The umbrella was probably purposely placed on the floor. You were right to leave it there initially, as the owner could have returned for it.[2] However, after a while, it was fine to pick it up, as security forbade people who left the stadium to return. If there were no siman (identifying sign), you could keep it (if we could assume the owner realized the umbrella was missing before you picked it up[3]). However, between the name and the location (there were seat numbers), there are simanim. Therefore, at first glance, you would be required to effectively publicize your find, and if no one responded, you would have to keep it indefinitely until you find the owner.[4]
One might argue that since people came to this event from many different places and people are usually not especially careful with their umbrellas, the owner would not believe he could retrieve it, despite the simanim. It is a good question whether we would follow standard rules or accept such a claim; consider that in Talmudic times, it also was difficult to retrieve lost objects.[5] However, the claim is not clearly applicable in practice either. MetLife Stadium has a lost-and-found service and asks people to give found articles to a worker. (When that is the most practical system, one need not be concerned that workers do not know the proper rules for returning lost objects.[6]) Also, the owner might have called a friend still at the stadium and asked him to retrieve it.
In general, if one loses something in a public place with a majority of non-Jews, we assume the owner had yeiush,[7] and there is no obligation to return it.[8] In your case, this is a somewhat tricky question. On the one hand, the area was occupied overwhelmingly by Jews at the time of the loss, so perhaps we should not assume yeiush. On the other hand, if Jews are there only infrequently, whereas a majority of non-Jews are usually in attendance (including the workers who cleaned up after the siyum was over) and are more likely to notice the object than the Jews were, perhaps we should assume yeiush.[9] Moreover, we can consider whether in this case, the participants were riveted to the event,[10] such that they were not looking to the floor for mitzvot (or free umbrellas), and it is more likely that people who were there after the event would find a lost object. Also, an umbrella on the floor is not initially a sign of a lost object, so Jews might not know they have to return it. Thus, it can be plausibly argued that there was yeiush in this case.
It is important to note that the Rama[11] says that when local law requires returning lost objects, one is obligated even when classical Halacha does not require it. It is not clear what New Jersey law is in a case like this. In the final analysis, you probably may keep the umbrella, but there is an element of going beyond the letter of the law to return it when feasible.[12]
Can you find the owner? Announcing the find in a local shul[13] is all but futile, and it is very unlikely that MetLife would have details of someone looking for an umbrella that was lost on a particular date. There is a website for reporting lost and/or found objects in Israel, but I did not find one in the NY/NJ area.[14]
What if we could not assume initial yeiush, but you do not want a stash of objects waiting for Mashiach with unrealistic chances of returning them now? Since an umbrella is readily replaceable, you could record its value and simanim to cover the remote possibility someone will step forward with simanim.[15] It might be nobler, however, to give it to someone in need so that the owner can receive some merit of tzedaka.[16]