Research

As a theoretical mage, Robin is the scholar in the party. At certain points you are able to direct her research efforts, producing various effects - some minor, some greater.

Trin's contract
After Trin joins the party in chapter 1, ask Robin to research improvements to her contract. Talk to Robin in the Yhilin Palace's library and select "Research". After the start of chapter 2, talk to Trin to update the contract. This increases both Trin's and Robin's relationship points.

There is no downside or opportunity cost to doing this, so it is highly recommended.

During the reunion
If Simon has acquired a transformation spell for Aka during his stay in the Orgasmic Empire, Robin can start researching one of three research options (see below); otherwise, Robin is too occupied with researching Aka's transformation to do anything else.

You are able to choose each of the ignored topics later in the game. If you got help with Aka's cure while in the Orgasmic Empire, you are able to do all chapter 3 research projects. Otherwise you have to sacrifice one. (And if you were wondering, no, you can't just let Aka die.)

After the Battle of Yhilin
After picking a research topic during the Reunion, Robin can pick one of the remaining topics to research after the invasion of Yhilin, or the new option of examining Dari.

After visiting the elven kingdoms
In addition to whichever of the previous topics you haven't completed, Robin now presents the option of studying to become a full member of the Order of Yhilini Thaumaturgy.

After visiting Zirantia
In addition to the previous topics, Qum offers a new project. Qum's "research" is a one-time opportunity and will consume Robin's next research period rather than this one. If you've kept up with research so far, there's little reason not to let her, but you should decide before speaking to her.

After receiving succubus refugees
If you approved Qum's project, Robin will be occupied with that during this time, and thus you cannot assign her another project.

After visiting Aram
Robin can manage one last project, and unlike previous visits to Yhilin, you will see the effects immediately in the walking-around part. Also, due to a bug that will be fixed, she will be able to complete an additional project in time for the war, but not in time for the player to see the effects.

Robin's Mystery Project
If by the end of the Third Arclent War you have completed all research projects and the total collateral damage is 99 or lower (thus Sarai don't ask help to cleanse the corruption widespread during the war) then Robin has a scene where she briefly reflects and decides that since she has no other research to do, she can start a personal project of her own...

As of version 0.29.x we don't have any idea what this mystery project is.