TL;DR
For this kit to even be considered for use in the game, the damage would have to be reduced to the point that he will be pretty much useless for everything but a peel support mage.
As of right now, your damage output has enough to theoretically one-shot the baron at the 35 minute mark.
(is not like Viktor)
Did you really have to add that into the first sentence of your concept post? That makes me question the usefulness of this concept before I even read the rest of it.
Mana: 340-860
This is kind of off topic, but why add in the amount of mana your concept has if you don't add in the proposed mana cost of your abilities? The only thing that even mentions mana is his passive, and that's referring to the amount he gains back.
Passive: Reverse Power
Brogan and his nearby allies are immune to displacement effects. After an ally blocks one, Brogan regenerates 8% of his maximum health and mana.
The first sentence alone makes this guy broken AF in general, yet alone to assassins. As for the second sentence, this is the kind of passive you'd see in a peel support tank like Tahm or Braum or a Sustain Bruiser like Darius.
Sustainability is not something that should be on an Assassin, since they are designed as opportunists -- to get in, kill the weak ones, and get out as quickly as possible. The only defense against this is CC and/or quick removal. Because of this, this passive simply does not belong on an assassin. Rather change the goal of your champion, or change the passive. And you'll have to change the passive anyways because permanent complete immunity to any form of CC simply can't be allowed into League.
Q: Force Pulse
Brogan causes an area to pulse with high gravity, slowing enemies in it by 30-52/54/56/58/60% depending on how close they are to the center. When he is in the area, it overloads and pulses, dealing 30/50/70/90/110 (+20% AP) magic damage to all enemies every 0.5 seconds. The field lasts 3 seconds.
So like a combination of Ekko's and Viktor's W? Don't get me wrong; there's nothing wrong with an ability of one champion being similar to another's. I'm just making sure I understand it correctly.
Anyways, your Q is compatible to -- if not better than -- Katarina's ult (One champion's basic ability should not be comparable to another's ult -- especially when the ult can be pretty powerful if the person's fed).
Brogan's Q at level 5; as you said, does 110 +20% AP every 0.5 seconds for 3 seconds. This ticks 6 times mean that it does:
660 +120% AP during the cast time.
-- 220 +40% AP a second.
Thing is though is that you can get an ability to level 5 when the champion is at level 9 (I think; I may be wrong with this number, but I know it's before 11). This means that you can have the Q do its full potential damage before Katarina can even get to level 2 with her ult.
At level 1, her ult does 25 Base Damage with +19% AP and +22% Bonus AD every 0.166 seconds -- 1/6 a second -- for 2.5 seconds. This means that it can tick 15 times per champion -- up to 3 champions -- during it's channel time.
During the full cast time, she can do 375 Base +285% AP +330% Bonus AD. That's 150 Base +114% AP +132% Bonus AD per second.
"Oh, but Brogan's Q is maxed, while Katarina can get her ult up 2 more levels."
At level 3, her ult does 50 Base.
Calculated out, this means that she does 750 base damage during the cast time. That's 300 per second.
Some other things to keep in mind that would push the scales over if the pure damage hasn't already:
1.) Katarina's ult can be canceled with almost any CC.
2.) His Q can't be canceled.
3.) I'm assuming you can place his Q at a distance similar to Ekko's and Viktor's W.
4.) His Q slows by default.
5.) A Q's CD is typically lower than an Ult.
Basically, for a DoT Q, it needs to do almost no damage.
W: Air Lift
Brogan immediately teleports to his Pulse Field, knocking up enemies for 1.2/1.3/1.4/1.5/1.6 seconds and causes a pulse. For every enemy champion in the field when he teleports, the duration of the field is increased by 1 second.
So, basically you can get the already completely broken Q to deal up to 5 more seconds of damage, on top of being able to keep up to 5 more people in itfor over a second? That means it can last up to 8 seconds.
That's 1,760 +320% AP at full power. That's......more than what Veigar's ult can do to a single target -- 325 +75% AP (x2 if the target is below 1/3 health). Good God.
"But they can just walk out of it."
Don't forget:
1.) They are knocked up for over a second.
2.) ADCs -- typically the targets of Assassins -- are usually fairly immobile with the exception of a few (i.e. Tristana) compared to other champs and require the peel and protection of their allies to survive.
3.) Brogan's Q + W isn't the only thing dealing damage in a team fight.
E: Gravitational Implosion
Brogan causes an implosion at an area, pulling nearby enemies to the center and dealing 60/90/120/150/180 (+45% AP) to 100/150/200/250/300 (+75% AP) magic damage based on how far they traveled to the center.
So now if they try to get away from the Q, they will just be pulled back in with E and do more damage than Orianna's ult in the process? Yea, there's nothing much more I can say about it. I've already spent most of my energy talking about the Q and W; I figured by this point you get the picture.
R: Gravity's Calling
Brogan causes all enemies (global, affects minions also) to be pulled to the center of the map, receiving 100/150/200 (+40% AP) magic damage at the start of the pull and the same damage at the end of the pull. If an enemy dies to the skill, the cooldown of Air Lift will be refreshed.
And it's now time for a 3 For 1 Deal! Not only will Air Lift lock them in place for over a second; and E pull them back to it, but now ANOTHER ability that can pull them back into the Force Pulse that will in turn allow you to reuse Air Lift again just to get in that 8th penta-kill for the game!
I'm sorry. Did you even think this concept through? Your whole kit is based around your Q. This champion is an AoE "assassin" that will fuck up the entire enemy team by himself without any issue? Also, you typically don't have a champion now a days that has their whole kit based around a single ability that they have. Sure, that used to be the case, but Riot has blatantly said that they are moving away from that as well as they can. I seriously doubt they would introduce a champion that would reinvent the issue they have been trying to hard to remove.
As a side note; I've also read some of your responses. You say "Give me your thoughts!" but you've seemed to chew out everybody that's responded negatively to your concept.
EDIT
I forgot to also mention that the Ult also refreshes the W. Because of this, Brogan's Q can be active for up to 12 seconds (Knock up 5 people, kill 1 with ult, knock up 4 people).
This means that the Q can do up to 2,640 Base +480% AP per person.
Now, if we go with a full AP build:
(Yes, I know the passive doesn't stack)
he will have 720 Item AP. The 35% extra AP from the Rabadon's UNIQUE Passive will be added in later.
-Now of course one would be able to pull off even more damage by doing a more meta build, but I'm just doing this to make the calculations easier. Because of that, this is actually the low end of what it can potentially pull off.
Also, there is 
, bringing the AP up to 770 AP.
A full scaling AP rune page will give_
Marks: 16.2
Seals: 16.2
Glyphs: 27.54
Quints: 23.22
Total: 83.16 Rune AP
Masteries:
Sorcery: +2% Ability Damage (will be adding it in at the very end)
Natural Talent: +3 AP
Bounty Hunter: +5% Damage (will be adding it in at the very end)
--I would like to add that I purposefully left some out that would have made the calculations a bit too annoying, even for me.
Neutral Buffs:
3 Fire Dragons: +36% AP with Elder (adding in with Rabadons)
Hand of Baron: +40 AP
He will have a total of:
1,532.4336 AP (I tend not to round until the very end of a calculation)
With that much AP, Brogan's Q will do 832.97344 damage a second before Elder.
Elder Dragon: 45x+225 where X is the number of elder dragons killed. Since you typically can't get a full Rabadon build until about an hour in, I'm going to assume 3. This brings it up to 360 over 3 seconds. So it's 120 damage a second.
Now, if you combine those two with the Sorcery and Bounty Hunter (I'm going to assume it translates into +7% Damage), that is:
1,020 Damage a second per champion (I went ahead and rounded at this point).
Here is the formula I will be using to calculate the total damage that Brogan's Q will do at full potential:
1,020 DPS x ((5 Champions x 3 Seconds) + (5 Champions x 5 Seconds) + (4 Champions x 4 Seconds))
TOTAL Q DAMAGE OUTPUT: 57,120
As for the other abilities?
His E does:
((300 base + 75% AP + 360 from Elder) + 7%) x 5 Champions
TOTAL E DAMAGE OUTPUT: 9,680
His R does:
((200 + 40% AP + 360 from Elder) +7%) x 5 Champions
TOTAL R DAMAGE OUTPUT: 6,275
As a side note I just noticed; why does his Q do more damage than his E and R combined?
FULL SINGLE COMBO DAMAGE OUTPUT: 21,667 (on all 5 champions combined)
To put this number into perspective, this is enough to 1-shot Baron Nashor at the 35 minute mark (this is including his magic resist).
This isn't even the worst of it:
1.) If you go with a more meta build, damage output would be far greater due to item passives.
2.) This is excluding damage-based Masteries.
