When we choose a metric on a manifold we determine the spectrum of the Laplace operator. Thus an eigenvalue may be considered as a functional on the space of metrics. For example the first eigenvalue would be the fundamental vibrational frequency. In some cases the normalized eigenvalues are bounded independent of the metric. In such cases it makes sense to attempt to find critical points in the space of metrics. For surfaces, the critical metrics turn out to be the induced metrics on certain special classes of minimal (mean curvature zero) surfaces in spheres and Euclidean balls. The eigenvalue extremal problem is thus related to other questions arising in the theory of minimal surfaces. In this talk we will give an overview of progress that has been made for surfaces with boundary, and discuss recent results on higher eigenvalue optimization.

Ailana Fraser is Professor of Mathematics at the University of British Columbia. She received her BSc from the University of Toronto and her PhD in Mathematics from Stanford University. She was a Courant Instructor at the Courant Institute (NYU) and a Tamarkin Assistant Professor at Brown University before joining the faculty at the University of British Columbia in 2002. Her research interests include differential geometry and geometric analysis. She was awarded the 2012 Krieger Nelson Prize and the 2021 Cathleen Synge Morawetz Prize from the Canadian Mathematical Society, and she is a Fellow of the AMS and CMS. She serves on several editorial boards, including Transactions and Memoirs of the AMS.

Zoom link: Contact Dmitry Zenkov for the zoom link.