This following is a brief summary of our research meeting on 2024-08-06.

Summary of discoveries

At this meeting we mostly chatted about big-picture ideas. We speculated about some possibilities and noted that a lot of ideas have been ruled out. Some things we're currently missing:

• A good method of quickly finding tropical equations that define a given congruence variety
• An algebraic method for determining when a tropical line (i.e., bend locus of a linear polynomial) is "tropically tangent" to a tropical conic (i.e., bend locus of a given quadratic polynomial).
• An understanding of the analysis in the classical case, in which tangency is detected via some minors of some matrices, ultimately leading to a formula for the dual conic.
  At our next meeting I'll reveal some facts about congruence varieties that I haven't mentioned yet (which might help us with one of our issues above), and briefly outline the tangent/duality argument in the classical case.

Tasks for next meeting

• Put our current investigations temporarily on hold and instead take a few days to prepare for the next "share out".
• Directions from Professor Champney for this "share out" are to ponder answers to the following questions:
  1. What is some work you've had to do that you were not expecting?
  2. What went faster/slower than you anticipated?
  3. How is your team handling roadblocks or unexpected things that happen in your work?
     and then to give an 10-minute presentation on the project.