The purpose of the present article is to demonstrate that by adopting a unifying differential geometric perspective on certain themes in physics one reaps remarkable new dividends in both microscopic and macroscopic domains. By replacing algebraic objects by tensor-transforming objects and introducing methods from the theory of differentiable manifolds at a very fundamental level we obtain a Kottler-Cartan metric-independent general invariance of the Maxwell field, which in turn makes for a global quantum superstructure for Gauss-Amp`ere and Aharonov-Bohm “quantum integrals.” Beyond this, our approach shows that postulating a Riemannian metric at the quantum level is an unnecessary concept and our differential geometric, or more accurately topological yoga can substitute successfully for statistical mechanics.

]]>Thus began the month-long, collaborative project at Loyola Marymount University between the honors underclassmen in HNRS 140, On Motion and Mechanics, taught by Alissa S. Crans, and the senior applied mathematics majors in MATH 495, Mathematical Modeling, taught by Robert Rovetti. During a period of four weeks, six teams of freshman and sophomore liberal arts honors students, each led by a senior math major, set out to reconstruct an old photograph using a mathematical technique based on straightforward geometry. Along the way they would run into inaccessible landscapes, blocked views, and busy schedules, but ultimately they emerged with both a finished product and a clearer understanding of what it means to apply a theoretical method to a real-world problem. We begin by describing the courses and the assigned project itself, and then we reflect on the pedagogical goals of the project and various observations made by both students and instructors.

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