Learn English – “There are no shortage of applications”

All the answers so far miss the mark by a country mile, or are oversimplified to the point of being plain wrong.

What you are proposing here is called notional concord:

As Quirk et al. 1985 explains it, notional agreement (called notional concord by Quirk and others) is agreement of a verb with its subject or of a pronoun with its antedecent in accordance with the notion of number rather than with the presence of an overt grammatical marker for that notion. Another way to look at the matter is that of Roberts 1954, who explains that notional agreement is agreement based on meaning rather than form.

The corresponding Wikipedia entry is synesis:

Synesis [...] is effectively an agreement of words with the sense, instead of the morphosyntactic form. [...] Such use in English grammar is often called notional agreement (or notional concord), because the agreement is with the notion of what the noun means, rather than the strict grammatical form of the noun (the normative formal agreement). The term situational agreement is also found[.]

Notional agreement for collective nouns is very common in British English. It is less customary in American English, but may sometimes be found after phrases of the type "a collective noun of plural nouns", e.g.,

• ... a multitude of elements were intertwined. (New York Review of Books)
• ... the majority of all the shareholdings are in the hands of women. (Daedalus)
• ... a handful of bathers were bobbing about in the waves. (Philip Roth)

Notional concord is quite common in English:

• A lot of people are.
• A number of cars are.
• A variety of species are.
• A few folks are.
• A couple cats are.
• A total of seven students are.

Conversely, quoting one of Edwin Ashworth's comments on this page (emphasis added):

'Piles (/oodles/truckloads/lots) of money is sent abroad each year' (quantifier; number-transparent noun rather than collective noun).

So, the question is not whether notional agreement is a thing. The question is, whether it is a thing in this one particular construction you are looking at here, "There are no shortage".

And there is no way at all to answer that question on a theoretical level, without actually looking at what native speakers actually say and write. Any answer that does not look at the reality of the language is not an answer, but mere opinion.

So, after this long-drawn-out preamble, let us look at the reality of the language, then.

Here are the actual usage stats from the British National Corpus (BNC) and the Corpus of Contemporary American English (COCA):

                                   BNC    COCA

there is no shortage of [NN2]       20      96
there are no shortage of [NN2]       6       6

We can clearly see that notional agreement in this particular construction is indeed possible. So you are not wrong in saying you can keep it.

At the same time, we can clearly see that singular agreement is preferred in this particular construction, by an impressive margin in the UK, and by an order of magnitude in the US. So your colleague is not wrong in preferring it in your sentence as well.