Professor Pedro Poitevin of the Mathematics Department was recently published in Positivity, a research journal in the field of mathematics. The Center for Research and Creative Activities (CRCA) was able to interview Professor Poitevin to discuss the article in greater depth.
How did the idea for this article come about, what inspired you to write about this topic?
Poitevin: The objects that mathematicians work with are so-called structures. (The rise of structuralism in XX century linguistics was very influential in the development of the philosophy of mathematics.) The kind of structures that one works with depends on what one’s area of expertise is: group theorists work with groups, ring theorists with rings, complex analysts with the complex plane, number theorists with the natural numbers, and so on. Functional analysis is an area of mathematics where there are lots of different examples of interesting structures to play around with, and one of these examples is Orlicz lattices.
There are some questions that mathematicians always ask about the structures they work with. One of these natural questions is: “What kind of substructures can this structure have?” In the case of Orlicz lattices, there is a venerable theorem that says that Orlicz lattices can be represented in a certain nice way, and that sub-Orlicz lattices, however, are generally slightly less well-behaved. If an Orlicz lattice can be represented in a particularly nice way, its sublattices will be representable in a less nice way, but the degree to which this way is less nice can be controlled to a reasonable degree.
I had used this theorem many times in my own research, but at some point, I asked myself what one could say about the opposite situation: suppose that you have an Orlicz lattice with somewhat complex behavior, can you prove that it lives inside a bigger Orlicz lattice that is nicer? I asked a couple of experts in the field, and both told me that they didn’t know the answer to my question and that nobody did. Naturally, this made it more interesting to me. I noticed early on that the techniques I would need to address this question were techniques I understood well, but I was not confident I would succeed. I love challenging problems, so soon I was rapt.
What did the process of being published in Positivity look like for you?
Poitevin: I was very excited about submitting to Positivity because it is one of the top journals in the world in this research area. I was also excited because the paper I was submitting was in an area that is not my nominal subject of expertise. I knew that it wouldn’t be easy to publish a paper in Positivity, so I was prepared for the possibility of a rejection.
When the referee report arrived, along with an acceptance contingent on revisions, I was over the moon, naturally. I had to make a few corrections (the referee was extremely helpful), and I had to add a section to the paper that would make the paper significantly stronger. I did so, and within a couple of very intense weeks, I submitted the corrections and additions, and the rest was a breeze.
Can you explain what “Orlicz lattices” are for those who may not already know?
Poitevin: Orlicz lattices are infinite-dimensional spaces (not two-dimensional, not three-dimensional, not four-dimensional, not even ten-dimensional, like some physicists believe the universe may be—infinite-dimensional, which means that it is very hard to visualize them) in which the distance between points is measured in a somewhat erratic way.
In one segment of the trajectory between points, you might use a standard ruler, but in another segment, you might use a different ruler (for example, you might calculate distance the way a taxi driver does rather than the way a straight ruler does), and the way in which these different systems of measurement are chosen depends on some random process.
Why would anyone be interested in spaces of this sort? Well, there are some physical phenomena that are best understood as happening in spaces like so. For example, if a physical system’s properties change upon the action of an external force, the geometry of the physical system can be modeled with a relatively well-behaved Orlicz lattice.
One nice example of a physical system whose properties change under the action of an external force is a mix of starch and water. As anyone with a bit of experience in the kitchen knows, if one stirs the mixture, the mixture thickens, but if one fails to stir, the mixture stays runny. There also are some fluid systems whose physical properties (like viscosity) change upon the action of an electric field. These sorts of systems turn out to be well modeled mathematically within certain kinds of Orlicz lattices.
Can you give us a brief description of what the article entails?
Poitevin: One of the most important contributions that logic has made to mathematics is a particular way of solving some kinds of mathematical problems.
Suppose that you are working in a certain mathematical structure, and you are trying to solve a problem there, but you run into trouble. Logic has certain tools to create a larger mathematical structure in such a way that whatever was true in the original structure you were working on is still true in the larger one, and whatever you can say in the larger structure about the stuff in the original structure is true only if it was already true in the original structure.
These kinds of larger structures are called elementary extensions. The lovely thing about the larger structure is that it has more room for you to solve the problem that you were having trouble solving in the original, more constrained structure. My contribution in this article is in the spirit of this approach. I essentially proved that every mildly annoying-looking Orlicz lattice can be found as a substructure of a better-behaved Orlicz lattice.
What does having this article published mean for you?
Poitevin: As someone who has occasionally felt, for the past few years, underappreciated for my work, I breathed a sigh of relief that my work had been evaluated by careful and discerning people whom I admire and whose judgment I respect. I also enjoyed being an outsider (a mathematical logician) publishing an article in functional analysis.
Will you continue doing work on this subject or are you currently working on anything else (or both)?
Poitevin: I have my eyes on another problem, this time in linear algebra, another subject that falls outside of my area of expertise. I’m also working on two books: one book of poems in Spanish (to be published in Mexico) and one book of palindromic poetry, also in Spanish (to be published in Argentina).
Is there anything you want us to include about your article or anything you specifically want people to know about this project?
Poitevin: I would love to say that there is no feeling in the world like pushing the boundary of human knowledge just a little bit farther. I am thankful for mathematics because it is a subject that allows one to know for certain—when one is lucky enough to have done so—that one has done exactly that.
Congratulations to Professor Poitevin on this major accomplishment! We look forward to seeing more of your work in the future.
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