Any honest attempt to navigate and understand the world, both in the large and in the small, is met with fierce resistance by the presence of <em> boundary value problems </em>.
As they lived out their lives, even the earliest humans could not avoid cliffs, shores, and other interfaces.
All of these are examples of boundaries.
The modern world demands that we rigorously understand natural phenomena through their description by <em> partial differential equations </em>.
Almost always, the setting is a bounded region within a geometric structure.

The aims of this course is to give a modern mathematical treatment of the linear and elliptic aspects of this subject.
Although relevant to the study of physical phenomena, the narration of this topic will be motivated by perspectives emerging from global analysis.
The setting is elliptic differential operators acting on Hermitian vector bundles over measured manifolds.
A description of the maximal domain of such an operator will be given in terms of a canonical  operator built out of the boundary trace map.
Rather than focus on particular boundary conditions, all possible boundary conditions described from a fundamental space on the boundary called the <em> Czech space </em>.

Although we commence with a description of general-order operators, we will focus our attention on first-order elliptic differential operators, expositing  recent developments due to Bär-Bandara in this setting.
The course will culminate with the proof of the famed index theorem of Atiyah-Patodi-Singer for Dirac-type operators.
Geometric applications of this theorem, including consequences to the study of positive scalar curvature metrics, will also be considered.