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I was given the opportunity to participate at
conversation on professional
norms in mathematics, a workshop (9/20-22, 2019) organized by Emily Riehl.
As a member of a group focusing on teaching and
support rather than research, I hoped to learn and listen.
The hope was more than fulfilled. Below are a few takeaways for me. The following
remarks are personal. I have an exotic job in a job class which is already exotic
and have often ideas which are not common. There is a lot of diversity of thought
even in my own family, at work and among the students we teach. One of the
most important things for me is to tolerate other points of view, to allow for
diversity in all kinds, especially in matters of teaching. What attracts me to pedagogy
is that there are so many different aspects and how teaching can work in many
different ways and how theoretical, ideological and practical considerations can collide.
The workshop has brought together a rather diverse pool of unusual mathematicians, who
not only care about mathematics as a subject itself but more globally about the profession
and even grander things like equity and climate change. |
|
 Eugenia Cheng |
^ Alex Diaz-Lopez |
^ Pamela Harris |
^ Denis Hirschfeldt |
^ Mike Hill |
^ Dagan Karp |
^ Oliver Knill |
 David Kung |
^ Izabella Laba |
^ Luis Leyva |
^ Michelle Manes |
^ Adriana Salerno |
^ Francis Su |
^ Aris Winger |
The picture below shows the ``6 little mice" I brought to the conference. There are exactly 6 positive curvature
graphs which are two dimensional discrete manifolds. I presented part of the "Mickey Mouse sphere theorem"
I currently finish up writing: every positive curvature d-graph is a d-sphere. It is a simple theorem
but it captures the essence of heavy sphere theorems in differential geometry: in the discrete, for rather silly
reasons, one does not have to assume orientability of the manifold, the reason being that the positive curvature species
are very small, too small to generate projective spaces.
[ See this page from 2011 for the two-dimensional case.
Euler characteristic of the projective plane is 1 and as positive curvature is larger or equal than 1/6,
implementing a positive curvature projective plane would need a graph with 6 vertices
which would just define a unit ball (wheel graph) in such a projective plane. Now to be a projective plane, we would have
to identify points on the 5 boundary points in a 1:1 manner without fixed point which is not possible.]
Also no pinching condition as the quantized curvature automatically
produces a sufficient curvature pinching [in the discrete there is 1/2 pinching even as only curvature 1/3 and 1/6 are allowed,
the octahedron being the constant curvature case with curvature 1/3 and the icosahedron the constant curvature case with
curvature 1/6]. The curvature assumption is much stronger than in the continuum.
I mentioned this result in a ``professional norms" set-up as there is often little appreciation for simple things in math
(as it is ``not deep enough"). It is the position like I'm in, which allows to do such things off the chart, away from
mainstream and also ignore general perception and more importantly, general opinion or fashion. [If one looks
at the history of math, many subjects were driven by fashion or professional norms at a time even so the topics turned
out to be of quite marginal value later on and sometimes, marginal parts have become fashion later on or exploded to entire
new fields. History cautions to take value systems of the majority as a gold standard.]
I like the Mickey Mouse world (a word coined by Raoul Bott as reported by Richard Stanley for this type of
mathematics [As pointed out by Stanley, Bott appreciated this type of math and obviously was teasing Stanley].
One must however also say that the math for the upper bound conjecture for example is comparable in difficulty with
classical sphere theorems (maybe even differentiable sphere theorems) which is definitely not Disney stuff).
The general Mickey mouse sphere theorem in the higher dimensional case just requires to use
clear definitions of what a discrete manifold and what a discrete sphere is. The proof can now be done with with what I call a
geomag lemma: take a two dimensional immersed two-dimensional surface embedded in
a discrete d-manifold. If it has a boundary point, we can strictly enlarge the surface at this point [not necessarily uniquely
of course similarly as in the continuum for Riemannian manifolds M, where we do not have geodesic two dimensional sheets,
S=exp(D) for a two dimensional disk D in the tangent space TxM is not geodesic at a different point unlike for
geodesic curves. In two dimensions already the non-commutativity of parallel transport
quantified as curvature is kicking in]. Back to geomag, the surface obtained by snapping on more and more magnets might
develop self-intersections and is not assumed to be an embedding, but we don't care. What is important that it
produces a two dimensional closed surface eventually due to the fact that we have only finitely
many vertices (magnetic balls) and edges (magnetic connectors) available.
Now, the positive curvature assumption implies that such an immersed two dimensional manifold is one of the six
mice. We can also show that if two points are given that there exists a shortest geodesic (not unique of course in general)
connecting them and that this curve can be extended (an other geomag argument) to a two dimensional surface containing the
geodesic. As the geodesic now must be sitting on one of the 6 mice, the diameter of the manifold is maximally 3
(in any dimension larger than 1) and G must be simply connected (the 6 mice are all simply connected!) implying Synge
(without orientability assumption). They can have high dimensional mighty mice but they are still mice.
Now, one can also show that positive curvature implies that a ball of radius 2 is always a ball in a technical sense
[this is not true in general, the union of all unit balls in a unit ball is not a ball in general;
(it is one of the pitfalls which need to be avoided when proving the 4-color theorem
constructively); but it is true in the positive curvature case]. So, now one can see that the manifold is the union of two
balls of radius 2. This means it is a sphere (a d-graph which when one vertex is removed becomes contractible).
It is a natural question to define positive curvature in a more
relaxed way so that one can get sphere theorems which resemble the classical case. Direct adaptations of the continuum do not
appear to be easy. One reason is that the injectivity radius in any d-graphs for d larger than 1 is always only 2.
There is no doubt that there is a formulation which implies the continuum case in a limiting situation
but it is also almost certain that it never can reach the simplicity and elegance of the Mickey mouse theorem. 
Panorama of of Johns Hopkins Campus. Click on the picture to see it large.
Mickey and friends
