Jonathan Lindbloom
Where did all the eigenvalues go? Some matrix analysis for sparsity-promoting inverse problems
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The majority of this talk will be of interest to anyone interested in eigenvalues. We will begin by going over some fundamental eigenvalue inequalities for symmetric matrices including the Courant-Fischer, Cauchy interlacing, Weyl, and Ostrowski theorems. We will sketch how they can all be proven using a simple “dimensional principle”. Then, we will apply these results to prove some new generalizations of the Ostrowski theorem. Some discussion will follow on how these theorems can be applied to prove a very general eigenvalue inequality that reveals the benefits of priorconditioning for sparsity-promoting inverse problems.
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Rohan Kapoor
A gentle introduction to random matrix theory
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Over winter break, I taught myself some of the basics of random matrix theory: the study of matrices where the entries are drawn from some probability distribution. In this talk, I will give a brief overview of the field of random matrices (based on a Princeton University colloquium by Terrence Tao), and then present one or two proofs (via combinatorics and complex analysis respectively) of the Wigner semicircle law of random matrices. There will be no measure theory in this talk.
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Michaela Polley
Departmental Check-in
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Casey Dowdle
Basis Set Discretization in Quantum Chemistry
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A common approach for discretizing molecules in quantum chemistry is to use a basis set comprised of linear combinations of Gaussian functions placed at the centers of the atoms. Due to analytic formulas for integrating products of Gaussians, they are a common basis set for molecular systems. Gausslets are gaussian basis set based on wavelet theory, which have many theoretical and practical applications due to their locality. In this talk I will discuss how Quiqbox.jl, a Julia package developed by Dartmouth physics graduate student Weishi Wang, is being used to explore the role of basis set discretization in quantum chemistry.
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Brian Mintz
How to count every possible symmetry
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Throughout history, symmetry has been almost universally revered. Mathematics allows us to classify and understand the ways in which objects and patterns can be symmetric. This presentation will introduce a system to describe symmetries popularized by John Conway that can be used to enumerate the 17 possible planar symmetries based on the classification of two-dimensional manifolds. We'll also have some hands-on practice with this, and see some beautiful applications to artistic tilings!
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Patrick Adonna
A Mathematical Basis for Synthetic Wavelength Holography
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Holography is an imaging technique that allows the observer to reconstruct a scene via a complex-valued wavefront. This wavefront retains both amplitude and phase information. By imaging at two slightly different wavelengths and exploiting correlated scattering profiles, we can recover this information even after light is scattered by optically rough surfaces. Modeling this physical phenomenon mathematically is not a trivial task, requiring consideration of both light transport and how optical fields are recorded by cameras. This talk will highlight how one of these models is constructed, with a brief overview of some fundamental principles of optics.
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