a lot of +single verb

問題出在 activity這個字
activity 可以做 countable noun 和 uncountable noun.

activity 如果是 uncountable noun, verb 一定是單數, noun 必定是 activity.

There has been a lot of activity. (uncountable noun)

activity 如果是 countable noun, 一定用 plural form, activities, verb 一定是雙數
There have been a lot of activities. (countable noun in plural form)

你的句子沒有提到什麼 activity, 你的句子假定 activity 為 uncountable noun.

Activity (countable noun) = (某一團體的)特殊活動; 帶消遣性的活動

Activity (uncountable noun) = 行動, 活動力, 活力,活躍 ~ abstract noun of “active”

如果是 social activities (社交活動), extracurricular activities (课外活動) 你的句子應該是:

There have been a lot of activities.

“A lot of “ 詳情在你另外的 question 我提到

A lot of activity has the verb (has) in singular form with uncountable nouns=activity.
In colloquial use in affirmative statement the phrase "a lot of " is used.eg:-He will have a lot of activity to run.She knows a lot of activity in the election hall.There has been a lot of activity in an election campaign.