This discussion was moderated by Lou Billera. Billera noted that, as it was the final discussion session, anything is a valid topic today. The discussion started off with some comments about the usefulness of the workshop:

- Billera began by
remarking that the organizers worried that having 4 different groups
of people at this workshop from different backgrounds could have been
disastrous. But he noted that it didn't appear to be... and had found
it interesting, and was pleased with how much discussion had taken
place.
- Diaconis emphasized the usefulness of this workshop for generating problems to work on, and cited an example of residuals for discrete data analysis, as something worth following up, as well as the discussion in the geometry session about what are natural measures on tree space. He noted that the workshop was valuable because he couldn't think of another setting in which all these ideas from different arenas could all come together.

Q. Concatenation as ``averaging''?

- Billera noted that in the discussion about averaging, he was struck
with the idea that concatenation was an average in sequence space, an
idea that he hadn't appreciated until this workshop. One question
He noted that the conference had touched on many ways to go from sequences to
trees, and one of the big questions was whether these methods were
coherent with concatenation-averaging.
- Penny: it is worrisome that there are many different ways of concatenating, and how to handle the number of parameters?

Q. Likelihood contours?

- There was further discussion about confidence sets and likelihood
contours, and how these might intersect transition points where the
tree space branches in several directions. For instance,
if a maximum likelihood process produces a tree with very short edge, then
it would be near a transition.
There was also discussion of whether the notion of curvature on tree
space would be helpful.

Q. Multivariate "metrics"?

- Holmes: in the date that arises for trees, you get multivariate
distances. We may want to speak of directions? is there anything
like that in metric topology?
- Diaconis: there are some distances that take values in partially ordered sets, etc., distances as a vector, etc.

Q. Alternate representations (not trees)?

- Epstein asked if are there other notions besides trees that would be helpful?
- St. John: Networks? Trees with several non-tree portions.

Q. Random walks on trees (generate letters, prodcue new trees)

- Vert: One can imagine a random walks in this space of trees, by starting a tree, generate data, estimate a tree (by maximum likelihood). Do this number of times, get a random walk? Might be interesting to study this walk?

Q. Would it be useful to study 2-colored trees to model orthology and paralogy? (Shareshian)

During the session,
Diaconis suggested that, as a follow up to this conference,
people can now get together in small groups and
communicate and start to work on a problem, whereas this wasn't true
before the beginning of the week. AIM could facilitate such
follow-up meetings.

There was some brief discussion about holding a follow-up conference to this one, since all the participants now have a solid foundation on which to try to work on some joint problems.

ARCC and the organizers (Lou Billera, Susan Holmes, Karen Vogtmann)
were all thanked enthusiastically for a stimulating week of talks and
discussions. The workshop was concluded over evening refreshments.

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