Beyond The Lecture

School Environment and Sustainable Development Goals Beyond 2030

This book on School Environment and the SDGs Beyond 2030 is a continuation of our maiden, second and third publications on School Environment in Nigeria and the Philippines, published in February, 2015; School Environment in Nigeria, Ghana and the Philippines published in March, 2017; and School Environment in Africa and Asia Pacific published in July, 2018. The philosophy being that since there is a shift from globalization to internationalization and to cross-border education, there is the urgent need to revisit some topical issues in our school environment towards the realization of an internationalized, qualitative and cross-border teaching and learning, using the Sustainable Development Goals as a yardstick.

From Groups to Geometry and Back

Groups arise naturally as symmetries of geometric objects, and so groups can be used to understand geometry and topology. Conversely, one can study abstract groups by using geometric techniques and ultimately by treating groups themselves as geometric objects. This book explores these connections between group theory and geometry, introducing some of the main ideas of transformation groups, algebraic topology, and geometric group theory. The first half of the book introduces basic notions of group theory and studies symmetry groups in various geometries, including Euclidean, projective, and hyperbolic. The classification of Euclidean isometries leads to results on regular polyhedra and polytopes; the study of symmetry groups using matrices leads to Lie groups and Lie algebras. The second half of the book explores ideas from algebraic topology and geometric group theory. The fundamental group appears as yet another group associated to a geometric object and turns out to be a symmetry group using covering spaces and deck transformations. In the other direction, Cayley graphs, planar models, and fundamental domains appear as geometric objects associated to groups. The final chapter discusses groups themselves as geometric objects, including a gentle introduction to Gromov's theorem on polynomial growth and Grigorchuk's example of intermediate growth. The book is accessible to undergraduate students (and anyone else) with a background in calculus, linear algebra, and basic real analysis, including topological notions of convergence and connectedness. This book is a result of the MASS course in algebra at Penn State University in the fall semester of 2009.

Beyond Basic Care