Conversation on Professional Norms in Mathematics

New: The book: A Conversation on Professional Norms in Mathematics will come out in November 2021. Title and back pages [PDF] ;
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.
There will be also a text with preparation notes, which will appear here also, once in a preliminary decent shape.
For the abstracts, see the workshop website. Here are just a few things which impressed me and which I remember most. I liked that most of the talks came with concrete takeaways. So, here are the impressions which I got. (Please consider that as an outsider and layperson in almost all of the topics, it can be easy to get things wrong.)
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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.

Update: October 6, 2019: A preprint about the sphere theorem and on the ArXiv.
Positive curvature graphs: 6 little mice.
Johns Hopkins University Panorama view, click for large version Panorama of of Johns Hopkins Campus. Click on the picture to see it large.
Mickey and friends
Oliver Knill, 9/20/2019 start, 9/24/2019 first thoughts, 9/26/2019 posting slides, 9/28/2019 a few more lines about the Mickey mouse theorem, 10/6/2019, link to sphere theorem paper, 10/13/2019, photos of campus and conference call, 10/15/2019, photo of 6 Disney figures, 10/16/2019, photo where the 6 friends admire a positive curvature graph