Gang Tian

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Gang Tian
Gang Tian.jpeg
Gang Tian at Oberwolfach in 2005
Born (1958-11-24) 24 November 1958 (age 59)
Nanjing, Jiangsu, China
Nationality China
Alma mater Harvard University
Peking University
Nanjing University
Known for Bogomolov–Tian–Todorov theorem
Awards Veblen Prize (1996)
Alan T. Waterman Award (1994)
Scientific career
Fields Mathematics
Institutions Princeton University
Peking University
Doctoral advisor Shing-Tung Yau
Doctoral students Nataša Šešum

Tian Gang (simplified Chinese :田刚; traditional Chinese :田剛; pinyin :Tián Gāng; born November 1958) [1] is a Chinese mathematician and an academician of the American Academy of Arts and Sciences. He is known for his contributions to geometric analysis and quantum cohomology especially Gromov-Witten invariants, among other fields.

Simplified Chinese characters standardized Chinese characters developed in mainland China

Simplified Chinese characters are standardized Chinese characters prescribed in the Table of General Standard Chinese Characters for use in mainland China. Along with traditional Chinese characters, they are one of the two standard character sets of the contemporary Chinese written language. The government of the People's Republic of China in mainland China has promoted them for use in printing since the 1950s and 1960s to encourage literacy. They are officially used in the People's Republic of China and Singapore.

Traditional Chinese characters Traditional Chinese characters

Traditional Chinese characters are Chinese characters in any character set that does not contain newly created characters or character substitutions performed after 1946. They are most commonly the characters in the standardized character sets of Taiwan, of Hong Kong and Macau, and in the Kangxi Dictionary. The modern shapes of traditional Chinese characters first appeared with the emergence of the clerical script during the Han Dynasty, and have been more or less stable since the 5th century.

Hanyu Pinyin, often abbreviated to pinyin, is the official romanization system for Standard Chinese in mainland China and to some extent in Taiwan. It is often used to teach Standard Mandarin Chinese, which is normally written using Chinese characters. The system includes four diacritics denoting tones. Pinyin without tone marks is used to spell Chinese names and words in languages written with the Latin alphabet, and also in certain computer input methods to enter Chinese characters.

Contents

He was born in Nanjing, and was a professor of mathematics at the Massachusetts Institute of Technology from 1995–2006 (holding the chair of Simons Professor of Mathematics from 1996), but now divides his time between Princeton University and Peking University. His employment at Princeton started from 2003, and now his title there is Higgins Professor of Mathematics; starting 2005, he has been the director of the Beijing International Center for Mathematical Research (BICMR); [2] he has also been Dean of School of Mathematical Sciences, Peking University since 2013. [3] He and John Milnor are Senior Scholars of the Clay Mathematics Institute (CMI). In 2011, Tian became director of the Sino-French Research Program in Mathematics at the Centre national de la recherche scientifique (CNRS) in Paris. In 2010, he became scientific consultant for the International Center for Theoretical Physics in Trieste, Italy. [4]

Nanjing Prefecture-level & Sub-provincial city in Jiangsu, Peoples Republic of China

Nanjing, formerly romanized as Nanking and Nankin, is the capital of Jiangsu province of the People's Republic of China and the second largest city in the East China region, with an administrative area of 6,600 km2 (2,500 sq mi) and a total population of 8,270,500 as of 2016. The inner area of Nanjing enclosed by the city wall is Nanjing City (南京城), with an area of 55 km2 (21 sq mi), while the Nanjing Metropolitan Region includes surrounding cities and areas, covering over 60,000 km2 (23,000 sq mi), with a population of over 30 million.

Massachusetts Institute of Technology University in Massachusetts

The Massachusetts Institute of Technology (MIT) is a private research university in Cambridge, Massachusetts. Founded in 1861 in response to the increasing industrialization of the United States, MIT adopted a European polytechnic university model and stressed laboratory instruction in applied science and engineering. The Institute is a land-grant, sea-grant, and space-grant university, with a campus that extends more than a mile alongside the Charles River. Its influence in the physical sciences, engineering, and architecture, and more recently in biology, economics, linguistics, management, and social science and art, has made it one of the most prestigious universities in the world. MIT is often ranked among the world's top universities.

Princeton University University in Princeton, New Jersey

Princeton University is a private Ivy League research university in Princeton, New Jersey. Founded in 1746 in Elizabeth as the College of New Jersey, Princeton is the fourth-oldest institution of higher education in the United States and one of the nine colonial colleges chartered before the American Revolution. The institution moved to Newark in 1747, then to the current site nine years later, and renamed itself Princeton University in 1896.

Biography

Tian qualified in the second college entrance exam after Cultural Revolution in 1978. He graduated from Nanjing University in 1982, and received a master's degree from Peking University in 1984. In 1988, he received a Ph.D. in mathematics from Harvard University, after having studied under Shing-Tung Yau. This work was so exceptional he was invited to present it at the Geometry Festival that year. In 1998, he was appointed as a Cheung Kong Scholar professor at the School of Mathematical Sciences at Peking University, under the "Cheung Kong Scholars Programme" (长江计划) of the Ministry of Education. Later his appointment was changed to Cheung Kong Scholar chair professorship. He was awarded the Alan T. Waterman Award in 1994, and the Veblen Prize in 1996. In 2004 Tian was inducted into the American Academy of Arts and Sciences.

Nanjing University university in Nanjing, China

Nanjing University, known as Nanda, is a major public university, the oldest institution of higher learning in Nanjing, Jiangsu, and a member of the elite C9 League of Chinese universities.

A master's degree is an academic degree awarded by universities or colleges upon completion of a course of study demonstrating mastery or a high-order overview of a specific field of study or area of professional practice. A master's degree normally requires previous study at the bachelor's level, either as a separate degree or as part of an integrated course. Within the area studied, master's graduates are expected to possess advanced knowledge of a specialized body of theoretical and applied topics; high order skills in analysis, critical evaluation, or professional application; and the ability to solve complex problems and think rigorously and independently.

Doctor of Philosophy Postgraduate academic degree awarded by universities in many countries

A Doctor of Philosophy is the highest university degree that is conferred after a course of study by universities in most English-speaking countries. PhDs are awarded for programs across the whole breadth of academic fields. As an earned research degree, those studying for a PhD are usually required to produce original research that expands the boundaries of knowledge, normally in the form of a thesis or dissertation, and defend their work against experts in the field. The completion of a PhD is often a requirement for employment as a university professor, researcher, or scientist in many fields. Individuals who have earned a Doctor of Philosophy degree may, in many jurisdictions, use the title Doctor or, in non-English-speaking countries, variants such as "Dr. phil." with their name, although the proper etiquette associated with this usage may also be subject to the professional ethics of their own scholarly field, culture, or society. Those who teach at universities or work in academic, educational, or research fields are usually addressed by this title "professionally and socially in a salutation or conversation." Alternatively, holders may use post-nominal letters such as "Ph.D.", "PhD", or "DPhil". It is, however, considered incorrect to use both the title and post-nominals at the same time.

Mathematical contributions

Much of Tian's earlier work was about the existence of Kähler–Einstein metrics on complex manifolds under the direction of Yau. In particular he solved the existence question for Kähler–Einstein metrics on compact complex surfaces with positive first Chern class, and showed that hypersurfaces with a Kähler–Einstein metric are stable in the sense of geometric invariant theory. He proved that a Kähler manifold with trivial canonical bundle has trivial obstruction space, known as the Bogomolov–Tian–Todorov theorem. [5]

In differential geometry, a Kähler–Einstein metric on a complex manifold is a Riemannian metric that is both a Kähler metric and an Einstein metric. A manifold is said to be Kähler–Einstein if it admits a Kähler–Einstein metric. The most important special case of these are the Calabi–Yau manifolds, which are Kähler and Ricci-flat.

In differential geometry, a complex manifold is a manifold with an atlas of charts to the open unit disk in Cn, such that the transition maps are holomorphic.

In mathematics Geometric invariant theory is a method for constructing quotients by group actions in algebraic geometry, used to construct moduli spaces. It was developed by David Mumford in 1965, using ideas from the paper in classical invariant theory.

Tian found an explicit formula for Weil-Petersson metric on moduli space of polarized Calabi-Yau manifolds. [6]

Tian made foundational contributions to Gromov-Witten theory. He (jointly with Jun Li) constructed virtual fundamental cycles of the moduli spaces of maps from curves in both algebraic geometry and symplectic geometry. He also (jointly with Y. Ruan) showed that the quantum cohomology ring of a semi-positive symplectic manifold is associative.

In algebraic geometry, a moduli space is a geometric space whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such spaces frequently arise as solutions to classification problems: If one can show that a collection of interesting objects can be given the structure of a geometric space, then one can parametrize such objects by introducing coordinates on the resulting space. In this context, the term "modulus" is used synonymously with "parameter"; moduli spaces were first understood as spaces of parameters rather than as spaces of objects. A variant of moduli spaces are formal moduli.

Algebraic geometry branch of mathematics

Algebraic geometry is a branch of mathematics, classically studying zeros of multivariate polynomials. Modern algebraic geometry is based on the use of abstract algebraic techniques, mainly from commutative algebra, for solving geometrical problems about these sets of zeros.

Symplectic geometry Branch of differential geometry and differential topology

Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Hamiltonian formulation of classical mechanics where the phase space of certain classical systems takes on the structure of a symplectic manifold.

He introduced the Analytical Minimal Model program which is known as Tian-Song program in birational geometry . In Kähler geometry he has a new theory which is known as Cheeger-Colding-Tian's theory. Tian's alpha-invariant was introduced by him and was later given an algebraic interpretation by János Kollár and Jean-Pierre Demailly.

He (with Yau and Donaldson) proposed the Yau-Tian-Donaldson conjecture. It was solved by Chen, Donaldson and Sun which was electronically published on March 28, 2014. [7] [8] [9] Tian also gave a proof electronically published on September 16, 2015. [10] [11]

In 2006, together with John Morgan of Columbia University (now at Stony Brook University), amongst others, Tian helped verify the proof of the Poincaré conjecture given by Grigori Perelman. [12]

Gang Tian was once one of the five members of the Abel Prize Committee. [13]

Gang Tian was also once one of the five members of the Ramanujan Prize selection committee.

In 2012, he became member of Leroy P. Steele Prize Committee in AMS. [14]

Editorial Positions

Gang Tian is member of the editorial boards of a number of journals in Mathematics.

1. Annals of Mathematics [15]

2. Annali della Scuola Normale Superiore [16]

3. Journal of Symplectic Geometry [17]

4. Journal of the American Mathematical Society,1995-1998. [18]

5. Geometry & Topology [19]

6. The Journal of Geometric Analysis [20]

7. Geometric and Functional Analysis [21]

8. Advances in Mathematics [22]

9. International Mathematics Research Notices [23]

10. Pacific Journal of Mathematics, 1994-1998.

11. Communications in Analysis and Geometry, 1994-2000.

12. Acta Mathematica Sinica, [24]

13. Mathematics Revista Matemática Complutense [25]

14. Communications in Mathematics and Statistics, [26]

15. Communication in Contemporary Mathematics, [27]

Selected publications

Tian, Gang. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), 629—646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987.

Tian, Gang. On Kähler-Einstein metrics on certain Kähler manifolds with . Invent. Math. 89 (1987), no. 2, 225—246.

Tian, G.; Yau, Shing-Tung. Complete Kähler manifolds with zero Ricci curvature. I. J. Amer. Math. Soc. 3 (1990), no. 3, 579—609.

Tian, G. On Calabi's conjecture for complex surfaces with positive first Chern class. Invent. Math. 101 (1990), no. 1, 101—172.

Tian, Gang. On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom. 32 (1990), no. 1, 99—130.

Ruan, Yongbin; Tian, Gang. A mathematical theory of quantum cohomology. J. Differential Geom. 42 (1995), no. 2, 259—367.

Tian, Gang. Kähler-Einstein metrics with positive scalar curvature. Invent. Math. 130 (1997), no. 1, 1--37.

Ruan, Yongbin; Tian, Gang. Higher genus symplectic invariants and sigma models coupled with gravity. Invent. Math. 130 (1997), no. 3, 455—516.

Li, Jun; Tian, Gang. Virtual moduli cycles and Gromov-Witten invariants of algebraic varieties. J. Amer. Math. Soc. 11 (1998), no. 1, 119—174.

Liu, Gang; Tian, Gang. Floer homology and Arnold conjecture. J. Differential Geom. 49 (1998), no. 1, 1--74.

Liu, Xiaobo; Tian, Gang. Virasoro constraints for quantum cohomology. J. Differential Geom. 50 (1998), no. 3, 537—590.

Tian, Gang. Gauge theory and calibrated geometry. I. Ann. of Math. (2) 151 (2000), no. 1, 193—268.

Tian, Gang; Zhu, Xiaohua. Uniqueness of Kähler-Ricci solitons. Acta Math. 184 (2000), no. 2, 271—305.

Cheeger, J.; Colding, T. H.; Tian, G. On the singularities of spaces with bounded Ricci curvature. Geom. Funct. Anal. 12 (2002), no. 5, 873—914.

Tao, Terence; Tian, Gang. A singularity removal theorem for Yang-Mills fields in higher dimensions. J. Amer. Math. Soc. 17 (2004), no. 3, 557—593.

Tian, Gang; Viaclovsky, Jeff. Bach-flat asymptotically locally Euclidean metrics. Invent. Math. 160 (2005), no. 2, 357—415.

Cheeger, Jeff; Tian, Gang. Curvature and injectivity radius estimates for Einstein 4-manifolds. J. Amer. Math. Soc. Vol. 19, No. 2 (2006), 487—525.

Morgan, John; Tian, Gang. Ricci flow and the Poincaré conjecture. Clay Mathematics Monographs, 3. American Mathematical Society, Providence, RI; Clay Mathematics Institute, Cambridge, MA, 2007, 525pp.

Song, Jian; Tian, Gang. The Kähler-Ricci flow on surfaces of positive Kodaira dimension. Invent. Math. 170 (2007), no. 3, 609—653.

Chen, X. X.; Tian, G. Geometry of Kähler metrics and foliations by holomorphic discs. Publ. Math. Inst. Hautes Études Sci. No. 107 (2008), 1--107.

Kołodziej, Sławomir; Tian, Gang A uniform estimate for complex Monge-Ampère equations. Math. Ann. 342 (2008), no. 4, 773–787.

Mundet i Riera, I.; Tian, G. A compactification of the moduli space of twisted holomorphic maps. Adv. Math. 222 (2009), no. 4, 1117–1196.

Rivière, Tristan; Tian, Gang The singular set of 1-1 integral currents. Ann. of Math. (2) 169 (2009), no. 3, 741–794.

Tian, Gang Finite-time singularity of Kähler-Ricci flow. Discrete Contin. Dyn. Syst. 28 (2010), no. 3, 1137–1150.

Related Research Articles

Calabi–Yau manifold Riemannian manifold with SU(n) holonomy

In algebraic geometry, a Calabi–Yau manifold, also known as a Calabi–Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics. Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi–Yau manifold, which led to the idea of mirror symmetry. Their name was coined by Candelas et al. (1985), after Eugenio Calabi who first conjectured that such surfaces might exist, and Shing-Tung Yau (1978) who proved the Calabi conjecture.

Shing-Tung Yau Chinese-born American mathematician

Shing-Tung Yau is a Chinese and naturalized American mathematician. He was awarded the Fields Medal for his mathematical research in 1982. He is currently the William Caspar Graustein Professor of Mathematics at Harvard University.

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 Arnoldus Schouten and David van Dantzig in 1930, and then introduced by Erich Kähler in 1933. The terminology has been fixed by André Weil.

Mirror symmetry (string theory) conjectured relation between pairs of Calabi–Yau manifolds; situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory

In algebraic geometry and theoretical physics, mirror symmetry is a relationship between geometric objects called Calabi–Yau manifolds. The term refers to a situation where two Calabi–Yau manifolds look very different geometrically but are nevertheless equivalent when employed as extra dimensions of string theory.

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric. They are named after Albert Einstein because this condition is equivalent to saying that the metric is a solution of the vacuum Einstein field equations, although both the dimension and the signature of the metric can be arbitrary, thus not being restricted to the four-dimensional Lorentzian manifolds usually studied in general relativity.

In differential geometry, a hyperkähler manifold is a Riemannian manifold of dimension and holonomy group contained in Sp(k). Hyperkähler manifolds are special classes of Kähler manifolds. They can be thought of as quaternionic analogues of Kähler manifolds. All hyperkähler manifolds are Ricci-flat and are thus Calabi–Yau manifolds.

Simon Donaldson English mathematician and Fields medalist

Sir Simon Kirwan Donaldson, is an English mathematician known for his work on the topology of smooth (differentiable) four-dimensional manifolds and Donaldson–Thomas theory. He is currently a permanent member of the Simons Center for Geometry and Physics at Stony Brook University and a Professor in Pure Mathematics at Imperial College London.

William Mark Goldman is a professor of mathematics at the University of Maryland, College Park. He received a B.A. in mathematics from Princeton University in 1977, and a Ph.D. in mathematics from the University of California, Berkeley in 1980.

Andreas Floer German mathematician

Andreas Floer was a German mathematician who made seminal contributions to the areas of geometry, topology, and mathematical physics, in particular the invention of Floer homology.

In mathematics, Floer homology is a tool for studying symplectic geometry and low-dimensional topology. Floer homology is a novel invariant that arises as an infinite-dimensional analog of finite-dimensional Morse homology. Andreas Floer introduced the first version of Floer homology, now called Hamiltonian Floer homology, in his proof of the Arnold conjecture in symplectic geometry. Floer also developed a closely related theory for Lagrangian submanifolds of a symplectic manifold. A third construction, also due to Floer, associates homology groups to closed three-dimensional manifolds using the Yang–Mills functional. These constructions and their descendants play a fundamental role in current investigations into the topology of symplectic and contact manifolds as well as (smooth) three- and four-dimensional manifolds.

In mathematics, the Calabi conjecture was a conjecture about the existence of certain "nice" Riemannian metrics on certain complex manifolds, made by Eugenio Calabi and proved by Shing-Tung Yau. Yau received the Fields Medal in 1982 in part for this proof.

John Willard Morgan is an American mathematician, with contributions to topology and geometry.

Huai-Dong Cao is A. Everett Pitcher Professor of Mathematics at Lehigh University and professor at Tsinghua University.

The Geometry Festival is an annual mathematics conference held in the United States.

Kefeng Liu American mathematician

Kefeng Liu, is a Chinese mathematician who is known for his contributions to geometric analysis, particularly the geometry, topology and analysis of moduli spaces of Riemann surfaces and Calabi-Yau manifolds. He is a professor of mathematics at University of California, Los Angeles, as well as the Executive Director of the Center of Mathematical Sciences at Zhejiang University.

Donaldson–Thomas theory

In mathematics, specifically algebraic geometry, Donaldson–Thomas theory is the theory of Donaldson–Thomas invariants. Given a compact moduli space of sheaves on a Calabi–Yau threefold, its Donaldson–Thomas invariant is the virtual number of its points, i.e., the integral of the cohomology class 1 against the virtual fundamental class. The Donaldson–Thomas invariant is a holomorphic analogue of the Casson invariant. The invariants were introduced by Simon Donaldson and Richard Thomas (1998). Donaldson–Thomas invariants have close connections to Gromov–Witten invariants of algebraic three-folds and the theory of stable pairs due to Rahul Pandharipande and Thomas.

Richard Thomas (mathematician) Pure mathematician at Imperial College London

Richard Paul Winsley Thomas FRS is a British mathematician working in several areas of geometry. He is a professor at Imperial College London. He studies moduli problems in algebraic geometry, and ‘mirror symmetry’ — a phenomenon in pure mathematics predicted by string theory in theoretical physics.

In differential geometry, the Kobayashi–Hitchin correspondence relates stable vector bundles over a complex manifold to Einstein–Hermitian vector bundles. The correspondence is named after Shoshichi Kobayashi and Nigel Hitchin, who independently conjectured in the 1980s that the moduli spaces of stable vector bundles and Einstein–Hermitian vector bundles over a complex manifold were essentially the same. This was proved by Simon Donaldson for algebraic surfaces and later for algebraic manifolds, by Karen Uhlenbeck and Shing-Tung Yau for Kähler manifolds, and by Jun Li and Yau for complex manifolds.

In differential geometry, a constant scalar curvature Kähler metric , is a Kähler metric on a complex manifold whose scalar curvature is constant. A special case is Kähler-Einstein metric, and a more general case is extremal Kähler metric.

References

  1. ↑ http://www.ams.org/notices/199603/comm-veblen.pdf
  2. ↑ Governing Board, Beijing International Center for Mathematical Research, http://www.bicmr.org/content/page/27.html
  3. ↑ History of School of Mathematical Sciences, Peking University, http://www.math.pku.edu.cn/static/lishiyange.html
  4. ↑ "ICTP - Governance". www.ictp.it. Retrieved 2018-05-28.
  5. ↑ The history about Tian-Todorov Lemma
  6. ↑ [Tian, Gang. Smoothness of the universal deformation space of compact Calabi-Yau manifolds and its Petersson-Weil metric. Mathematical aspects of string theory (San Diego, Calif., 1986), 629—646, Adv. Ser. Math. Phys., 1, World Sci. Publishing, Singapore, 1987. ]
  7. ↑ Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. I: Approximation of metrics with cone singularities. J. Amer. Math. Soc. 28 (January 2015), no. 1, 183–197.
  8. ↑ Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. II: Limits with cone angle less than 2π . J. Amer. Math. Soc. 28 (January 2015), no. 1, 199–234.
  9. ↑ Chen, Xiuxiong; Donaldson, Simon; Sun, Song Kähler-Einstein metrics on Fano manifolds. III: Limits as cone angle approaches 2π and completion of the main proof. J. Amer. Math. Soc. 28 (January 2015), no. 1, 235–278.
  10. ↑ Gang Tian: K-Stability and Kähler-Einstein Metrics. Communications on Pure and Applied Mathematics, Volume 68, Issue 7, pages 1085–1156, July 2015 http://onlinelibrary.wiley.com/doi/10.1002/cpa.21578/abstract
  11. ↑ Gang Tian: Corrigendum: K-stability and Kähler-Einstein metrics. Communications on Pure and Applied Mathematics, Volume 68, Issue 11, pages 2082–2083, September 2015 http://onlinelibrary.wiley.com/doi/10.1002/cpa.21612/full
  12. ↑ Morgan, John W.; Gang Tian (25 July 2006). "Ricci Flow and the Poincaré Conjecture". arXiv: math.DG/0607607 .
  13. ↑ http://www.ams.org/notices/201304/rnoti-p480.pdf
  14. ↑ http://www.eusja.org/top-secret-of-abel-prize/
  15. ↑ http://annals.math.princeton.edu/board
  16. ↑ http://annaliscienze.sns.it/index.php?page=EditorialBoard
  17. ↑ http://intlpress.com/site/pub/pages/journals/items/jsg/_home/editorial/index.html
  18. ↑ http://www.ams.org/publications/journals/journalsframework/jams/editorial_history_jams
  19. ↑ http://www.msp.warwick.ac.uk/gt/about/journal/editorial.html
  20. ↑ https://www.springer.com/mathematics/geometry/journal/12220?detailsPage=editorialBoard
  21. ↑ https://www.springer.com/birkhauser/mathematics/journal/39?detailsPage=editorialBoard
  22. ↑ http://www.journals.elsevier.com/advances-in-mathematics/editorial-board/
  23. ↑ http://www.oxfordjournals.org/our_journals/imrp/editorial_board.html
  24. ↑ https://www.springer.com/mathematics/journal/10114
  25. ↑ https://www.springer.com/mathematics/journal/13163
  26. ↑ https://www.springer.com/mathematics/journal/40304
  27. ↑ http://www.worldscientific.com/page/ccm/editorial-board
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