Cork in a Glass

Relevant wiki: Surface Tension

The reason that a cork drifts to the side of a glass is that it floats to the highest point. Since water "clings" to the glass, the highest point is around the edge of the water. To get the cork to float in the middle of the glass, all you have to do is fill the glass as much as possible. The water will form a convex shape above the glass (with the highest point at its center). This is where the cork will settle.

Relevant wiki: Surface Tension

the cork will try to stay at the highest point to maintain its inertia.

so, if we fill the glass with water,then the upper portion of the glass will make about a convex shape and the center portion of the glass will be the highest point and the cork will be in the center of the glass.

When the weight of the object is less than the buoyancy force the object will accelerate in the direction opposite to the force of gravity until both forces become equal (as they're already opposite). The less of the object is submerged in the liquid the less the buoyant force. The volume of the cork submerged in the liquid will decrease as the cork will try to accelerate out of the liquid (due to $F_{buoyancy}>F_g$ ). At some point the volume of the cork submerged in the liquid will decrease such that $F_{buoyancy}=F_g$ for which the cork will have no net acceleration. Before $F_{buoyancy}=F_g$ becomes stable, the cork will lose it's velocity due to $F_{buoyancy}<F_g$ .

Due to water tension when the glass is filled completely the meniscus of the water becomes convex rather than concave and the cork floats in the middle this is just a fancy way of saying the same thing as others have said in their solutions.

The "problem" is surface tension. How do you eliminate it? If you put a drop of soap in the water, in theory you should significantly reduce it so that if the cork is placed in the middle of the glass it should stay there.

The glass must be filled up slightly above the top before it starts overflowing. At this point the height in the middle or somewhere close has to provide static pressure higher than the liquid height by the wall due to capillary pressure. The cork will try floating ABOVE the water due to buoyancy force and will seek the highest point.

I see two sensible points in the comments I have read: 1) try it! 2) What about surface tension?
I think the selection of answers for this question is incomplete.

Surface tension is caused by adhesion between water and a different material, glass for example, which causes a meniscus at the edge touching the glass. But a meniscus can also form around the cork! If the cork moves, or floats, to the highest point, it will not go to the edge if its own meniscus is higher than the meniscus at the glass. Filling the glass completely is clever, but unnecessary for a "No" answer. However, the cork will most often go to the edge even when it is carefully placed in the center of the water because minute residual motion in the water can move the cork until it is "captured" by the meniscus at the glass edge.

Add a substance to the water like surfactant which lowers the surface tension of the water, just like surfactant dies in the alveoli of the lungs. Soap will do the same thing.