Ailana Fraser

Fraser was born in Toronto, Ontario. She received her Ph.D. from Stanford University in 1998 under the supervision of Richard Schoen.{{mathgenealogy|name=Ailana Margaret Fraser|id=37458}}. After postdoctoral studies at the Courant Institute of New York University, she taught at Brown University before moving to UBC.

Fraser is well-known for her 2011 work with Schoen on the first "Steklov eigenvalue" of a compact Riemannian manifold-with-boundary. This is defined as the minimal nonzero eigenvalue of the "Dirichlet to Neumann" operator which sends a function on the boundary to the normal derivative of its harmonic extension into the interior. In the two-dimensional case, Fraser and Schoen were able to adapt Paul Yang and Shing-Tung Yau's use of the Hersch trick in order to approximate the product of the first Steklov eigenvalue with the length of the boundary from above, by topological data.{{cite journal

| last1=Hersch | first1=Joseph

| title=Quatre propriétés isopérimétriques de membranes sphériques homogènes

| journal=Comptes Rendus de l'Académie des Sciences, Série A

| volume=270

| date=1970

| pages=1645–1648|mr=0292357|zbl=0224.73083|url=https://gallica.bnf.fr/ark:/12148/bpt6k480298g/f1655.item}}{{cite journal|zbl=0446.58017|last1=Yang|first1=Paul C.|author-link1=Paul C. Yang|last2=Yau|first2=Shing Tung|title=Eigenvalues of the Laplacian of compact Riemann surfaces and minimal submanifolds|journal=Annali della Scuola Normale Superiore di Pisa. Classe di Scienze|series=Serie IV|volume=7|year=1980|issue=1|pages=55–63|mr=0577325|url=http://www.numdam.org/item/ASNSP_1980_4_7_1_55_0/|author-link2=Shing-Tung Yau}}

Under an ansatz of rotational symmetry, Fraser and Schoen carefully analyzed the case of an annulus, showing that the metric optimizing the above eigenvalue-length product is obtained as the intrinsic geometry of a geometrically meaningful part of the catenoid. By use of the uniformization theorem for surfaces with boundary, they were able to remove the condition of rotational symmetry, replacing it by certain weaker conditions; however, they conjectured that their result should be unconditional.

In general dimensions, Fraser and Schoen developed a "boundary" version of Peter Li and Yau's "conformal volume."{{cite journal|last1=Li|first1=Peter|author-link1=Peter Li (mathematician)|last2=Yau|first2=Shing Tung|zbl=0503.53042|title=A new conformal invariant and its applications to the Willmore conjecture and the first eigenvalue of compact surfaces|journal=Inventiones Mathematicae|volume=69|year=1982|issue=2|pages=269–291|mr=0674407|doi=10.1007/BF01399507|bibcode=1982InMat..69..269L|s2cid=123019753|url=http://www.digizeitschriften.de/dms/resolveppn/?PID=GDZPPN002098652|author-link2=Shing-Tung Yau}} By building upon some of Li and Yau's arguments, they gave lower bounds for the first Steklov eigenvalue in terms of conformal volumes, in addition to isoperimetric inequalities for certain minimal surfaces of the unit ball.

Fraser won the Krieger–Nelson Prize of the Canadian Mathematical Society in 2012 and became a fellow of the American Mathematical Society in 2013.[https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved 2013-01-21.

In 2018 the Canadian Mathematical Society listed her in their inaugural class of fellows{{citation|url=https://cms.math.ca/MediaReleases/2018/Fellows|title=Canadian Mathematical Society Inaugural Class of Fellows|publisher=Canadian Mathematical Society|date=December 7, 2018}}

and in 2021 awarded her, along with Marco Gualtieri, the Cathleen Synge Morawetz Prize.{{citation|url=https://cms.math.ca/news-item/professors-ailana-fraser-and-marco-gualtieri-to-receive-the-2021-cms-cathleen-synge-morawetz-prize/|title=Professors Ailana Fraser and Marco Gualtieri to receive the 2021 CMS Cathleen Synge Morawetz Prize|publisher=Canadian Mathematical Society|date=February 10, 2021}} In 2022 she was awarded a Simons Fellowship.{{Cite web |date=2022-02-18 |title=2022 Simons Fellows in Mathematics and Theoretical Physics Announced |url=https://www.simonsfoundation.org/2022/02/18/2022-simons-fellows-in-mathematics-and-theoretical-physics-announced/ |access-date=2022-07-04 |website=Simons Foundation |language=en-US}}

• {{cite journal

| last1=Fraser | first1=Ailana

| last2=Schoen | first2=Richard|author-link2=Richard Schoen

| title=The first Steklov eigenvalue, conformal geometry, and minimal surfaces

| journal=Advances in Mathematics

| volume=226

| date=2011

| issue=5

| pages=4011–4030

| doi=10.1016/j.aim.2010.11.007 | doi-access=free|zbl=1215.53052|mr=2770439| arxiv=0912.5392

}}

• {{cite journal|mr=3461367|last1=Fraser|first1=Ailana|last2=Schoen|first2=Richard|author-link2=Richard Schoen|title=Sharp eigenvalue bounds and minimal surfaces in the ball|journal=Inventiones Mathematicae|volume=203|year=2016|issue=3|pages=823–890|doi=10.1007/s00222-015-0604-x|zbl=1337.35099|arxiv=1209.3789|bibcode=2016InMat.203..823F|s2cid=119615775}}

{{reflist|colwidth=30em}}

{{Authority control}}

{{DEFAULTSORT:Fraser, Ailana}}

Category:Year of birth missing (living people)

Category:Living people

Category:Canadian mathematicians

Category:Canadian women mathematicians

Category:Stanford University alumni

Category:Brown University faculty

Category:Academic staff of the University of British Columbia

Category:Fellows of the American Mathematical Society

Category:Fellows of the Canadian Mathematical Society

Category:Scientists from Toronto

Category:Courant Institute of Mathematical Sciences alumni