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We consider the moduli space Mr of polygons with fixed side lengths in five-dimensional Euclidean space. We analyze the local structure of its singularities and exhibit a real-analytic equivalence between Mr and a weighted quotient of n-fold products of the quaternionic projective line HP 1 by the diagonal P SL(2, H)-action. We explore the relation between Mr and the fixed point set of an anti-symplectic involution on a GIT quotient Gr C (2, 4) n /SL(4, C). We generalize the Gel ′ fand-MacPherson correspondence to more general complex Grassmannians and to the quaternionic context, and realize our space Mr as a quotient of a subspace in the quaternionic Grassmannian Gr H (2, n) by the action of the group Sp(1) n . We also give analogues of the Gel ′ fand-Tsetlin coordinates on the space of quaternionic Hermitean matrices and briefly describe generalized action-angle coordinates on Mr.

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This paper shows that the AL (Ablowitz-Ladik) hierarchy of (integrable) equations can be explicitly viewed as a hierarchy of commuting flows which: (a) are Hamiltonian with respect to both a standard, local Poisson operator J , and a new non-local, skew, almost Poisson operator K, on the appropriate space; (b) can be recursively generated from a recursion operator R = KJ −1 . In addition, the proof of these facts relies upon two new pivotal resolvent identities which suggest a general method for uncovering bi-Hamiltonian structures for other families of discrete, integrable equations.

Purpose The aim of this tutorial is to share lessons learned from a speech, language, and hearing sciences department at a land-grant, Hispanic-serving institution (HSI) after revising the graduate admissions review process and subsequent discussions related to equity in assessment. This tutorial describes the department as a case example and includes guiding questions that may be helpful for other organizations. Method An adaptive case study approach was used and centered on supporting students at Hispanic-serving institutions. The approach was characterized by structured collaborative reflection throughout the process and engagement of relevant stakeholders at multiple levels. The adaptive method allowed for interim synthesis of stakeholder discussions to inform subsequent phases of the reflection process. Results This tutorial shares critical motivations, barriers, facilitators, and phases that were identified in moving toward holistic evaluation for graduate admissions. Targeted areas for continued improvement related to diversity, equity, and inclusion are described. Conclusions This tutorial outlines lessons learned from changing graduate admissions practices toward holistic review. Self-reflection prompts are provided for institutions and organizations considering changes to their review process. In order to best support diverse communities, increased workforce diversity is needed in the speech, language, and hearing sciences professions, and holistic review practices are recommended as one way to support increased diversity, equity, and inclusion.