- Lecture 1
- Introduction; structure of jet bundles.

- Lecture 2
- Structure of jet bundles (cont'd); the
*C*-compact-open topology.^{k}

- Lecture 3
- The
*C*-compact-open topology; Lipschitz and locally Lipschitz mappings; the^{∞}*C*^{lip}-compact-open topology for spaces of locally Lipschitz mappings between metric spaces; the*C*^{k+lip}-compact-open topology for spaces of*C*-mappings with locally Lipschitz^{k}*k*th derivative; the*C*^{hol}-compact-open topology; germs of real holomorphic mappings about a subset of a real analytic manifold.

- Lecture 4
- Final and initial topologies for spaces of
*C*^{ω}-mappings; uniform structure for spaces of mappings between manifolds using explicit semimetrics; topologies for spaces of sections of vector bundles using explicit seminorms; weak-PB descriptions for topologies for spaces of mappings between manifolds.

- Lecture 5
- Local, i.e., coordinate, characterisations of topologies for
spaces of mappings; background on locally convex topologies in preparation
for talking about time-varying vector fields, mainly continuity,
boundedness, measurability, and integrability of mappings taking values in
locally convex spaces, with an emphasis on the rôle of the Suslin
property for measurability.

- Lecture 6
- The weak-
*L*topology for spaces of vector fields;*C*^{ν}-Carathéodory vector fields; locally integrally and locally essentially bounded*C*^{ν}-vector fields; flows of locally integrally bounded*C*^{ν}-vector fields; holomorphic extension of locally integrally bounded real analytic vector fields.

- Lecture 7
*C*^{ν}-control systems; holomorphic extension of real analytic control systems; presheaves and sheaves of sets; étalé spaces and the étalé topology.

- Lecture 8
- Morphisms of presheaves; correspondence between sheaves and spaces of
local sections of the étalé space; the stalk topology; sheaf of
time-varying vector fields; local description of sections of sheaf of
time-varying vector fields; bare bones introduction to groupoids; the
groupoid of
*C*^{ν}-local diffeomorphisms.

- Lecture 9
- More about the sheaf of time-varying vector fields; elementary
description of the flow of a time-varying vector field; absolute continuity
of curves of local diffeomorphisms; groupoid characterisation of flow of a
time-varying vector field; the stalk topology and the exponential map for
flows.

- Lecture 10
- Definition of tautological control system; examples of tautological
control systems: (1) tautological control systems from control systems and
vice versa, (2) tautological control systems from differential inclusions and
vice versa, (3) differential inclusions from control systems and vice versa,
(4) tautological control systems from piecewise defined families of vector
fields, (5) tautological control systems from distributions; sheafification
of presheaves of sets of vector fields; implications of sheafification for
(control system)-(differential inclusion) equivalence and for (tautological
control system)-(differential inclusion) equivalence.

- Lecture 11
- Differential inclusions of class
*C*^{ν}; global generators for distributions; open-loop systems; open-loop subfamilies; trajectories; linear control systems (these are not what you think); trajectory correspondence between control systems and their tautological control systems.

- Lecture 12
- Direct images of tautological control systems; morphisms of tautological
control systems; natural morphisms; isomorphisms and natural isomorphisms;
étalé open-loop systems; étalé open-loop subfamilies;
étalé trajectories; orbits (briefly).

- Lecture 13
- Tangent and vertical lifts of vector fields; linearisation of
tautological control systems as tautological control systems on the tangent
bundle; trajectories of linearisation about a reference trajectory;
trajectories of linearisation about a reference flow; linearisation about an
equilibrium point; example from Lecture 1 revisited; linear
controllability.