I recently purchased some carbon rims for my road bike and there appears to be some buldging forming around the rim. I'm no expert in carbon fibre and I was wondering whether the wheel is still safe to ride on. The last thing I want to happen is the whole wheel just snap! I have attached some photos for reference
Carbon fiber structures are made from a bunch of sheets of carbon fiber and resin. Manufacturers will lay up a bunch of sheets on a mandrel (like a mold), and then bake that under pressure in an oven to set the resin. That will cross-link all the hydrocarbon molecules in the resin, causing it to harden. To Nathan's comment on the original question, if you get the finished structure too hot, that can cause the resin to break down - it's not a thermoplastic that will retain its original properties once it cools down, it will be permanently damaged. Depending on the resin's transition temperature, that can happen under prolonged braking on a long descent (for rim brake carbon wheels, which look to be what the OP has), or sometimes if you leave the wheels in the path of your car's exhaust pipe (i.e. bicycle on hitch rack).
In the photos presented, the rim sidewalls look warped. To one of the other comments on the original post, I'm not sure that overinflation alone would do this; I'd expect the tire to blow off before the rim cracked. And I'd expect the rim to crack rather than warp. I'd guess that warping like in the photo would either occur from the factory or it would be caused by a temperature excursion (two potential causes outlined in the first paragraph).
If the rim came this way from the factory, it should not have passed quality control. This amount of warping will probably cause rim brakes to thump when they pass the warped area. I'm not sure if this is a structural problem (assuming the resin wasn't damaged from a temperature excursion), but it would definitely be a QC issue because it would impair braking.
On the other hand, if this did occur after a temperature excursion, the manufacturer might deny warranty on the grounds of user error - these days, some major wheel manufacturers are moving towards no-fault warranties, and I don't know what type of warranty applies here.