Large-Scale Structures
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Talk given to first- and second-year physics and astronomy majors during the Rutgers University Society of Physics Students' Winter Research Symposium. I presented key findings from my research on systematic errors in galaxy cluster lens models due to cosmological large-scale structures.
When Mathematics meets Astrophysics: Analyzing the Number of Solutions to the Gravitational Lens Equation
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Senior honors thesis defense presented in front of the Department of Physics and Astronomy's Honors Thesis Committee. On basis of my work, I was awarded highest honors in astronomy, was designated as a Rutgers School of Arts and Sciences Paul Robeson Scholar, and was named one of the four Henry Rutgers Scholar Award winners in the department.