I have been invited to give 2 lectures at the most prestigious meeting in my field.
However, they fall on Shabbat.
Is it permissible for me to give the lecture if a goy changes the slides for me? Can we consider that he is changing the slides for the audience and not for me so that he is not technically doing a melacha for me?
Would this be considered tzorech rabbim in order for the permission of shvut deshvut to apply?
I assume that the meeting is a medical conference, with health ramifications.
The technical problem of having the gentile change the slide, while not simple, is surmountable based on the reasons you mentioned. In addition, since the slide serves everyone, and the majority of attendees are gentile, the rationale of ner l'echad ner l'meiah might be applicable. (Shulchan Aruch O.C. 276:2) Regardless, you should not say, "Change the slide," but arrange a hint without a command, such as: "We are finished with this slide," or, "As we can see in the next slide," etc. (M.B. 307:76)
However, there is serious concern of mar'is ayin and chilul Hashem in having your name published as giving two lectures on Shabbos. Furthermore, many people there will not be aware of your care in utilizing a non-Jew. For this reason it is much more severe than attending a conference as a participant. (See O.C. 252:3)
You should make a concerted effort to have the date changed. If it cannot be changed, it seems to me that for any reasons of personal advancement, etc., it is preferable to decline. If you feel that your lecture is critical for the advancement of health to those attending, you could accept the invitation, with the limitations above. You should also make it obvious to those attending that you are utilizing the services of the gentile to avoid desecration of Shabbos.
Rabbi Meir Orlian