Evaluation of PhD Thesis by Ronen Katz,
    "Elliptic Associators and the LMO Functor"
    ==========================================

Dear Hebrew University Officials,

This note is to recommend the acceptance of Ronen Katz's excellent
dissertation.

In deep and difficult paper, Calaque, Enriquez, and Etingof (CEE)
had proven that the group of "elliptic braids", namely braids on a
genus 1 surface, has a multiplicative and co-multiplicative expansion,
or "universal finite type invariant". Their proof involved the "KZB
connection", a generalization of the KZ connection to genus 1 which
in itself is a difficult construct. Later in a paper titled "Elliptic
Associators", Enriquez had shown how to reduce the computation of a the
CEE invariant to the computation of just a pair of elements within the
target space, and named such a pair an "Elliptic Associator".

In a parallel development, Cheptea, Habiro, and Massuyeau (CHM),
had shown how to use the ideas that go into the construction of the
"LMO Invariant" (a universal finite type invariant of rational homology
spheres) in order to construct an similarly-universal invariant of what
they call "Lagrangian Cobordisms". Their construction can be tweaked in
a minor way (as done by Katz) to yield a construction of a (similarly
universal) invariant of tangles, and hence also of braids, in a torus
times an interval. Their construction can also be used to construct
something similar to those "Elliptic Associators", except that the
CHM-Katz elliptic associators take values in a different space; roughly,
the space that corresponds to tangles rather than the space for braids.

In his thesis, Mr. Katz explains how the two constructions of elliptic
associators become equivalent once the target space of the CHM-Katz
construction is reduced modulo "homotopy relations". A significant amount
of non-trivial work goes into the proof, by Katz, of the equivalence
of the relevant target spaces once the "bigger one" is reduced modulo
homotopy.

This result is significant. It provides a link between two
previously-unlinked topics (CHM and CEE), and it suggests means by which
it may be possible in the future to generalize the CEE construction to
higher genera. The overall idea of Katz's thesis is novel, the topology
is non-trivial, and so is the combinatorics. It is an excellent thesis
overall.

The main shortcoming in Katz's thesis, in my opinion, is the lack of
topological interpretation to many of the combinatorial constructions
and arguments within. This by no means should delay the acceptance of
the thesis; merely it is an indication that Katz is leaving some work
for his future followers. Yet this lack is the main reason I do not wish to
recommend this thesis for a Hebrew University prize.

This evaluation is open and available at
http://drorbn.net/AcademicPensieve/People/Katz

Dror Bar-Natan,
Toronto, March 8 2015.