Three Puzzles on Mathematics, Computation, and Games

In this lecture I will talk about three mathematical puzzles involving mathematics and computation that have preoccupied me over the years. The first puzzle is to understand the amazing success of the simplex algorithm for linear programming. The second puzzle is about errors made when votes are counted during elections. The third puzzle is: are quantum computers possible?Comment: ICM 2018 plenary lecture, Rio de Janeiro, 36 pages, 7 Figure

Exploiting Polyhedral Symmetries in Social Choice

A large amount of literature in social choice theory deals with quantifying the probability of certain election outcomes. One way of computing the probability of a specific voting situation under the Impartial Anonymous Culture assumption is via counting integral points in polyhedra. Here, Ehrhart theory can help, but unfortunately the dimension and complexity of the involved polyhedra grows rapidly with the number of candidates. However, if we exploit available polyhedral symmetries, some computations become possible that previously were infeasible. We show this in three well known examples: Condorcet's paradox, Condorcet efficiency of plurality voting and in Plurality voting vs Plurality Runoff.Comment: 14 pages; with minor improvements; to be published in Social Choice and Welfar

Review of high-contrast imaging systems for current and future ground- and space-based telescopes I. Coronagraph design methods and optical performance metrics

The Optimal Optical Coronagraph (OOC) Workshop at the Lorentz Center in September 2017 in Leiden, the Netherlands gathered a diverse group of 25 researchers working on exoplanet instrumentation to stimulate the emergence and sharing of new ideas. In this first installment of a series of three papers summarizing the outcomes of the OOC workshop, we present an overview of design methods and optical performance metrics developed for coronagraph instruments. The design and optimization of coronagraphs for future telescopes has progressed rapidly over the past several years in the context of space mission studies for Exo-C, WFIRST, HabEx, and LUVOIR as well as ground-based telescopes. Design tools have been developed at several institutions to optimize a variety of coronagraph mask types. We aim to give a broad overview of the approaches used, examples of their utility, and provide the optimization tools to the community. Though it is clear that the basic function of coronagraphs is to suppress starlight while maintaining light from off-axis sources, our community lacks a general set of standard performance metrics that apply to both detecting and characterizing exoplanets. The attendees of the OOC workshop agreed that it would benefit our community to clearly define quantities for comparing the performance of coronagraph designs and systems. Therefore, we also present a set of metrics that may be applied to theoretical designs, testbeds, and deployed instruments. We show how these quantities may be used to easily relate the basic properties of the optical instrument to the detection significance of the given point source in the presence of realistic noise.Comment: To appear in Proceedings of the SPIE, vol. 1069

Statistical Modeling of Influenza-Like-Illness in Montana using Spatial and Temporal Methods

Studying air pollution and public health has been a historically important question in science. It has long been hypothesized that severe air pollution conditions lead to negative implications in basic human health. Primarily, areas thats are prone to severe degrees of human pollution are the focus of such studies. Such research relating to less populated areas are scarce, and this scarcity raises the question of how such pollution dynamics (human-made and natural) influence human health in more rural areas. The aim of this study is to explore this hole in research; in particular we explore possible links between air pollution and Influenza-like-illness in Montana. We begin with a discussion of our starting hypotheses, the data we have accumulated to test these hypotheses, and some exploratory analysis of these data. The body of this research is based on modeling of the natural factors that influence influenza dynamics in general and how these factors apply in the state of Montana. Here, we will explore different modeling approaches and how to apply them to the given data. To conclude this research, a summary is provided and the implications this has for the state of Montana

The X-ray Power Density Spectrum of the Seyfert 2 Galaxy NGC 4945: Analysis and Application of the Method of Light Curve Simulations

The study of the power density spectrum (PDS) of fluctuations in the X-ray flux from active galactic nuclei (AGN) complements spectral studies in giving us a view into the processes operating in accreting compact objects. An important line of investigation is the comparison of the PDS from AGN with those from galactic black hole binaries; a related area of focus is the scaling relation between time scales for the variability and the black hole mass. The PDS of AGN is traditionally modeled using segments of power laws joined together at so-called break frequencies; associations of the break time scales, i.e., the inverses of the break frequencies, with time scales of physical processes thought to operate in these sources are then sought. I analyze the Method of Light Curve Simulations that is commonly used to characterize the PDS in AGN with a view to making the method as sensitive as possible to the shape of the PDS. I identify several weaknesses in the current implementation of the method and propose alternatives that can substitute for some of the key steps in the method. I focus on the complications introduced by uneven sampling in the light curve, the development of a fit statistic that is better matched to the distributions of power in the PDS, and the statistical evaluation of the fit between the observed data and the model for the PDS. Using archival data on one AGN, NGC 3516, I validate my changes against previously reported results. I also report new results on the PDS in NGC 4945, a Seyfert 2 galaxy with a well-determined black hole mass. This source provides an opportunity to investigate whether the PDS of Seyfert 1 and Seyfert 2 galaxies differ. It is also an attractive object for placement on the black hole mass-break time scale relation. Unfortunately, with the available data on NGC 4945, significant uncertainties on the break frequency in its PDS remain