In the sentence: Today we learned that the university is going to close the math department it is necessary to include that, but in the sentence The work I do is very important it is not included. Shouldn't it be: The work that I do...?
There are a number of instances in English, where it is possible, even desirable, to omit that.
that as conjunction with reporting verbs
In your first example sentence, that is used as a conjunction, joining two parts of the sentence. After verbs like learned, discovered, found (out), knew, felt, thought, it is quite natural to omit that, especially in informal speech:
I discovered Julian had borrowed my car without my permission.
I felt he was wrong to do this, but he thought it would be all right.
After the more common reporting verbs, (e.g. say, tell) it is also entirely natural to omit that in informal speech:
I told him I'd be back by ten o'clock but he said he needed me here by nine.
After certain verbs (e.g. replied, shouted) that cannot be omitted and it is not normally dropped after nouns:
The Dean of the Humanities Faculty informed the students that the drama dept was going to close.
He left a message on my voice mail that he was leaving immediately for Vienna.
I replied (to his message) that he should remain in Britain.
He shouted at me that he was fed up with living in Britain.
omitting that in two-word conjunctions
There are a number of two-word conjunctions where that may be omitted. These include so that and now that which we can use to talk about purpose and result and providing that and provided that which we can use to talk about imposing conditions.
In a more formal style we may prefer to retain that, but in an informal style it is often omitted. Compare the following:
We intend to send her to Brazil so that she can perfect her Portuguese.
I spent Easter with Anneke in Switzerland so I could learn to ski.
Now that we've joined the EU, prices are sure to rise.
Now the exams are over I can lie in bed all morning.
Provided that / providing that you sign the contract before we join the EU, you won't have to pay VAT.
You can borrow my DVD player, providing / provided you return it on Monday.
omitting that as relative pronoun
In your second example sentence, that is used as a relative pronoun, introducing a relative clause. When that is the object in a relative clause, as in your example, we normally leave it out:
The work (that / which) she does for this company is much appreciated.
The representatives of the company (that / who) I met in Portugal were very helpful.
Note from the above examples that that can be used to refer to both things and people, whereas which as a relative pronoun can only refer to things and who can only refer to people.
Note also that when the relative pronoun is the subject of a relative clause, it has to be included. It cannot be omitted then:
Menorca is one of the Balearic Islands that / which lies to the north east of Mallorca.
We have a number of friends who / that have built holiday homes on the island.