Heard a physics professor say that when an aircraft flies inverted, the wing assumes a significant angle of attack (by pushing the stick forward) to continue to generate the necessary "upward" lift vector to maintain level flight. I'm confused. I thought the lift "vector" is actually reversed and points towards the ground (not upward) during inverted SUSTAINED straight-level flight? What am I missing here? The horizontal stabilizer is basically an inverted airfoil, and its "lift vector" points downwards, giving the airplane tail-down force. How is the wing any different when it's inverted?
In sustained horizontal inverted flight, the wing is generating a skyward lift force. In most aircraft the pilot will be exerting a significant "push" force on the stick or yoke, causing the stick or yoke to be well forward of the position it would be in level upright flight at the same airspeed (or at any airspeed for that matter.) Regardless of the airfoil of the horizonal tail, it is capable of creating an upforce in the aircraft's reference frame when the elevator is sufficiently deflected. Due to downwash from the wing, the required deflection may be less than you might otherwise expect.
Most horizontal tails have symmetrical airfoils.
"How is the wing any different when it's inverted?" -- the position of the elevator is key. With sufficient forward stick deflection, the top of the wing rather than the bottom will be presented to the airflow.
Even a flat-bottom, curved-top airfoil can sustain inverted flight in this manner. Examples are Gentle Lady and Radian radio-controlled model sailplanes, which I'd guestimate can sustain a glide angle of 20 degrees below the horizon or better in power-off inverted flight, given that they are set up with sufficient elevator travel. It is also possible to demonstrate constant-altitude sustained inverted flight in a radio-controlled model airplane with only moderate engine power (e.g. not capable of a sustained vertical climb) even with flaps strongly deployed (about 45 degrees) in the usual direction in the aircraft's reference frame. Again, one key is to have sufficient elevator travel.
Remember, in unaccelerated linear flight, net force must add to zero. The forces in sustained linear inverted flight are weight, thrust, drag and lift. The vector sum of these forces could never be zero if the wing were generating an earthwards force.
Remember also that (except for apparent forces created by rapid rotation rates perceived by a pilot sitting far from the CG) the force a pilot "feels" in flight is nothing other than the net aerodynamic force generated by the aircraft, which is normally dominated by the lift force generated by the wings. As you hang in the seat belts during sustained inverted flight, that force you feel from the seat belts digging into your shoulders is really the skyward force generated by the wings. Conversely when you feel the seat bottom pushing lightly against you as you float over the top of a loop at a reduced but still positive G-loading, that is really the earthward force generated by the wings.
This is easiest to understand with a fully symmetrical wing, which indeed are popular with aerobatic aircraft. The direction of "lift" has nothing to do with sky or ground and everything to do with your angle of attack to the relative wind.
When inverted, you simply (as the professor said) push forward on the stick, which makes the wind strike the "top" of your wing. Inverted, this does the trick as lift is now towards your floor, but gravity is toward your head.
Right side up, you make the wind strike the bottom of your wing by pulling back on the stick.