running and acrobatics

let's say that you have an ability that grants you the ability to pass through threatened space without halving your speed.

like p.e.:

Swashbuckler's Grace (Ex):
At 7th level, while the swashbuckler has at least 1 panache point, she takes no penalty for moving at full speed when she uses Acrobatics to attempt to move through a threatened area or an enemy's space.

can you RUN through them?

Speed

Your speed tells you how far you can move in a round and still do something, such as attack or cast a spell. Your speed depends mostly on your size and your armor.

If you use two move actions in a round (sometimes called a "double move" action), you can move up to double your speed. If you spend the entire round running, you can move up to quadruple your speed (or triple if you are in heavy armor).

acrobatics:

Move Through Threatened Squares

In addition, you can move through a threatened square without provoking an attack of opportunity from an enemy by using Acrobatics. When moving in this way, you move at half speed. You can move at full speed by increasing the DC of the check by 10. You cannot use Acrobatics to move past foes if your speed is reduced due to carrying a medium or heavy load or wearing medium or heavy armor. If an ability allows you to move at full speed under such conditions, you can use Acrobatics to move past foes. You can use Acrobatics in this way while prone, but doing so requires a full-round action to move 5 feet, and the DC is increased by 5. If you attempt to move through an enemy’s space and fail the check, you lose the move action and provoke an attack of opportunity.

What are you asking?

If you mean: Using a full-round action to move while also using acrobatics to move through threatend area or an enemy space with that 7th level Swashbuckler ability: Yes you can.
Unless your speed is reduced due to carrying a medium or heavy load or wearing medium or heavy armor, then you can't do it at all.

You're focusing on the wrong sentence. Try this:

In addition, you can move through a threatened square without provoking an attack of opportunity from an enemy by using Acrobatics. When moving in this way, you move at half speed. You can move at full speed by increasing the DC of the check by 10. You cannot use Acrobatics to move past foes if your speed is reduced due to carrying a medium or heavy load or wearing medium or heavy armor. If an ability allows you to move at full speed under such conditions, you can use Acrobatics to move past foes. You can use Acrobatics in this way while prone, but doing so requires a full-round action to move 5 feet, and the DC is increased by 5. If you attempt to move through an enemy’s space and fail the check, you lose the move action and provoke an attack of opportunity.

You can move through squares at full speed without any additional abilities, but the DC is 10 higher. As Rub-Eta said, you can't do this if your speed is reduced by armor or encumbrance. (Be warned: some people may say "increasing the DC by 10" =/= "taking a penalty", so this ability could be worthless at some tables.)

As far as running goes, Run says "You can run as a full-round action. If you do, you do not also get a 5-foot step. When you run, you can move up to four times your speed in a straight line (or three times your speed if you're in heavy armor). You lose any Dexterity bonus to AC unless you have the Run feat."

So to combine the two, you would need to run in a straight line only, and you would still lose your Dex bonus to AC (which would be bad for a swashbuckler, I imagine).

I don't think there's any way to run without provoking.

Say your base speed is 30 - if you double-moved using Acrobatics normally, you could move 30 feet. If you double-move and add 10 to the DC, you can move 60 feet. Presumably, if you were running and moving 120 feet, the DC would be more difficult still.

I don't think "full speed" means "at a run", but rather "at normal walking speed". Because, it really makes very little sense that someone sprinting in a straight line could avoid an AoO.

I don't think there's any way to run without provoking.

Say your base speed is 30 - if you double-moved using Acrobatics normally, you could move 30 feet. If you double-move and add 10 to the DC, you can move 60 feet. Presumably, if you were running and moving 120 feet, the DC would be more difficult still.

There's nothing in the rules for additional penalties to Acrobatics (or Stealth, for that matter) when using the Run action. You're always free to rule that way as a GM, but it would be a house rule.

I don't think "full speed" means "at a run", but rather "at normal walking speed". Because, it really makes very little sense that someone sprinting in a straight line could avoid an AoO.

You're correct that "full speed" doesn't mean "using the Run action", but I haven't see anything in the rules that says you can't try to avoid an attack of opportunity while using the Run action. "Straight line" in the game only means "in a series of 5 foot squares", which gives you a lot of room to duck and weave without breaking the straight line.

I think hurdles provides a good real world analogy to using Acrobatics while using the Run action. In a hurdle race, the runners are running flat out, and yet they have to make Acrobatics checks to avoid obstacles. The current record for 400 meter hurdles is 46.7 seconds, which would work out to about 19 miles per hour. Technically, in game mechanics, this is completely impossible:

You can't run across difficult terrain or if you can't see where you're going.

A run represents a speed of about 13 miles per hour for an unencumbered human.

Most hurdlers would have taken the Run Feat, and probably Fleet at least once by the time they were Olympians. The world record holder has taken Fleet twice, and Run. They now have a base move of 40 feet, and run five times their normal speed.

Hurdling might count as acrobatics, but I don't think the hurdles are attempting an Attack of Opportunity on the hurdler.

i ask purely academically.

i was checking dares, and saw:

Run Like Hell (Ex)

While this dare is active, your speed increases by 10 feet, and you retain your Dexterity bonus to your AC while running. You regain 1 grit or panache point the first time you are more than 100 feet away from your closest enemy.

and:

Derring-Do (Ex):

At 1st level, a swashbuckler can spend 1 panache point when she makes an Acrobatics, Climb, Escape Artist, Fly, Ride, or Swim check to roll 1d6 and add the result to the check. She can do this after she makes the check but before the result is revealed. If the result of the d6 roll is a natural 6, she rolls another 1d6 and adds it to the check. She can continue to do this as long as she rolls natural 6s, up to a number of times equal to her Dexterity modifier (minimum 1).

and thought:

hmm, i wonder if one can combine them:

assuming that the swashbuckler is at low HP and has only 1 panache.
Can he use Derring-do (spending the 1 panache), gaining a +1d6 on acrobatics, roll DC+10, and Run the hell away at 160ft, while avoiding AoO, keeping his Dex on AC, and at the end of his movement recover the spent panache?

edit:changed it a bit because you can't use a dare and still use swashbuckler's grace. duh, that was stupid of me.

but the question remains:
Can you use acrobatics +10 to DC to nullify AoO while RUNNING?