This following is a brief summary of our research meeting on 2024-07-26.

Summary of discoveries

This was a brief meeting, mostly checking in to see how the "share-out" yesterday went and how initial explorations have gone.

Nicholas briefly illustrated an example (of a tropical conic of type $B_2$). Using the interactive feature of Desmos, Nicholas showed the tropical lines that should probably qualify as being tangent to the given conic, as well as where the dual points to those lines would be. This led to a theoretical dual curve/congruence, which he also had displayed.

The finding was something very similar to this piece of investigations from my own work:

Tasks for next meeting

• We need lots of examples! Let's try to at least sketch (conjectural) dual regions for each of the twenty combinatorial types of tropical conics.

• We also needs lots of examples how how the dual locus changes as we slowly adjust the coefficients of the tropical conic; i.e., for each given type, a bunch of explicit examples of what happens to tropical conics of that type.
• We need explicit formulas for the conjectural dual regions (whether that be tropical polynomials for which that is the bend locus, or tropical equations if the conjectural dual region is a congruence variety). In the image above, you can see where I've given at least one tropical equation that produces the shaded congruence variety.
• Look for patterns! With enough examples, we should be able to slowly piece together a single formula/rule that produces the conjectured dual picture. There will probably be cases (e.g., type $A_1$) for which our formula/rule doesn't work. But we'll cross that bridge when we come to it!