Research in Groups: Steklov problem on orbifolds
Whilst we have been busy over the spring and summer running maths workshops, it is not the only activity at ICMS. We also run a popular and successful Resarch in Groups (RIGS) programme. The RIGS programme enables small groups of researchers to come together for an intensive period of research in Edinburgh.
Over the summer of 2017, we will welcome several groups to Edinburgh. During the middle of July we welcomed a group researching, Steklov problem on orbifolds. Researchers were, Teresa Arias-Marco (Professor at Universidad de Extremadura, Spain), Emily Dryden (Professor at Bucknell University, US), Carolyn Gordon (Professor at Dartmouth College, US), Asma Hassannezhad (Lecturer at University of Bristol, UK, and EPDI fellow at the Mittag-Leer Institute, Sweden), Allie Ray (Visiting Assistant Professor at Trinity College, CT, US) and Elizabeth Stanhope (Professor at Lewis & Clark College, US)
RIGS participants, Steklov problem on orbifolds, July 2017
"The opportunity for the team members, who span three countries in two continents, to come together at ICMS was invaluable to our work."
Whilst the group were in Edinburgh, we had a quick chat with them.
Can you tell me a little about the research project that you are working on?
Our work is focused on the Steklov problem, typically in two dimensions. This problem models the vibration of a free membrane with all of its mass concentrated along the boundary and has applications to electrical impedance tomography. The frequencies of vibration correspond to the Steklov eigenvalues. During our stay at ICMS we concentrated on finding a sharp upper bound on the lowest Steklov eigenvalue, and characterizing the maximizers, when our membranes are invariant with respect to group actions.
For those interested in a more technical answer:
The classical Weinstock inequality gives a sharp upper bound for the lowest Steklov eigenvalue of a simply-connected planar domain of fixed boundary length and states that the bound is attained only by the round disk. We explored a possible equivariant version of this result: Let G be a finite subgroup of the general linear group of 2x2 matrices over R. Within the family of all G-invariant simply-connected bounded planar domains, what are the maximizers of the lowest non-zero G-invariant Steklov eigenvalue? (We say an eigenvalue is G-invariant if the corresponding eigenspace contains a G-invariant function.) This problem can be restated as a question about the maximizers, within a suitable class of Riemannian orbisurfaces, of the lowest non-zero Steklov eigenvalue. We also discussed possible generalizations of this question.
What brought you to this area of research?
We are six people with several responses. Some of us were inspired by a woman mathematician in the field who is an excellent mentor. Some were inspired by a particular class; a graduate course on Lie groups, a survey of modern geometry, for example. Some were attracted to the fact that geometry uses tools from many areas including topology, analysis, and algebra. Some met a welcoming researcher in the area and were encouraged to work with that person.
What do you think the main benefits of the Research in Groups project are?
The primary benefits for our team were the financial and logistical supports that allowed us to be together for an extended period of time in a location where we could do mathematics. Our team of six includes researchers from institutions that are thousands of miles apart. Although technology does allow us to collaborate remotely, we make much faster progress when we are together. As a team of all women, it is important to point out that this travel support helps us resolve the problem of being isolated from other women working within our subfield.
Would you have been able to do this research without ICMS funding?
The ability to meet together as a group has been essential to the success of our team. The ICMS was an excellent host, as it provided a good geographic fit for our travel needs, library and technological support, expertise with local accommodations, and an overall welcoming environment.
Now for some general questions about mathematics
If you could solve one maths problem, what would it be?
In spectral geometry, knowing whether a convex, or smooth, planar domain is determined by its Laplace spectrum, and the well-known Polya conjecture on the sharp asymptotic bounds for Laplace eigenvalues of a bounded Euclidean domain, are of keen interest. However, “The product of mathematics is clarity and understanding. Not theorems, by themselves” as Thurston said. This clarity and understanding does not often happen as a result of solving one particular problem or conjecture, but rather it is the accumulation of efforts in understanding patterns, curiosity in finding connections, and creativity in asking questions and solving them.
Do you have any thoughts regarding how we can raise the profile of maths?
Advocate for teaching elementary school mathematics through inquiry and experimentation. Bring the research of the ICMS to high school teachers: Consider the IMU Klein Project. Partner with the Edinburgh Math Circle. Work to get math topics into newspapers.
Do you have any thought on how the gender balance/diversity in mathematics can be improved?
Thank you for hosting us, including financial and logistical support for mathematicians with family obligations. Be flexible with researchers who might have a gap in publication due to personal commitments (the birth of a child, elder care needs, health challenges). Conferences for women only to build networks between women in a particular field. Make sure that when conferences are organized women and people from underrepresented backgrounds are on the speaker list. Specifically set aside funding in each institute program for women and people from underrepresented backgrounds. Make efforts to understand why women and people from underrepresented backgrounds sometimes choose to leave mathematics.
Who is your favourite mathematician and why?
We all have different responses to this question. However common characteristics are creativity, the ability to open new avenues of inquiry and to connect different areas, someone who takes mentoring newer mathematicians seriously, someone who is concerned about communicating well, and someone who has a sense of professional and social responsibility.