A theorem of Göttsche establishes a connection between cohomological invariants of a complex projective surface S and corresponding invariants of the Hilbert scheme of n points on S. This relationship is encoded in certain infinite product q-series which are essentially modular forms. Here we make use of the circle method to arrive at exact formulas for certain specializations of these q-series, yielding convergent series for the signature and Euler characteristic of these Hilbert schemes. We also analyze the asymptotic and distributional properties of the q-series' coefficients.
COVID-19 has presented society with a unique set of challenges, including seeking a scientific understanding of the novel coronavirus, modeling its epidemiology, and inferring appropriate societal response. In this article, we posit that fighting a pandemic is as much a social endeavor as a medicinal and scientific one and focus on developing a platform for understand the social pulse of the United States during the COVID-19 crisis. We collected a multitude of data that includes longitudinal trends of news topics, social distancing behaviors, community mobility changes, web searches, and other descriptors of the COVID-19 pandemic’s effects on the United States. Our preliminary results show that the number of COVID-19-related news articles published immediately after the World Health Organization declared the pandemic on March 11 have steadily decreased—regardless of changes in the number of cases or public policies. Additionally, we found that politically moderate and scientifically grounded sources have, relative to baselines measured before the beginning of the pandemic, published a lower proportion of COVID-19 news articles than more politically extreme sources—a fact that has implications for the spread and consequences of misinformation during the pandemic. We suggest that further analysis of these multi-modal signals could produce meaningful social insights and present an interactive dashboard to aid further exploration. 1
We present and begin to explore a collection of social data that represents part of the COVID-19 pandemic's effects on the United States. This data is collected from a range of sources and includes longitudinal trends of news topics, social distancing behaviors, community mobility changes, web searches, and more. This multimodal effort enables new opportunities for analyzing the impacts such a pandemic has on the pulse of society. Our preliminary results show that the number of COVID-19-related news articles published immediately after the World Health Organization declared the pandemic on March 11, and that since that time have steadily decreased-regardless of changes in the number of cases or public policies. Additionally, we found that politically moderate and scientifically-grounded sources have, relative to baselines measured before the beginning of the pandemic, published a lower proportion of COVID-19 news articles than more politically extreme sources. We suggest that further analysis of these multimodal signals could produce meaningful social insights and present an interactive dashboard to aid further exploration. 1
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