There are six other topics that we could count among the machlokot (disputes) where Beit Hillel is more strict and Beit Shammai is more lenient. On a Torah level, having meat and milk together is a problem, but that's only with actual meat and milk. Poultry—chicken or turkey—with dairy would only be asur de-rabbanan (rabbinically prohibited). So the machlokot here will discuss how far we go with that de-rabbanan prohibition on chicken, poultry, and dairy. Poultry is allowed to be put on the same table as cheese or other dairy, but it's not allowed to be eaten together, even if they were not cooked together. As long as they're eaten separately, it's okay according to Beit Shammai. But according to Beit Hillel, poultry is not allowed to be put on the same table as cheese, because we're nervous that eventually it's going to lead to mixing regular meat with dairy products. And, of course, it's not allowed to be eaten with cheese.
The second machloket we'll have here has to do with Terumah, which is the produce that's separated in Eretz Yisrael and given to the Kohanim (priests). A person is not allowed to take Terumah from one product. For example, you can't take Terumah of wheat on behalf of barley. The question becomes: What about processed and unprocessed forms of a single variety of produce? Is that permissible? The Mishnah says, “You're allowed to separate olives as Terumah for olive oil, and grapes as Terumah for wine.” Olives and olive oil are from the same item, both from olives. And wine and grapes are from the same thing. So, even though these are different forms of the same item, Beit Shammai hold, k'divrei (according to the words of) Beit Shammai, that they're permissible. You can do that. But Beit Hillel say, no, you're not allowed to separate olives as Terumah for olive oil or grapes for wine, because unprocessed is different from processed. Grapes are different from wine, and olive oil is different from olives.
The third machloket we'll have is about a grapevine. A grapevine planted too close to actual grain creates a problem, and both have to be burned. If the grapevine was part of a full vineyard, then that entire vineyard has to be burned. The question becomes: What is a vineyard? How do we describe a vineyard? If it's part of a larger field, we're not going to say that you have to get rid of the whole field. So what is considered to be a single vineyard?
The fourth machloket we'll have is about a vineyard. If a person planted grain within four amot (cubits) of a vineyard, and the vineyard has two or more rows, Beit Shammai says that one full row of vines opposite the grain has to be burned; it becomes prohibited because a single row is considered to be a vineyard. And Beit Hillel says, no, two rows of the vineyard opposite the grain are considered to be a vineyard, and therefore that's the minimum size that has to be burned.
The Torah gives us a mitzvah (commandment) of challah: On every dough or bread that's made, a portion has to be given over to the Kohen, and only the Kohanim are allowed to eat it. The question becomes: What about a certain type of dough called me'isa, made by taking flour and adding it to boiling water, then cooking it until it thickens, and then putting it into an oven to bake? Ha-me'isa, Beit Shammai potrin, Beit Shammai exempts the me'isa from the mitzvah of challah because it was cooked before it was baked. According to them, once something is already cooked, baking it afterwards doesn't obligate it in challah. Beit Hillel requires that challah be separated from it; as long as it's baked afterwards, it's considered to be bread.
The next machloket is about dipping a utensil into a Mikveh of rainwater that's running down a mountain. As long as there's a combination of 40 se'ah (a volume measure) of water from the top of that stream on top of the mountain to the bottom, it's considered to be a kosher Mikveh, and you can dip a utensil in there. And Beit Hillel says no, you're not allowed to dip a utensil in that type of stream. In order to be considered a Mikveh, you need 40 se'ah gathered in one spot.
Every member of the Jewish people has to take part in the Korban Pesach (Passover offering) on Erev Pesach (Passover eve). In the Beit HaMikdash, they would slaughter the Korban Pesach. On the afternoon of Erev Pesach, which is the 14th of Nisan, the meat is then taken and eaten that night at the Seder (Passover seder meal). A person who converts to Judaism is also obligated to take part in the Korban Pesach. But if somebody converted on Yud-Dalet (the fourteenth), the day that the Korban is brought, we have a machloket about whether he could participate in the Korban Pesach.
If a person converted to Judaism on Erev Pesach, he got the Milah (circumcision) on that day. Say Beit Shammai: He dips into a Mikveh to complete his conversion right then on Erev Pesach, and then he could eat the Korban Pesach that night. And Beit Hillel says, ha-poresh min ha-orlah, one who separates from the foreskin—say Beit Hillel—no, if somebody gets a Brit Milah (covenant of circumcision), he's like he's separating from the kever. A convert who gets a Brit Milah and separates from his foreskin is like somebody who separates from a grave. In other words, when a person separates from a grave, he becomes tameh (ritually impure) because he's involved with a human corpse, and he's not allowed to eat any Kodashim (sacred offerings) until he purifies himself with the Mei Hatat, with the water of the Parah Adumah, on the third and the seventh day of his Tumah and Tahara (ritual impurity and purity). So the Hachamim (rabbis) were nervous. They said that a person who converted to Judaism is also not allowed to eat from the Korban Pesach right away. He has to go through a seven-day purification process first, after completing his conversion. They were nervous that maybe the person, while he was a non-Jew the day before, had contact with the corpse. And that contact with the corpse wouldn't make him impure because he was non-Jewish at the time. So the following year, he may also have contact and not realize that it was a problem. He says, “You know what? Last year I had contact with a corpse, and I ate the Korban Pesach the day later, after I converted,” not knowing that there's a difference between a Jew and a non-Jew with Tumah and Tahara. The following year he'll be near a corpse, and he won't get the proper purification process. So Hachamim said, “You need to go through the proper purification of seven days after the conversion to make sure that the person is fully tahor (ritually pure), in order to allow him to eat from the Korban Pesach.” The fact that he completed his conversion on Erev Pesach doesn't allow him to eat from the Korban.