Steven Morris Zelditch (13 September 1953 – 11 September 2022) was an American mathematician, specializing in global analysis, complex geometry, and mathematical physics (''e.g.''

quantum chaos Quantum chaos is a branch of physics focused on how chaotic classical dynamical systems can be described in terms of quantum theory. The primary question that quantum chaos seeks to answer is: "What is the relationship between quantum mechanics ...

). Zelditch received in 1975 from

Harvard University Harvard University is a Private university, private Ivy League research university in Cambridge, Massachusetts, United States. Founded in 1636 and named for its first benefactor, the History of the Puritans in North America, Puritan clergyma ...

his bachelor's degree in mathematics and in 1981 from the

University of California, Berkeley The University of California, Berkeley (UC Berkeley, Berkeley, Cal, or California), is a Public university, public Land-grant university, land-grant research university in Berkeley, California, United States. Founded in 1868 and named after t ...

his Ph.D. under

Alan Weinstein Alan David Weinstein (born 17 June 1943) is a professor of mathematics at the University of California, Berkeley, working in the field of differential geometry, and especially in Poisson manifold, Poisson geometry. Early life and education ...

with thesis ''Reconstruction of singularities of solutions for Schrödinger's equations''. From 1981 to 1985 Zelditch was

Ritt Ritt is a given name and a surname. Notable people with the name include: *Eugène Ritt (1817–1898), French theatermaker *Joseph Ritt (1893–1951), American mathematician at Columbia University *Martin Ritt (1914–1990), American director, ac ...

Assistant Professor at

Columbia University Columbia University in the City of New York, commonly referred to as Columbia University, is a Private university, private Ivy League research university in New York City. Established in 1754 as King's College on the grounds of Trinity Churc ...

. At

Johns Hopkins University The Johns Hopkins University (often abbreviated as Johns Hopkins, Hopkins, or JHU) is a private university, private research university in Baltimore, Maryland, United States. Founded in 1876 based on the European research institution model, J ...

he was from 1985 to 1989 an assistant Professor, from 1989 to 1992 an associate professor, and from 1992 to 2010 a professor. In 2010 he moved to

Northwestern University Northwestern University (NU) is a Private university, private research university in Evanston, Illinois, United States. Established in 1851 to serve the historic Northwest Territory, it is the oldest University charter, chartered university in ...

, where he was Wayne and Elizabeth Jones Professor of Mathematics. In 1987/88 he was at MIT and in 1988 a visiting professor at MSRI. He has done research on the spectral and scattering theory of the Laplace operator on

Riemannian manifold In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surf ...

s and especially the asymptotic and distribution of its eigenfunctions (''e.g.'' quantum ergodicity, equidistribution of eigenfunctions in billiard geometries, quantum ergodic restriction theorems to separating hypersurfaces). He has also done research on the inverse spectral problem. (This problem is described in ''Can you hear the shape of a drum?'' by

Mark Kac Mark Kac ( ; Polish: ''Marek Kac''; August 3, 1914 – October 26, 1984) was a Polish-American mathematician. His main interest was probability theory. His question, " Can one hear the shape of a drum?" set off research into spectral theory, th ...

.) In a seminal paper in 2009, Zelditch showed that one can recover the shape of a convex, analytic planar domain with up-down symmetries from its Laplace spectrum. In 2019, with his coauthor, Zelditch showed that ellipses of small eccentricity are spectrally determined amongst all smooth, convex planar domains. Among Zelditch's other research topics are

Bergman kernel In the mathematical study of several complex variables, the Bergman kernel, named after Stefan Bergman, is the reproducing kernel for the Hilbert space ( RKHS) of all square integrable holomorphic functions on a domain ''D'' in C''n''. In det ...

Kähler metric Kähler may refer to: People *Birgit Kähler (born 1970), German high jumper * Erich Kähler (1906–2000), German mathematician * Heinz Kähler (1905–1974), German art historian and archaeologist * Luise Kähler (1869–1955), German trade union ...

s, Gaussian analytic functions, and random metrics. In a famous paper, Zelditch applied semiclassical methods to complex algebraic geometry with the semiclassical parameter playing the role of the reciprocal power of an

ample line bundle In mathematics, a distinctive feature of algebraic geometry is that some line bundles on a projective variety can be considered "positive", while others are "negative" (or a mixture of the two). The most important notion of positivity is that of ...

over a

Kähler manifold In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure. The concept was first studied by Jan Arnol ...

. The Tian-Yau-Zelditch theorem in this case gives a complete asymptotic expansion of the Bergman kernel near the diagonal. For example, the Catlin-De Angelo-Quillen theorem easily follows from this. In 2002 he was an invited speaker with talk ''Asymptotics of polynomials and eigenfunctions'' at the

International Congress of Mathematicians The International Congress of Mathematicians (ICM) is the largest conference for the topic of mathematics. It meets once every four years, hosted by the International Mathematical Union (IMU). The Fields Medals, the IMU Abacus Medal (known before ...

in Beijing. He was elected a Fellow of the

American Mathematical Society The American Mathematical Society (AMS) is an association of professional mathematicians dedicated to the interests of mathematical research and scholarship, and serves the national and international community through its publications, meetings, ...

in 2012. In 2013, he and Xiaojun Huang shared the

Stefan Bergman Prize The Stefan Bergman Prize is a mathematics award, funded by the estate of the widow of mathematician Stefan Bergman and supported by the American Mathematical Society. The award is granted for mathematical research in: "1) the theory of the kernel ...

for research done independently; Zelditch was cited for his research on the

. Prior to his death, he was on the editorial boards of ''

Communications in Mathematical Physics ''Communications in Mathematical Physics'' is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, sta ...

'', ''Analysis & PDE'', and the ''Journal of Geometric Analysis''.

Selected publications

Articles

Reconstruction of singularities for solutions of Schrödinger's equation
Communications in Mathematical Physics, Vol. 90, 1983, pp. 1–26 * Uniform distribution of eigenfunctions on compact hyperbolic surfaces, Duke Mathematical Journal, Vol. 55, 1987, pp. 919–941 * with Maciej Zworski
Ergodicity of eigenfunctions for ergodic billiards
Comm. Math. Phys., Vol. 175, 1996, 673–682
Quantum Ergodicity of C* Dynamical Systems, Comm. Math. Phys., 177, 1996, 507–528
* "Szegö kernels and a theorem of Tian." International Mathematics Research Notices, Vol. 1998, no. 6, 1998, 317–331 * with Bernard Shiffman: Distribution of Zeros of Random and Quantum Chaotic Sections of Positive Line Bundles, Communications in Mathematical Physics, Vol. 200, 1999, pp. 661–683
arXiv preprint
* with Pavel Bleher and B. Shiffman: Universality and scaling of correlations between zeros on complex manifolds, Inventiones mathematicae, vol. 142, 2000, pp. 351–395
arXiv preprint

From random polynomials to symplectic geometry, Proc. Internat. Congress Math. Phys. 2000

Survey of the inverse spectral problem, surveys in Diff. Geom., 2004
* Complex zeros of real ergodic eigenfunctions, Invent. Math., Vol. 167, 2007, 419–443
Arxiv

Local and Global Analysis of Natural Functions, Handbook of Geometric Analysis, Volume 1, 2008

Recent developments in mathematical quantum chaos, Current Developments in Mathematics 2009
* with Frank Ferrari and Semyon Klevtsov
Random Geometry, Quantum Gravity and the Kähler Potential, Phys. Lett. 705, 2011

IAS / Park City Lectures on Eigenfunctions 2013

Eigenfunctions and nodal sets, Surveys in Differential Geometry, Vol. 18, 2013, 237–308

Books

American Mathematical Society 1992 * ttps://books.google.com/books/about/Eigenfunctions_of_the_Laplacian_on_a_Rie.html?id=XwVDDwAAQBAJ Eigenfunctions of the Laplacian on a Riemannian manifold American Mathematical Society 2017

References

External links

C.V. with Selected Publications
* {{DEFAULTSORT:Zelditch, Steven 1953 births 2022 deaths 20th-century American mathematicians 21st-century American mathematicians Partial differential equation theorists Harvard College alumni UC Berkeley College of Letters and Science alumni Northwestern University faculty Johns Hopkins University faculty Fellows of the American Mathematical Society