View source for Elliott H. Lieb

{{short description|American mathematical physicist}}
{{Use mdy dates|date=November 2022}}
{{Infobox scientist
| name              = Elliott H. Lieb
| image             = Elliott-Lieb-07.jpeg
| birth_date        = {{birth date and age|1932|07|31|mf=y}}
| birth_place       = [[Boston]], Massachusetts, U.S.
| residence         = 
| fields            = Mathematics, Physics
| doctoral_advisor  = [[Samuel Frederick Edwards]]<br>[[Gerald Edward Brown]]
| doctoral_students = [[Rafael Benguria]]<br>[[Jennifer Tour Chayes]]<br>[[Robert McCann (mathematician)|Robert McCann]]<br>[[Jan Philip Solovej]]<br>[[Horng-Tzer Yau]]
| known_for         = [[Lieb conjecture]]<br>[[Artificial lattice|Lieb lattice]]<br>[[Trace_inequality#Lieb's_concavity_theorem|Lieb's concavity theorem]]<br>[[Lieb's square ice constant]]<br>[[Trace_inequality#Lieb's_theorem|Lieb's theorem]]<br>[[Lieb–Liniger model]]<br>[[Lieb–Oxford inequality]]<br>[[Lieb–Robinson bounds]]<br>[[Lieb–Thirring inequality]]<br>[[Adiabatic accessibility|Lieb–Yngvason entropy principle]]<br>[[AKLT model|Affleck-Kennedy-Lieb-Tasaki model]]<br>[[Borell–Brascamp–Lieb inequality]]<br>[[Brascamp–Lieb inequality]]<br>[[Brezis–Lieb lemma]]<br>[[Strong_subadditivity_of_quantum_entropy#Carlen-Lieb_Extension|Carlen–Lieb extension]]<br>[[Temperley–Lieb algebra]]<br>[[Dicke model]]<br>[[Stability of matter]]<br>{{no wrap|[[Strong subadditivity of quantum entropy]]}}
| footnotes         = 
| work_institution  = [[Princeton University]]
| author_abbreviation_bot = 
| author_abbreviation_zoo = 
| prizes            = [[Dannie Heineman Prize for Mathematical Physics|Heineman Prize for Mathematical Physics]] {{small|(1978)}}<br>[[Max Planck medal]]<br>[[Birkhoff Prize]] {{small|(1988)}}<br>[[Boltzmann medal]] {{small|(1998)}}<br>[[Rolf Schock Prizes]] in Mathematics {{small|(2001)}}<br>[[Levi L. Conant Prize]] {{small|(2002)}}<br>[[Henri Poincaré Prize]] {{small|(2003)}}<br>[https://www.esi.ac.at/esi-medal/recipients/2021 Medal of the Erwin Schrödinger Institute] {{small|(2021)}}<br>[[APS Medal for Exceptional Achievement in Research]] {{small|(2022)}}<br>[[Carl Friedrich Gauss Prize]] {{small|(2022)}} <br>[[Dirac Medal (ICTP)|Dirac Medal]] {{small|(2022)}}<br>[[Kyoto Prize in Basic Sciences]] {{small|(2023)}}
| religion          = 
| education         = [[Massachusetts Institute of Technology]] ([[B. S.|BS]])<br /> [[University of Birmingham]] ([[PhD]])
}}

'''Elliott Hershel Lieb''' (born July 31, 1932) is an American [[mathematical physics#Mathematically rigorous physics|mathematical physicist]]. He is a professor of mathematics and physics at [[Princeton University]]. Lieb's works pertain to [[Quantum mechanics|quantum]] and [[n-body problem|classical many-body problem]],<ref name="statmech">{{cite book |title=Statistical mechanics : selecta of Elliott H. Lieb |date=November 29, 2004 |publisher=Springer |isbn=3-540-22297-9}}</ref><ref name="condmat">{{cite book |title=Condensed matter physics and exactly soluble models : selecta of Elliott H. Lieb |date=November 29, 2004 |publisher=Springer |isbn=3-540-22298-7}}</ref><ref name="stabmat">{{cite book |title=The stability of matter : from atoms to stars : selecta of Elliott H. Lieb |date=November 29, 2004 |publisher=Springer |isbn=3-540-22212-X |edition=4th}}</ref> [[atomic structure]],<ref name="stabmat" /> the [[stability of matter]],<ref name="stabmat" /> functional inequalities,<ref name="inequalities">{{cite book |title=Inequalities: Selecta of Elliott H. Lieb |date=2002 |doi=10.1007/978-3-642-55925-9|isbn=978-3-642-62758-3 |doi-access=free|editor-last1=Loss |editor-last2=Ruskai |editor-first1=Michael |editor-first2=Mary Beth }}</ref> the theory of [[magnetism]],<ref name="condmat" /> and the [[Hubbard model]].<ref name="condmat" />

==Biography==
Lieb was born in Boston in 1932, the family moved to New York when he was five. His father came from Lithuania and was an accountant, his mother came from [[Bessarabia]] and worked as a secretary.<ref name="aip">{{cite web |last1=Physics |first1=American Institute of |title=Elliott H. Lieb |url=https://www.aip.org/history-programs/niels-bohr-library/oral-histories/47475 |website=aip.org |access-date=8 November 2023 |language=en |date=15 December 2022}}</ref>

Lieb received his [[Bachelor of Science|B.S.]] in physics from the [[Massachusetts Institute of Technology]] in 1953<ref name=bio>{{cite web |title=Lieb, Elliott H. |url=https://history.aip.org/phn/11511008.html |website=American Institute of Physics |access-date = January 5, 2020}}</ref> and his PhD in mathematical physics from the [[University of Birmingham]] in England in 1956.<ref name="bio" /><ref>{{cite web |title=Elliott Lieb|url=https://www.genealogy.math.ndsu.nodak.edu/id.php?id=18857 |website=Mathematics Genealogy Project |access-date = January 5, 2020}}</ref> Lieb was a [[Fulbright Fellow]] at [[Kyoto University]], Japan (1956–1957),<ref name="bio" /> and worked as the Staff [[Theoretical physics|Theoretical Physicist]] for [[IBM]] from 1960 to 1963.<ref name="bio" /> In 1961–1962, Lieb was on leave as professor of applied mathematics at [[Fourah Bay College]], the [[University of Sierra Leone]].<ref name="bio" /> In 1963, he joined the [[Yeshiva University]] as an associate professor.<ref name="aip"/> He has been a professor at Princeton since 1975,<ref name="bio" /> following a leave from his professorship at MIT.

Lieb is married to fellow Princeton professor [[Christiane Fellbaum]].

For years, Lieb has rejected the standard practice of transferring copyright of his research articles to [[academic publishers]]. Instead, he would only give publishers his consent to publish.<ref name="ste22"/>

==Awards==
Lieb has been awarded several prizes in mathematics and physics, including the [[Dannie Heineman Prize for Mathematical Physics|Heineman Prize for Mathematical Physics]] of the [[American Physical Society]] and the [[American Institute of Physics]] (1978),<ref>{{cite web |title=1978 Dannie Heineman Prize for Mathematical Physics |url=https://www.aps.org/programs/honors/prizes/prizerecipient.cfm?last_nm=Lieb&first_nm=Elliott&year=1978 |website=American Physical Society |language=en |access-date = January 5, 2020}}</ref> the [[Max Planck Medal]] of the [[German Physical Society]] (1992),<ref>{{cite web |title=Preisträgerinnen und Preisträger, Max Planck Medaille |url=https://www.dpg-physik.de/auszeichnungen/dpg-preise/max-planck-medaille/preistraeger |website=Deutsche Physikalische Gesellschaft |language=de |access-date = January 5, 2020}}</ref> the [[Boltzmann medal]] of the [[International Union of Pure and Applied Physics]] (1998),<ref>{{cite web |title=The Boltzmann Award |url=http://iupap.cii.fc.ul.pt/Boltz_Award/BA1998.html |archive-url=https://web.archive.org/web/20150220045439/http://iupap.cii.fc.ul.pt/Boltz_Award/BA1998.html |url-status=dead |archive-date=February 20, 2015 |website=The Web Archive |date=February 20, 2015}}</ref> the [[Schock Prize]] (2001),<ref>{{cite web |title=Schock Prize 2001 |url=https://www.kva.se/en/priser/pristagare/elliott-h-lieb-2 |website=Kungl. Vetenskaps-Akademien |access-date=January 5, 2020 }}{{Dead link|date=September 2023 |bot=InternetArchiveBot |fix-attempted=yes }}</ref> the [[Henri Poincaré Prize]] of the [[International Association of Mathematical Physics]] (2003),<ref>{{cite web |title=The Henri Poincaré Prize |url=http://www.iamp.org/page.php?page=page_prize_poincare |website=International Association of Mathematical Physics |access-date = January 5, 2020}}</ref> and the [https://www.esi.ac.at/esi-medal/recipients/2021 Medal of the Erwin Schrödinger Institute for Mathematics and Physics] (2021).<ref>{{cite web |title=ESI Medal |url=https://www.esi.ac.at/esi-medal/recipients/2021 |website=ESI |access-date = July 2, 2022}}</ref>

In 2022 Lieb was awarded the [[List of American Physical Society prizes and awards|Medal for Exceptional Achievement in Research]] from the [[American Physical Society]] for ″major contributions to theoretical physics through obtaining exact solutions to important physical problems, which have impacted condensed matter physics, quantum information, statistical mechanics, and atomic physics″<ref>{{cite web | url = https://www.aps.org/programs/honors/prizes/prizerecipient.cfm?last_nm=Lieb&first_nm=Elliott&year=2022 | title = 2022 APS Medal for Exceptional Achievement in Research Recipient | language = en | access-date = June 15, 2022 }}</ref>
and the [[Carl Friedrich Gauss Prize]] at the [[International Congress of Mathematicians]] ″for deep mathematical contributions of exceptional breadth which have shaped the fields of quantum mechanics, statistical mechanics, computational chemistry, and quantum information theory.″<ref>{{cite web |title=Gauss Prize |url=https://www.mathunion.org/imu-awards/carl-friedrich-gauss-prize/carl-friedrich-gauss-prize-2022|access-date=July 5, 2022}}</ref> Also in 2022 he received the [[Dirac Medal (ICTP)|Dirac Medal]] of the ICTP<ref>{{cite web |title=Dirac Medal |url=https://www.ictp.it/about-ictp/media-centre/news/2022/8/2022-dirac-medal-winners-announced.aspx#close |website=ICTP |access-date = August 8, 2022}}</ref> jointly with [[Joel Lebowitz]] and [[David Ruelle]]. 

Lieb is a member of the [[U.S. National Academy of Sciences]]<ref>{{cite web |title=Elliott Lieb |url=http://www.nasonline.org/member-directory/members/53258.html |website=U.S. National Academy of Sciences |access-date = January 5, 2020}}</ref> and has twice served (1982–1984 and 1997–1999) as the president of the [[International Association of Mathematical Physics]].<ref>{{cite web |title=About the IAMP – Past presidents |url=http://www.iamp.org/page.php?page=page_about |website=International Association of Mathematical Physics |access-date = January 5, 2020}}</ref> Lieb was awarded the [[Austrian Decoration for Science and Art]] in 2002.<ref>{{cite web | url = http://www.parlament.gv.at/PAKT/VHG/XXIV/AB/AB_10542/imfname_251156.pdf | title = Reply to a parliamentary question | language = de | page=1517 | access-date = November 19, 2012 }}</ref> In 2012 he became a fellow of the [[American Mathematical Society]]<ref>[https://www.ams.org/profession/fellows-list List of Fellows of the American Mathematical Society], retrieved January 27, 2013.</ref> and in 2013 a [[Foreign Member of the Royal Society]].<ref>{{cite web|url=http://royalsociety.org/about-us/fellowship/new-fellows-2013/|title=New Fellows 2013|publisher=Royal Society|access-date = July 30, 2013}}</ref>

In 2023 Lieb received [[Kyoto Prize in Basic Sciences]] for his achievements in many-body physics.<ref>{{cite web | title = Elliott H. Lieb | publisher = Inamori Foundation | url = https://www.kyotoprize.org/en/laureates/elliott_h_lieb/ | accessdate=2023-06-16}}</ref>

==Works==
Lieb has made fundamental contributions to both theoretical physics and mathematics. Only some of them are outlined here. His main research papers are gathered in four Selecta volumes.<ref name="statmech" /><ref name="condmat" /><ref name="stabmat" /><ref name="inequalities" /> More details can also be found in two books published by [[EMS Press]] in 2022 on the occasion of his 90th birthday.<ref name="Lieb-90">{{cite book |isbn=978-3-98547-019-8|editor1-last=Frank |editor1-first=Rupert |editor2-last=Laptev |editor2-first=Ari |editor3-last=Lewin |editor3-first=Mathieu |editor4-last=Seiringer |editor4-first=Robert |title=The Physics and Mathematics of Elliott Lieb, The 90th Anniversary Volume (vol 1 and 2)|date=2022 |volume=2 |edition=EMS Press |url=https://ems.press/books/standalone/234|doi=10.4171/90|arxiv=2202.01867 |last1=Wanless |first1=Ian M. }}</ref> His research is reviewed there in more than 50 chapters.

===Statistical mechanics, soluble systems===

Lieb is famous for many groundbreaking results in [[statistical mechanics]] concerning, in particular, soluble systems. His numerous works have been collected in the Selecta ''″Statistical mechanics″''<ref name="statmech" /> and ''″Condensed matter physics and exactly soluble models″'',<ref name="condmat" /> as well as in a book with Daniel Mattis.<ref name="Lieb-Mattis" /> They treat (among other things) [[Ising model|Ising-type models]], models for [[ferromagnetism]] and [[ferroelectricity]], the exact solution of the [[six-vertex model]] of ice in two dimensions, the one-dimensional delta Bose gas (now called the [[Lieb-Liniger model]]) and the [[Hubbard model]].

Together with Daniel Mattis and Theodore Schultz, Lieb solved in 1964 the two-dimensional [[Ising model]] (with a new derivation of the exact solution by [[Lars Onsager]] via the [[Jordan-Wigner transformation]] of the transfer matrices) and in 1961 the [[XY model]], an explicitly solvable one-dimensional spin-1/2 model. In 1968, together with [[Fa-Yueh Wu]], he gave the exact solution of the one-dimensional Hubbard model.

In 1971 Lieb and [[Harold Neville Vazeille Temperley|Neville Temperley]] introduced the [[Temperley-Lieb algebra]] in order to build certain transfer matrices. This algebra also has links with [[knot theory]] and the [[braid group]], [[quantum groups]] and subfactors of [[von Neumann algebras]].

Together with [[Derek W. Robinson]] in 1972, Lieb derived bounds on the propagation speed of information in non-relativistic spin systems with local interactions. They have become known as [[Lieb-Robinson bounds]] and play an important role, for instance, in error bounds in the [[thermodynamic limit]] or in [[quantum computing]]. They can be used to prove the exponential decay of correlations in spin systems or to make assertions about the gap above the [[ground state]] in higher-dimensional spin systems (generalized Lieb-Schultz-Mattis theorems).

In 1972 Lieb and [[Mary Beth Ruskai]] proved the [[strong subadditivity of quantum entropy]], a theorem that is fundamental for [[quantum information theory]]. This is closely related to what is known as the [[data processing inequality]] in quantum information theory. The Lieb-Ruskai proof of strong subadditivity is based on an earlier paper where Lieb solved several important conjectures about operator inequalities, including the Wigner-Yanase-Dyson conjecture.<ref>{{cite journal |last1=Lieb |first1=Elliott H |title=Convex trace functions and the Wigner-Yanase-Dyson conjecture |journal=[[Advances in Mathematics]] |date=December 1973 |volume=11 |issue=3 |pages=267–288 |doi=10.1016/0001-8708(73)90011-X|doi-access=free}}</ref>

In the years 1997–99, Lieb provided a rigorous treatment of the increase of entropy in the [[second law of thermodynamics]] and [[adiabatic accessibility]] with [[Jakob Yngvason]].<ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Yngvason |first2=Jakob |title=The physics and mathematics of the second law of thermodynamics |journal=Physics Reports |date=March 1999 |volume=310 |issue=1 |pages=1–96 |doi=10.1016/S0370-1573(98)00082-9|arxiv=cond-mat/9708200 |bibcode=1999PhR...310....1L |s2cid=119620408 }}</ref>

===Many-body quantum systems and the stability of matter===
{{see also|stability of matter}}

In 1975, Lieb and [[Walter Thirring]] found a proof of the [[stability of matter]] that was shorter and more conceptual than that of [[Freeman Dyson]] and Andrew Lenard in 1967. Their argument is based on a new inequality in spectral theory, which became known as the [[Lieb-Thirring inequality]]. The latter has become a standard tool in the study of large fermionic systems, e.g. for (pseudo-)relativistic fermions in interaction with classical or quantized electromagnetic fields. On the mathematical side, the [[Lieb-Thirring inequality]] has also generated a huge interest in the spectral theory of Schrödinger operators.<ref name="LT">{{cite book |last1=Frank |first1=Rupert L. |last2=Laptev |first2=Ari |last3=Weidl |first3=Timo |title=Schrödinger Operators: Eigenvalues and Lieb–Thirring Inequalities |date=2022 |publisher=Cambridge University Press}}</ref> This fruitful research program has led to many important results that can be read in his Selecta ''″The stability of matter : from atoms to stars″''<ref name="stabmat" /> as well as in his book ''″The stability of matter in quantum mechanics″'' with [[Robert Seiringer]].<ref name="stabmat-Seiringer">{{cite book |last1=Lieb |first1=Elliott H. |last2=Seiringer |first2=Robert |title=The Stability of Matter in Quantum Mechanics |date=November 5, 2009 |doi=10.1017/CBO9780511819681|isbn=9780521191180 }}</ref>

Based on the original Dyson-Lenard theorem of stability of matter, Lieb together with [[Joel Lebowitz]] had already provided in 1973 the first proof of the existence of thermodynamic functions for quantum matter. With Heide Narnhofer he did the same for [[Jellium]], also called the [[homogeneous electron gas]], which is at the basis of most functionals in [[Density Functional Theory]].

In the 1970s, Lieb together with [[Barry Simon]] studied several nonlinear approximations of the many-body Schrödinger equation, in particular [[Hartree-Fock theory]] and the [[Thomas-Fermi model]] of atoms. They provided the first rigorous proof that the latter furnishes the leading order of the energy for large non-relativistic atoms. With Rafael Benguria and [[Haïm Brezis]], he studied several variations of the [[Thomas-Fermi model]].

The ionization problem in mathematical physics asks for a rigorous upper bound on the number of electrons that an atom with a given nuclear charge can bind. Experimental and numerical evidence seems to suggest that there can be at most one, or possibly two extra electrons. To prove this rigorously is an open problem. A similar question can be asked concerning molecules. Lieb proved a famous upper bound on the number of electrons a nucleus can bind. Moreover, together with [[Israel Michael Sigal]], [[Barry Simon]] and [[Walter Thirring]], he proved, for the first time, that the excess charge is asymptotically small compared to the nuclear charge.

Together with [[Jakob Yngvason]], Lieb gave a rigorous proof of a formula for the ground state energy of dilute Bose gases. Subsequently, together with [[Robert Seiringer]] and [[Jakob Yngvason]] he studied the Gross-Pitaevskii equation for the ground state energy of dilute bosons in a trap, starting with many-body quantum mechanics.<ref name="BEC">{{cite journal|doi=10.1090/S0273-0979-07-01147-0|title=Book Review: ''The mathematics of the Bose gas and its condensation'' |year=2007 |last1=Hoffmann-Ostenhof |first1=T. |journal=Bulletin of the American Mathematical Society |volume=44 |issue=3 |pages=493–497 |doi-access=free }}</ref> Lieb's works with Joseph Conlon and [[Horng-Tzer Yau]] and with [[Jan Philip Solovej]] on what is known as the <math>N^{7/5}</math> law for bosons provide the first rigorous justification of Bogoliubov's pairing theory.

In quantum chemistry Lieb is famous for having provided in 1983 the first rigorous formulation of Density Functional Theory using tools of convex analysis. The universal Lieb functional gives the lowest energy of a Coulomb system with a given density profile, for mixed states. In 1980, he proved with Stephen Oxford the [[Lieb-Oxford inequality]]<ref>{{cite journal |last1=Lieb |first1=Elliott H. |last2=Oxford |first2=Stephen |title=Improved lower bound on the indirect Coulomb energy |journal=International Journal of Quantum Chemistry |date=March 1981 |volume=19 |issue=3 |pages=427–439 |doi=10.1002/qua.560190306}}</ref> which provides an estimate on the lowest possible classical Coulomb energy at fixed density and was later used for calibration of some functionals such as PBE and SCAN. More recently, together with [[Mathieu Lewin]] and [[Robert Seiringer]] he gave the first rigorous justification of the [[Local-density approximation]] for slowly varying densities.<ref>{{cite journal |last1=Lewin |first1=Mathieu |last2=Lieb |first2=Elliott H. |last3=Seiringer |first3=Robert |title=The local density approximation in density functional theory |journal=Pure and Applied Analysis |date=January 1, 2020 |volume=2 |issue=1 |pages=35–73 |doi=10.2140/paa.2020.2.35|arxiv=1903.04046 |s2cid=119176239 }}</ref>

===Analysis===

In the 70s Lieb entered the mathematical fields of [[calculus of variations]] and [[partial differential equations]], where he made fundamental contributions. An important theme was the quest of best constants in several inequalities of [[functional analysis]], which he then used to rigorously study nonlinear quantum systems. His results in this direction are collected in the Selecta ''″Inequalities″''.<ref name="inequalities" /> Among the inequalities where he determined the sharp constants are Young's inequality and the Hardy-Littlewood-Sobolev inequality, to be further discussed below. He also developed tools now considered standard in analysis, such as [[Riesz rearrangement inequality|rearrangement inequalities]] or the [[Brezis-Lieb lemma]] which provides the missing term in [[Fatou's lemma]] for sequences of functions converging almost everywhere.

With Herm Brascamp and [[Joaquin Luttinger]], Lieb proved in 1974 a generalization of the [[Riesz rearrangement inequality]], stating that certain multilinear integrals increase when all the functions are replaced by their [[symmetric decreasing rearrangement]]. With [[Frederick Almgren]], he clarified the continuity properties of rearrangement. Rearrangement is often used to prove the existence of solutions for some nonlinear models.

In two papers (one in 1976 with Herm Brascamp and another one alone in 1990), Lieb determined the validity and the best constants of a whole family of inequalities that generalizes, for instance, the [[Hölder's inequality]], [[Young's convolution inequality|Young's inequality for convolutions]], and the [[Loomis-Whitney inequality]]. This is now known as the [[Brascamp-Lieb inequality]]. The spirit is that the best constant is determined by the case where all functions are Gaussians. The [[Brascamp-Lieb inequality]] has found applications and extensions, for instance, in harmonic analysis.

Using rearrangement inequalities and compactness methods, Lieb proved in 1983 the existence of optimizers for the [[Hardy-Littlewood-Sobolev inequality]] and of the [[Sobolev inequality]]. He also determined the best constant in some cases, discovering and exploiting the conformal invariance of the problem and relating it, via [[stereographic projection]], to a conformally equivalent, but more tractable problem on the sphere. A new rearrangement-free proof was provided later with Rupert Frank, allowing to treat the case of the Heisenberg group.<ref>{{cite journal |last1=Frank |first1=Rupert L. |last2=Lieb |first2=Elliott H. |title=Sharp constants in several inequalities on the Heisenberg group |journal=Annals of Mathematics |date=July 1, 2012 |volume=176 |issue=1 |pages=349–381 |doi=10.4007/annals.2012.176.1.6 |doi-access=free}}</ref>

In a 1977 work, Lieb also proved the uniqueness (up to symmetries) of the ground state for the Choquard-Pekar equation, also called [[Schrödinger–Newton equation]],<ref>{{cite journal |last1=Lieb |first1=Elliott H. |title=Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation |journal=Studies in Applied Mathematics |date=October 1977 |volume=57 |issue=2 |pages=93–105 |doi=10.1002/sapm197757293}}</ref> which can describe a self gravitating object or an electron moving in a polarizable medium ([[polaron]]). With Lawrence Thomas he provided in 1997 a variational derivation of the [[Schrödinger–Newton equation|Choquard-Pekar equation]] from a model in quantum field theory (the [[polaron|Fröhlich Hamiltonian]]). This had been solved earlier by [[Monroe Donsker]] and [[Srinivasa Varadhan]] using a probabilistic path integral method.

In another work with Herm Brascamp in 1976, Lieb extended the [[Prékopa-Leindler inequality]] to other types of convex combinations of two positive functions. He strengthened the inequality and the [[Brunn-Minkowski inequality]] by introducing the notion of essential addition.

Lieb also wrote influential papers on harmonic maps, among others with [[Frederick Almgren]], [[Haïm Brezis]] and [[Jean-Michel Coron]]. In particular, Algrem and Lieb proved a bound on the number of singularities of energy minimizing harmonic maps.

Finally, his textbook ''″Analysis″'' with [[Michael Loss]] should be mentioned.<ref name="analysis">{{cite book |last1=Lieb |first1=Elliott H. |last2=Loss |first2=Michael |title=Analysis : Second Edition |year=2001 |publisher=American Mathematical Society |isbn=978-0-8218-2783-3}}</ref>
It has become a standard for graduate courses in mathematical analysis. It develops all the traditional tools of analysis in a concise, intuitive and eloquent form, with a view towards applications.

== Selected publications ==

;Books
* Lieb, Elliott H.; [[Robert Seiringer|Seiringer, Robert]]. ''The stability of matter in quantum mechanics''. [[Cambridge University Press]], 2010 {{ISBN|978-0-521-19118-0}}<ref name="stabmat-Seiringer"/>
* Lieb, Elliott H.; [[Michael Loss|Loss, Michael]]. ''Analysis''. [[Graduate Studies in Mathematics]], 14. American Mathematical Society, Providence, RI, 1997. xviii+278 pp. {{ISBN|0-8218-0632-7}}<ref name="analysis"/>
*  Lieb, Elliott H.; Seiringer, Robert; Solovej, Jan Philip; Yngvason, Jakob. ''The mathematics of the Bose gas and its condensation''. Oberwolfach Seminars, 34. Birkhäuser Verlag, Basel, 2005. viii+203 pp. {{ISBN|978-3-7643-7336-8}}; 3-7643-7336-9<ref name="BEC"/>

; Articles
* ''Statistical mechanics. Selecta of Elliott H. Lieb''. Edited, with a preface and commentaries, by B. Nachtergaele, J. P. Solovej and J. Yngvason. Springer-Verlag, Berlin, 2004. x+505 pp. {{ISBN|3-540-22297-9}}<ref name="statmech" />
* ''Condensed matter physics and exactly soluble models. Selecta of Elliott H. Lieb''. Edited by B. Nachtergaele, J. P. Solovej and J. Yngvason. Springer-Verlag, Berlin, 2004. x+675 pp. {{ISBN|3-540-22298-7}}<ref name="condmat" />
* ''The Stability of Matter: From Atoms to Stars. Selecta of Elliott H. Lieb'' (4th edition). Edited by W. Thirring, with a preface by F. Dyson. Springer-Verlag, Berlin, 2005. xv+932 pp. {{ISBN|978-3-540-22212-5}}<ref name="stabmat" />
* ''Inequalities. Selecta of Elliott H. Lieb''. Edited, with a preface and commentaries, by M. Loss and M. B. Ruskai. Springer-Verlag, Berlin, 2002. xi+711 pp. {{ISBN|3-540-43021-0}}<ref name="inequalities" />

;As editor
* Lieb, Elliott H. and Mattis, Daniel C., editors. ''Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles'', Academic Press, New York, 1966. {{ISBN|978-0-12-448750-5}}<ref name="Lieb-Mattis">{{cite journal|doi=10.1063/1.3034501|title=Review of ''Mathematical Physics in One Dimension: Exactly Soluble Models of Interacting Particles'' by Elliott H. Lieb and Daniel C. Mattis |year=1967 |last=Dyson |first=Freeman J. |author-link=Freeman Dyson |journal=Physics Today |volume=20 |issue=9 |pages=81–82 }}</ref>

;Other
* [https://ems.press/books/standalone/234 ''The Physics and Mathematics of Elliott Lieb'']. Edited by R. L. Frank, A. Laptev, M. Lewin and R. Seiringer. EMS Press, July 2022, 1372 pp. {{ISBN|978-3-98547-019-8}}<ref name="Lieb-90"/>

These are two books published by [[EMS Press]] on the occasion of Lieb's 90th birthday, which contain around 50 chapters about his impact on a very broad range of topics and the resulting subsequent developments. Many contributions are of an expository character and thus accessible to non-experts.

==See also==
{{Div col|colwidth=20em}}
*[[Adiabatic accessibility]]
*[[AKLT model]]
*[[Trace inequality#Araki–Lieb–Thirring inequality|Araki–Lieb–Thirring inequality]]
*[[Borell–Brascamp–Lieb inequality]]
*[[Brascamp–Lieb inequality]]
*[[Brezis–Lieb lemma]]
*[[Strong subadditivity of quantum entropy#Carlen-Lieb Extension|Carlen-Lieb extension]]
*[[Entropy#Entropy and adiabatic accessibility|Entropy]]
*[[Ice-type model]]
*[[Lieb conjecture]] on the [[Wehrl entropy]]
*[[Lieb–Liniger Model]]
*[[Lieb–Oxford inequality]]
*[[Lieb-Robinson bounds|Lieb–Robinson bounds]]
*[[Lieb–Thirring inequality]] 
*Lieb-Wu equation for the [[Hubbard model]]
*[[Lieb's square ice constant]]
*[[Trace inequality#Lieb's theorem|Lieb's concavity theorem]]
*[[Stability of matter]]
*[[Strong subadditivity of quantum entropy]]
*[[Temperley–Lieb algebra]]
*[[Von Neumann entropy]]
{{Div col end}}

== References ==
{{Reflist|refs=
<ref name="ste22">{{cite journal | last=Sternheimer | first=Daniel | title=Some Multifaceted Aspects of Mathematical Physics, Our Common Denominator with Elliott Lieb | journal=Axioms | date=April 5, 2022 | volume=11 | issue=10 | page=522 | doi=10.3390/axioms11100522 | arxiv=2204.02160| doi-access=free }}</ref>
}}

==External links==
* [http://www.math.princeton.edu/~lieb/ Faculty page] at Princeton.
* {{cite web|url=https://www.youtube.com/watch?v=HqE9dJ2sXOA|title=Part 8 - Elliott Lieb: "Understanding entropy without probability"|date=February 28, 2019|publisher=Erwin Schrödinger International Institute for Mathematics and Physics (ESI)|website=YouTube|postscript=; Lieb's talk, Oct. 29, 2018, as part of Symposium »Concepts of Probability in the Sciences«, held October 29–30, 2018 at the [[University of Vienna]]}}
* {{cite web|url=https://www.youtube.com/watch?v=j3if0zjBDXs&t=140s|date=May 8, 2024|title=Kyoto Prize at Oxford 2024 – Dr Elliott H Lieb: My Journey Through Physics and Mathematics|publisher=Blavatnik School of Government|website=YouTube}}

{{Schock Prize laureates}}
{{FRS 2013}}
{{Authority control}}

{{DEFAULTSORT:Lieb, Elliott}}
[[Category:1932 births]]
[[Category:Living people]]
[[Category:Mathematical physicists]]
[[Category:20th-century American mathematicians]]
[[Category:21st-century American mathematicians]]
[[Category:Members of the United States National Academy of Sciences]]
[[Category:Alumni of the University of Birmingham]]
[[Category:20th-century American physicists]]
[[Category:21st-century American physicists]]
[[Category:Massachusetts Institute of Technology School of Science alumni]]
[[Category:Massachusetts Institute of Technology faculty]]
[[Category:Princeton University faculty]]
[[Category:Recipients of the Austrian Decoration for Science and Art]]
[[Category:Fellows of the American Mathematical Society]]
[[Category:Rolf Schock Prize laureates]]
[[Category:Foreign members of the Royal Society]]
[[Category:Scientists from Boston]]
[[Category:American mathematical physicists]]
[[Category:Winners of the Max Planck Medal]]
[[Category:Presidents of the International Association of Mathematical Physics]]
[[Category:Kyoto laureates in Basic Sciences]]
[[Category:Recipients of the Boltzmann Medal]]