Would it be okay to give a professional conference lecture on Sabbath? Someone told me it would be okay as long as the speaker doesn't touch the microphone or the slides computer, but that doesn't seem right. Wouldn't there be a problem with the lecture itself?
(Of course, for any practical ruling, please consult your local Orthodox Rabbi)
Assuming that all technical prohibitions are avoided (which is in itself quite difficult; even if, using your example, you don't change the slides on the computer, it might be forbidden to have a non-Jew move through the slides on your behalf as you speak. But the details of that are beyond the scope of this answer), this would likely still be forbidden.
There are a number of "catch-all" prohibitions on Shabbos that prohibit one from doing actions that, while no Melachos are being violated, is forbidden for being not in the "Spirit of Shabbos". This is learnt out from the verse in Isaiah 58:13
If thou turn away thy foot because of the sabbath, from pursuing thy business on My holy day; and call the sabbath a delight, and the holy of the LORD honourable; and shalt honour it, not doing thy wonted ways, nor pursuing thy business, nor speaking thereof;
One of the things we learn from that verse is that one is forbidden from speaking about business related matters on Shabbos. Presumably, your example would fall under the general umbrella of speech that is forbidden on Shabbos.
See Shulchan Aruch Orach Chaim Siman 306 for a more detailed treatment of the prohibition of forbidden speech on Shabbos.
The interdict "nor pursuing thy business, nor speaking thereof" (trans. from @Salmononius2) is indeed a general umbrella. For example, business-related documents are forbidden to read on Sabbath (TB Sab. 149a). The reason for this is of dispute; some apply the aforementioned interdict (Rosh, Sab. 23:1) while others forbid it as a precaution so one does not mistakenly feel it's weekday and erase [writing] (Rambam, Sab., 23:19). Some even advance that it is only forbidden to orally read such material (Mordechai, Sab., §254), yet, most authorities side with the Shulchan Aruch that even not orally is forbidden (OC 307:13).
However, IMHO a lecture would fall into more defined parameters. SA (ibid. 307:17) advances an opinion that it is forbidden to learn anything -including educational material- except Torah on Sabbath and holidays. Then, he cites a dissenting opinion which allows such study. The Mishneh Berurah (loc. cit. 65) notes that the lenient view is more prevalent.
Furthermore, OP indicated in a comment that the topic of this lecture would be medical-related. The study of such material is even considered by some to be a mitzvah (as it can enable one to save a life; cf. Rashba 1:415, Chazon Ovadiah, Shabbat vol. 3 pg. 32). R. Yitzchok Zilberstien permits one to partake in a medical lecture provided the technical issues are taken care of/avoided (Shiurei Torah LeRof'im, vol. 1 pg. 103). Rabbi Dr. Abraham (Lev Avraham vol. 2 pg. 58) also permits one to partake in such a lecture (though advises not to ask a non-Jew, on Sabbath, to take notes for him). R. Elchanan Prins (Avnei Derech vol. 8 314) discusses a question where a rabbi was giving a medical lecture at a seminar on Sabbath and during summation noticed a [Jewish] member in the audience taking notes. The rabbi was posed with the question how to proceed. R. Prins cites numerous personal correspondence that he conducted with rabbis (containing different suggestions) while none raised an issue with having the lecture.