This page of The Last of Us game guide contains a walkthrough of the Science Building stage as part of The University chapter. This level consists of a solid dose of exploration (look for collectibles) and encounters with the bandits. During the final part of the stage Joel will be seriously injured and that outcome cannot be avoided. Our walkthrough will tell you how to escape from the university.
You need to get to the door that the yellow and the red cable go under. Before you cross the door, search through the room at the end of the corridor. You need to use your knife to open it.
As you follow the cables, you will eventually reach a desk with a slip of paper on it. Read the document and climb up the stairs.
Go through the door to the left and go ahead. Search through the rooms that you pass by.
This way, you will reach a room with infected monkeys.
In the next location, Joel will find a voice recorder, which will start a short cutscene.
After it finishes, backtrack to the exit by following the red flares.
Unfortunately, on your way out, you will be constantly nagged at by bandits. The first two enemies appear right after you walk into the corridor. Kill them, and then pick up the hatchet dropped by one of the assailants.
More bandits will come from the opposite end of the corridor. It is worthwhile to eliminate as many of them as you can at the distance, still before they take cover.
After you eliminate the enemies downstairs, try and open the door on the terrace. There will be a bandit to come through, and he will grab Joel by his throat. To loosen his grip, start tapping the square.
Eventually, you will make it to a lower level and you will be pierced with the metal rod sticking out from the floor.
You now need to keep following Ellie and kill the enemies you encounter.
Once you mount the horse, the chapter will draw to an end.