By Timothy Batt and Professor John Clifford
In late May, I began participating in an eight-week mathematics research project, known as an REU (Research Experience for Undergraduates). I worked with two other students, Georgia Corbett from Bucknell University and Lauren Forbed from St. Thomas University, under the mentorship of Professor John Clifford. Our goal was to characterize the numerical range of a composition operator symbolized by an anti-diagonal matrix. The composition operator is defined as
where A is an n × n anti-diagonal matrix and f is a function defined on n-dimensional complex Euclidean space. In this instance, the composition operator takes a function f and gives out another function f ◦ A. Since our group worked with Composition Operators we became known as The CoOp.
Due to the complexities of COVID restrictions, most of our projects needed to be conducted at home over Zoom. We met twice every day, once in the morning and once in the afternoon. Our morning meetings would begin with ice-breakers to help us form team camaraderie. After a few laughs together, we would get down to business by presenting our progress on different facets of the project or learning new mathematical ideas and mechanics which would help with the project. After breaking for a few hours to eat lunch and get our thoughts together, we would resume collaborative work with an afternoon group meeting. Throughout the week, we also attended various math talks geared towards exposing students to different mathematical applications. On Fridays, my group presented our results from the week to the other members of the large group, consisting of our group and the other four research groups.
The first two weeks of our project was devoted to learning new material important in our research. Personally, I found this part to be very challenging and, at times, overwhelming due to the amount of complex information. However, as the research began in the third week, everything seemed to fit together and become more clear. This is when the fun began. Also, at this time, Georgia, Lauren, and I started working on different aspects of the project, which would contribute to the overall project. I thoroughly enjoyed working on and thinking about my problem independently and then presenting it to my group for advice. The different perspectives each group member brought served to deepen everyone’s understanding and, more often than not, proved invaluable in solving our individual problems. I focused most of my time on computing the numerical range of anti-diagonal composition operators symbolized by 2 × 2 and 3 × 3 matrices. This was made possible by Georgia and Lauren’s work in characterizing the composition operator as the direct sum of matrices.
The numerical range involves taking what is called an inner product (this is taken by multiplying the components of two vectors pairwise and then adding them all together) of the matrix representation of our operator acting a vector (which yields a new vector) with the original vector. Each vector put into the numerical range gives a single complex number, but there are infinitely many vectors to be used as inputs. With so many possible outputs, we set out to characterize the numerical range in an easy-to understand manner by using boundaries which encapsulate the points in the numerical range. Interestingly, the boundary of the numerical range of our composition operator can be characterized nicely by an infinite number of rotating ellipses. As such, our project sought to characterize this collection of ellipses. An example of the numerical range of an anti-diagonal composition operator symbolized by a 2 × 2 matrix is shown below and is the region in light blue. This numerical range can be visualized as the set of all points inside a rubber band wrapped around the red ellipsis and the point (1, 0).
For experts reading this article, the composition operator CA is acting on the Hardy space of analytic functions on the open unit ball in n-dimensional Euclidean space. The numerical range of a composition operator CA is the set
During the final week of the research project, and due to the University’s Covid restrictions becoming more lax, we were able to meet in person at a hotel in Dearborn. Interacting in person with my team was a wonderful culmination to our research project. During this time, we tied up any loose ends on our project and prepared for our final presentation to the SUMMR Math Conference, hosted by Michigan State University. It was exciting, and a little nerve-wracking, to present to so many different people; however, it was an experience of a lifetime, and I am grateful to have been a part of it.
From Professor Clifford:
Professor’s Yunus and Hyejin Kim started the University of Michigan-Dearborn REU (Research Experience for Undergraduates) in 2017. In five short years it has become a nationally recognized program. I am extremely proud of the University of Michigan Dearborn’s summer REU. I have participated as a mentor four different times and am always amazed by the talented students. Every year, I learn more about the interesting work of my colleagues. The last two REUs, 2020 and 2021, have been held virtually over zoom. Although I was hesitant to participate in a on-line REU, and still prefer in person REUs, I was surprised how much I enjoyed the virtual REU experience. Reasons why the virtual REU was nice were the ease with which students could share their work, convenience of scheduling meetings, and the productivity of the students. I believe the University of Michigan-Dearborn’s REU in Mathematics is a high impact experience for both students and mentors.
This last summer we had six mentors and five groups of students (pictured below). The mentors where me (John Clifford) and professors Yulia Hristrova, Kelly Jabbusch, Aditya Viswanathan, Tian An Wong, and Yunus Zeytuncu. The five projects where in analysis (2 projects), number theory, algebraic geometry, and inverse problems.
My group focused on the spectrum and numerical range of a composition operator on the Hardy Space. My group consisted of three students (from left to right): Timothy Batt, Georgia Corbett, and Lauren Forbes, pictured below.
They presented their work at Summer Michigan Undergraduate Conference hosted by Michigan State University and at the highly selective Young Mathematicians Conference hosted by Ohio State University. We are presently working on writing up our results for publication.