A reduction for the distinct distances problem in R^d

We introduce a reduction from the distinct distances problem in R^d to an incidence problem with (d−1)-flats in R^(2d−1). Deriving the conjectured bound for this incidence problem (the bound predicted by the polynomial partitioning technique) would lead to a tight bound for the distinct distances problem in R^d. The reduction provides a large amount of information about the (d−1)-flats, and a framework for deriving more restrictions that these satisfy. Our reduction is based on introducing a Lie group that is a double cover of the special Euclidean group. This group can be seen as a variant of the Spin group, and a large part of our analysis involves studying its properties

A reduction for the distinct distances problem in R^d

We introduce a reduction from the distinct distances problem in R^d to an incidence problem with (d−1)-flats in R^(2d−1). Deriving the conjectured bound for this incidence problem (the bound predicted by the polynomial partitioning technique) would lead to a tight bound for the distinct distances problem in R^d. The reduction provides a large amount of information about the (d−1)-flats, and a framework for deriving more restrictions that these satisfy. Our reduction is based on introducing a Lie group that is a double cover of the special Euclidean group. This group can be seen as a variant of the Spin group, and a large part of our analysis involves studying its properties

Scattering diagrams from holomorphic discs in log Calabi-Yau surfaces

We construct special Lagrangian fibrations for log Calabi-Yau surfaces, and scattering diagrams from Lagrangian Floer theory of the fibres. Then we prove that the scattering diagrams recover the scattering diagrams of Gross-Pandharipande-Siebert and the canonical scattering diagrams of Gross-Hacking-Keel. With an additional assumption on the non-negativity of boundary divisors, we compute the disc potentials of the Lagrangian torus fibres via a holomorphic/tropical correspondence. As an application, we provide a version of mirror symmetry for rank two cluster varieties.First author draf

The Pedagogical Value of Polling: A Coordinated 2012 Exit Poll Project across Diverse Classrooms

Several previous studies have demonstrated that student exit polling has educational value and promotes civic engagement (Berry and Robinson 2012, Evans and Lagergren 2007, Lelieveldt and Rossen 2009, and others). The authors of this paper have created assignments and an instructor\u27s manual on running student exit polls in undergraduate courses. Three institutions used these assignments during the fall 2012 semester. Working together, these instructors created an opportunity for their students to participate collaboratively with others in survey design and data analysis. This effort further provided assessment data on the effectiveness of this pedagogical approach for student engagement outside of the classroom in different communities and course contexts. Student surveys measured the impact that this experience had on their understanding of their own community, their relationship to the national community, their understanding of survey methodology, and descriptive statistics. Do students learn more about their community or the scientific process? Does it matter whether the course is designed primarily around politics, statistics, or public opinion? This paper addresses these questions and how these effects vary across different types of students and classrooms