# Academy of Mathematics and Systems Science, CAS: explore the sky and the earth with mathematics and systems science, and pursue truth with diligence and earnestness

Academy of Mathematics and Systems Science, CAS: explore the sky and the earth with mathematics... View largeDownload slide View largeDownload slide The Academy of Mathematics and Systems Science (AMSS) of the Chinese Academy of Sciences (CAS) was founded in December 1998 with the integration of four institutes: the Institute of Mathematics (established 1952), the Institute of Applied Mathematics (established 1979), the Institute of Systems Science (established 1979) and the Institute of Computational Mathematics and Scientific/Engineering Computing (established 1995). Nowadays, besides the four institutes, the AMSS also houses six key laboratories and five centers (details shown in the next page). As a national comprehensive research center of mathematics and systems science, the goal of the AMSS is to become a world-renowned center for scientific research, talent training and scholarly exchanges in the field of mathematics and systems science, and an advisory center on national strategic issues. In the past 60 years, the AMSS has made outstanding contributions to the development of Chinese mathematics and systems science as well as to the national economy. During 2013–17, 3970 academic articles have been published by AMSS researchers. AMSS scientists have won more than 500 scientific awards, including one State Supreme Science and Technology Award (Wentsun Wu, 2001), four First Class Prizes of National Natural Science Award (the theory of functions of several complex variables over classical domains by Loo-Keng Hua, 1956; studies on characteristic classes and embedding classes by Wentsun Wu, 1956; studies on the Goldbach conjecture by Jingrun Chen, Yuan Wang and Chengdong Pan, 1982; symplectic geometric algorithm of Hamiltonian systems by Kang Feng et al., 1997), 30 Second Class Prizes of National Natural Science Award (consecutively getting this award from 2006 to 2017), as well as other important national and international prizes and awards. View largeDownload slide View largeDownload slide MAJOR RESEARCH CENTERS Center for Excellence in Mathematical Sciences The Center for Excellence in Mathematical Sciences (CEMS) of CAS was established on 12 December 2014 as a part of the CAS Pioneer Initiative (2015–30). CEMS aims at excellence in fundamental research and education of mathematical sciences by integrating the cutting-edge research in AMSS and the mathematical education in the University of Chinese Academy of Sciences. National Center for Mathematics and Interdisciplinary Sciences Based at AMSS, the National Center for Mathematics and Interdisciplinary Sciences (NCMIS) was founded on 24 November 2010 as a part of the CAS ‘Innovation 2020’ program. NCMIS aims to establish a national research platform for interdisciplinary cooperation between mathematics and other research fields, including information technology, economics and finance, advanced manufacturing, materials and environment, biology and medicine, physics and engineering. Hua Loo-Keng Center for Mathematical Sciences Hua Loo-Keng Center for Mathematical Sciences (HCMS) was set up on 20 March 2017. HCMS was supported by the project of National Natural Science Foundation of China, ‘Geometry, analysis, and computation on manifolds’. HCMS has 45 faculty members, including five Academicians of CAS. The main purpose of HCMS is to carry out high-level research on some of main directions of mathematical science through cooperation and exchange. Morningside Center of Mathematics The Morningside Center of Mathematics, CAS, was founded in 1996 with the endowment from CAS and Hong Kong Morningside Company. The Center is going to promote Chinese mathematics research by promoting mathematical communication with other countries and also Hong Kong, Macao, Taiwan and other areas. Taking the successful model of the Princeton Institute for Advanced Study as the reference, the Center is accomplishing an open, mobile and international research institution. View largeDownload slide View largeDownload slide ‘Never forget why you started, and your mission can be accomplished. In the past decades, the AMSS has gained plenty significant research outputs, and many talented researchers have been working or trained here. We obtained these achievements by standing on the shoulders of innumerable mathematical giants. Looking forward, we are confident to build the AMSS into a world-class research institute in the field of mathematics and systems science, and to make more contributions to scientific research and talents cultivation’ (Nanhua Xi, President of AMSS) RECENT RESEARCH HIGHLIGHTS Solutions of the optimal L2 extension problem and Demailly's strong openness conjecture Completed by: Xiangyu Zhou et al. Publications: Qi’an Guan and Xiangyu Zhou. A solution of an L2 extension problem with an optimal estimate and applications. Ann Math 2015; 181: 1139–208. Qi’an Guan and Xiangyu Zhou. A proof of Demailly's strong openness conjecture. Ann Math 2015; 182: 605–16. Several complex variables is a traditionally strong research area of the AMSS, which was initiated by Loo-Keng Hua and developed by Qikeng Lu. Xiangyu Zhou, a member of AMSS, has recently made important breakthroughs in this field with collaborators: (i) The solution of the optimal L2 extension problem. As applications, they completely solved the Suita conjecture and some open problems posed by Ohsawa et al. This work is commented on as ‘remarkable achievements’ by Ohsawa. (ii) The proof of Demailly's strong openness conjecture. As corollaries, they proved some conjectures and questions posed by Demailly, Ein, Lazarsfeld et al. and gave the new vanishing theorem for the pseudo-effective line bundle on compact Kähler manifolds. This work is commented on in the Math. Review as ‘among the greatest achievements’ in the intersection of ‘complex analysis and algebraic geometry in recent years’. Progress on the congruent number problem Completed by: Ye Tian Publications: Ye Tian. Congruent numbers with many prime factors. Proc Natl Acad Sci USA 2012; 109: 21256–8. Ye Tian. Congruent numbers and Heegner points. Camb J Math 2014; 2: 117–61. A positive integer is called ‘congruent’ if it is the area of a right-angled triangle, all of whose sides have rational length. The congruent number problem, which is the oldest unsolved major problem in number theory, is simply the question of deciding which positive integers are, or are not, congruent numbers. By discovering an induction method, together with relations between L-values and Heegner points, AMSS researcher Ye Tian established the following results on the congruent number problem: Theorem 1.For any given positive integer k, there are infinitely many square-free congruent numbers with exactly k prime divisors. Theorem 2.Let N ≡ 5, 6, 7 mod 8 be a square-free positive integer all except one of whose odd prime factors are ≡ 1 mod 8. Assume that the field $$\mathcal{Q}({\sqrt {- N}})$$ has no ideal class of exactly order 4. Then N is a congruent number, furthermore, the BSD conjecture (rank part) holds for the elliptic curve Ny2 = x3 − x. Proof of the fundamental conjectures of local theta correspondences Completed by: Binyong Sun et al. Publications: Wee Teck Gan and Binyong Sun. The Howe duality conjecture: quaternionic case. In: Jim Cogdell, Ju-Lee Kim and Chen-Bo Zhu (eds). Representation Theory, Number Theory, and Invariant Theory. Basel: Birkhäuser, 2017, 175–92. Binyong Sun and Chen-Bo Zhu. Conservation relations for local theta correspondence. J Amer Math Soc 2015; 28: 939–83. The classical invariant theory is one of the greatest achievements in the early study of classical groups. In the 1970s, Roger Howe initiated the theory of local theta correspondences, which is a profound extension of the classical invariant theory to the setting of infinite dimensional representations. There were the two most fundamental conjectures in the history of local theta correspondences, namely the Howe duality conjecture and the conservation relation conjecture of Kudla-Rallis. Jointly with collaborators, AMSS researcher Binyong Sun proved the conservation relation conjecture in full generality, and completed proof of the Howe duality conjecture by treating the last remaining case of quaternionic groups. View largeDownload slide Celebration of Professor Roger Howe's 70th birthday View largeDownload slide Celebration of Professor Roger Howe's 70th birthday Deep insights into electronic structure models and calculations Completed by: Aihui Zhou et al. Publications: Huajie Chen, Xiaoying Dai and Xingao Gong et al. Adaptive finite element approximations for Kohn-Sham models. Multiscale Model Sim 2014; 12: 1828–69. Xiaoying Dai, Xingao Gong and Aihui Zhou et al. A parallel orbital-updating approach for electronic structure calculations. arXiv 2014; arXiv:1405.0260. It is significant to understand the existing models and their approximations as well as to design a novel efficient and supercomputer-friendly algorithm for electronic structure calculations. In this field, AMSS researcher Aihui Zhou and his group made several important proofs and justifications. They proposed a mathematically rigorous proof of the Hohenberg-Kohn theorem, which is the foundation of density functional theory, and provided mathematical justification for the convergence of the existing widely used methods, as well as innovative methods. In particular, they obtained the uniform convergence and optimal complexity of the adaptive finite element approximations to linear eigenvalue problems and to Kohn-Sham equations. Besides these basic works, Zhou's group has also developed the package RealSPACES (Real Space Parallel Adaptive Calculation of Electronic Structure), which has been successfully applied to quantum chemistry and nanometer materials computations. View largeDownload slide An example of full potential calculation by RealSPACES: the calculated convergence curve for the ground state energy (a.u.) of C35 H44 O9 N12. Insert: the configuration of C35 H44 O9 N12. View largeDownload slide An example of full potential calculation by RealSPACES: the calculated convergence curve for the ground state energy (a.u.) of C35 H44 O9 N12. Insert: the configuration of C35 H44 O9 N12. Dimension-free matrix theory Completed by: Daizhan Cheng et al. Publications: Daizhan Cheng, Hongsheng Qi and Zhiqiang Li. Analysis and Control of Boolean Networks—a Semi-Tensor Product Approach. London: Springer-Verlag, 2011. Daizhan Cheng. On equivalence of matrices. arXiv 2016; arXiv: 1605.09523v3. The matrix theory is one of the most powerful mathematical tools widely applied in scientific research and engineering designs. However, the classical matrix theory faces a severe barrier of the dimension restriction. AMSS researcher Daizhan Cheng and colleagues proposed the dimension-free matrix theory to overcome this obstruction. This theory is based on the semi-tensor product (STP or M-product), which generalized and extended the conventional matrix product to two matrices of arbitrary dimensions. This theory also contains the V-product, which makes an arbitrary matrix a linear mapping to a vector space of arbitrary dimension, as well as M-addition and V-addition. It is a powerful matrix theory for modeling and analysing dimension-varying systems. New progress on the Hilbert's sixth problem Completed by: Feimin Huang et al. Publications: Feimin Huang, Yi Wang and Yong Wang et al. The limit of the Boltzmann equation to the Euler equations for Riemann problems. SIAM J Math Anal 2013; 45: 1741–811. Feimin Huang, Yi Wang and Tong Yang. Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to a Riemann problem. Arch Ration Mech Anal 2012; 203: 379–413. In 1900, David Hilbert delivered 23 mathematical problems, which have been influencing mathematical research in the world ever since. The justification of the compressible Euler limit of the Boltzmann equation is a part of Hilbert's sixth problem, ‘Mathematical treatment of the axioms of physics’. Cooperating with professor Tong Yang, from City University of Hong Kong, the AMSS group of Feimin Huang justified the limit from the Boltzmann equation to the compressible Euler equation in the setting of Riemann solutions by introducing two kinds of hyperbolic waves with different solution backgrounds to capture the extra masses carried by the hyperbolic approximation of the rarefaction wave and the diffusion approximation of contact discontinuity. Global value chain and trade in value-added measurement Completed by: Cuihong Yang et al. Publications: Cuihong Yang, Dietzenbacher E and Jiansuo Pei et al. Processing trade biases the measurement of vertical specialization in China. Econ Syst Res 2015; 27: 60–76. Xuemei Jiang, Quanrun Chen and Dabo Guan et al. Revisiting the global net carbon dioxide emission transfers by international trade. J Ind Ecol 2016; 20: 506–14. The expansion of global trade is characterized by increasing international fragmentation of production and traditional trade statistics on gross exports no longer gives accurate information about the actual value added obtained by the exporting countries/regions. The team led by Xikang Chen and Cuihong Yang proposed the idea to use value added by export/import instead of gross trade value to measure the trade volume and bilateral trade balances, and developed an input-occupancy-output model capturing processing trade for China (abbreviated as the DPN model) in order to illustrate the heterogeneity of processing trade. Cuihong Yang's team further developed the World Input-output Model capturing China's processing trade. They have submitted 14 policy reports to the Ministry of Commerce of China on this issue and gained international recognition in the past several years. View largeDownload slide China--US trade balance in gross volume, DVA content. View largeDownload slide China--US trade balance in gross volume, DVA content. Sharp stability for a multiscale method in solids Completed by: Pingbing Ming et al. Publications: Jianfeng Lu and Pingbing Ming. Convergence of a force-based hybrid method in three dimensions. Comm Pure Appl Math 2013; 66: 83–108. Jianfeng Lu and Pingbing Ming. Convergence of a force-based hybrid method with planar sharp interface. SIAM J Numer Anal 2014; 52: 2005–26. The quasicontinuum method (QC) is a successful multiscale method in solid mechanics coupling the atomic model and the continuum model, but its stability in high dimension largely remains an open problem. The AMSS group of Pingbing Ming proposed a new multiscale hybrid paradigm in the framework of QC, which couples the force fields at micro and macro scales by a smooth blending function, and proved that this method has sharp discrete H2 stability as well as the optimal convergence rate. And, for the force coupling method with rough blending function, the researchers have also proved its sharp discrete H2 stability in three dimensions. As a byproduct of the stability results, they proved that, for the Bravais lattice, the Lindmann stability criterion implies the Born stability criterion, which seems unjustified folklore in solid-state physics. View largeDownload slide A cartoon for QC with sharp interface. View largeDownload slide A cartoon for QC with sharp interface. INTERNATIONAL COOPERATION AMSS offers plenty of opportunities for researchers to communicate with scientists throughout the world. In 2016, around 80 person-times of AMSS researchers served on leader positions in important international academic conferences and organizations. The newly founded research center CEMS has 13 distinguished international visiting professors who are invited to visit CEMS for one to three months each year. These visiting professors have made significant contributions to the training and recruitment of young scholars in CEMS and AMSS. View largeDownload slide The 8th International Congress on Industrial and Applied Mathematics was held in Beijing in August 2015, and it was the first time to be held in Asia. Over 3400 scientists from over 70 countries and regions attended the Congress. View largeDownload slide The 8th International Congress on Industrial and Applied Mathematics was held in Beijing in August 2015, and it was the first time to be held in Asia. Over 3400 scientists from over 70 countries and regions attended the Congress. ACADEMIC LEADERS Prof. Xiangyu Zhou, Academician of CAS, is a world leading expert in several complex variables and complex geometry. Zhou and his coauthor solved the extended future tube conjecture, the optimal L2 extension problem and Suita conjecture, and proved Demailly's strong openness conjecture. Zhou has received many significant awards including Tan Kah Kee Science Award (2016) and the State Natural Science Award of China (2004). Prof. Yuefei Wang, deputy director of CEMS, vice director of NCMIS and a member of HCMS, has devoted in the research of complex dynamical systems, non-Archimedean dynamical systems, conformal geometry and SLE. His group has focused on and made contributions to the study of dynamics of transcendental holomorphic maps in the complex field, p-adic fields and Berkovich spaces, minimal decompositions of p-adic rational maps, etc. Prof. Fuzhou Gong's research area is stochastic analysis and its application. His major contributions are the proof of Poincare inequalities for the weighted first order Sobolev spaces on loop spaces and the proof of Log-Sobove inequalities with the neat and explicit expressed potentials on loop spaces. He also pays attention to the applications of stochastic analysis on many fields. He has received several influential mathematical awards and national funds. Prof. Lei Guo's major research area is systems and control science. He has made fundamental contributions to the theory of adaptive control, adaptive filtering, feedback capability, flocks synchronization and PID control, etc. Prof. Guo was elected Academician of CAS, Fellow of IEEE, Fellow of TWAS, Fellow of IFAC and Foreign Member of Royal Swedish Academy of Engineering Science. Prof. Yaxiang Yuan works on numerical methods for non-linear optimization. He has made outstanding contributions to trust region algorithms, quasi-Newton methods and non-linear conjugate gradient methods and subspace methods. Yuan was elected as Academician of CAS, Fellow of SIAM and Fellow of TWAS. He has won numerous awards and is currently the president of the Chinese Mathematical Society. Professor Shouyang Wang, Institute of Systems Science, AMSS Prof. Shouyang Wang, director of Center for Forecasting Science, is a leading specialist in systems engineering and economic forecasting. He has made outstanding achievements in the areas of decision analysis, financial risk management, economic analysis and forecasting, etc. Besides academic research work, he has also submitted more than 180 policy research reports to the State Council. Many of his policy suggestions have been adopted by the government. Professor Xiao-Shan Gao, Institute of Systems Science, AMSS Prof. Xiao-Shan Gao's research area is computer mathematics. He established the theories of differential sparse resultant and differential Chow form, introduced complete and highly efficient algorithms of geometric constraint solving, solved basic problems in parallel robotics and computer vision, and established the Area Method for automated generation of short and readable proofs for geometric theorems. Gao is vice president of the Chinese Mathematics Society and vice president of the China Society of Industrial and Applied Mathematics. INTERVIEW View largeDownload slide Zhiming Chen, Director of Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS View largeDownload slide Zhiming Chen, Director of Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS NSR: What is the importance of mathematics and systems science research? How about China's support on this field? Chen: Mathematics plays a fundamental and decisive role in modern science and high-end technology. I think our government has recognized the importance of mathematics and the support to mathematics has been increased substantially in the last years. There is still much room of improvement in the funding structure and policy for mathematics, especially in the areas where mathematics interacts with the other scientific disciplines. View largeDownload slide Jinhu Lü, Professor at Institute of Systems Science, AMSS View largeDownload slide Jinhu Lü, Professor at Institute of Systems Science, AMSS Lü: Mathematics and systems science are critical to the development of a country and profoundly affect the comprehensive national strength. Developed countries tend to maintain the leading position of mathematics and systems science as their major strategic need. China pays high attention to the development of mathematics and systems science. National Natural Science Foundation of China, CAS, Ministry of Science and Technology and other departments have set up the Special Funds or Major Project of Mathematical and Systems Science to support the original research in these fields and yielded fruitful results. View largeDownload slide Xiangdong Li, Professor at Institute of Applied Mathematics, AMSS View largeDownload slide Xiangdong Li, Professor at Institute of Applied Mathematics, AMSS Li: AMSS is a national research institute in mathematics in China. It plays an important role in the fundamental research of mathematics as well as its applications in China. The Chinese government gives us very strong support, not only in finance, but also provides us with very good environment in research. View largeDownload slide Xin Wan, Professor at Institute of Mathematics, AMSS View largeDownload slide Xin Wan, Professor at Institute of Mathematics, AMSS Wan: I think mathematical science is the leading subject for the study of both natural science and social science. On one hand, it studies problems motivated from physics, biology, economics, etc., and in turn provides powerful tools for these study. On the other hand, mathematics itself has some parts with independent interest and principles, which, although may not see immediate applications at the moment, might be of crucial importance in the future. In my opinion, the development of mathematics normally goes ahead of the human knowledge system, and usually provides directions for other disciplines. Nowadays, there have been aggressive programs initiated by the Chinese government supporting developments of science including mathematics, like the ‘Recruitment Program for Global Experts’ (and some others). These not only provide plenty amount of grants, nice environments for scientists, but also come with life benefits so as to help us better focus on work. Observably there have been more and more good mathematicians attracted by these programs to come back work in China. NSR: Would you please describe the research environment of AMSS? Chen: AMSS encourages excellence and innovation, emphasizes originality in research that has long-lasting impact. These are the most attractive characteristics of the research environment in AMSS. Lü: AMSS provides a relaxed, free and open research environment to encourage us to explore freely and devote ourselves into research, including: (i) for different disciplines and different directions, instead of the past ‘one size fits all’ evaluation method, AMSS establishes the diversified international evaluation system; (ii) for young researchers, AMSS gives them much more freedom, so that they can concentrate on doing much more important scientific problems; (iii) AMSS gives us great support in manpower, material resources and other aspects, so that researchers can concentrate on devoting themselves into research for long-term in their areas. Li: AMSS provides us very good environment to do mathematical research work. We have the freedom to do our research in mathematics based on our own interest. AMSS also provides each researcher a basic grant to perform research activity. Wan: Besides the national level support, AMSS also provides additional amount of grants as complement. Inviting people has been pretty easy for me. If I want to organize a conference I can also apply for some special funding for this. AMSS is the leading institution in China on number theory, which is my field of study. I can discuss my mathematics with a lot of colleagues. I have benefited a lot from such communications for my own research. Mostly during summer, the department invites some famous mathematicians to visit for longer period. This helps us to keep connection with most recent developments. Administrative burden like things related to reimbursement has not influenced me much (such burden is complained by a few friends working in other universities). NSR: What is the most predominant characteristic of AMSS? Chen: AMSS is the melting pot of mathematics. People working in every aspects of the theory and application of mathematics find their places in AMSS. Lü: (i) AMSS has a glorious history and impressive achievements. The free academic exploration, painstaking research scientific spirits have been formed in its long history. (ii) AMSS is a high-level research platform in which mathematics and other disciplines work together. AMSS has a group of masters in mathematics and related disciplines. It is promising to make a major breakthrough on a number of key scientific questions. Li: We have very good researchers in different fields of mathematics. So we can discuss various topics in mathematics and we may do some collaboration with our colleagues. Moreover, we have many excellent visitors from China and abroad to exchange ideas and to do collaboration in mathematical research. We are also strongly supported to go abroad to attend conferences and to visit some leading mathematical institutes and universities in the world. This provides us a good opportunity to do research collaboration with many good mathematicians in the world. Wan: I think the most important characteristic of AMSS that attracted me is the flexibility, compared to universities. The system is to some extent similar to the CNRS in France or IAS in the US. I have more freedom arranging my time on research, on teaching time and teaching subjects, and on visits. Personally I like this freedom and I think it indeed helps bringing my ability of research into full play. NSR: What are your future expectations for your research and the development of AMSS? Chen: I expect that AMSS will be one of the very top research institutions in mathematics worldwide in the coming years. I hope I can contribute to the success of AMSS in a substantial way. Lü: My research fields include complex networks, non-linear circuits and systems, and big data. Combining theory and application, I will try to obtain a series of original results by overcoming a number of bottleneck problems. I hope that AMSS can achieve a series of essential breakthroughs in a number of major key scientific issues, which will lead the rapid development of related fields. As a result, AMSS will do much more contributions to our country. Li: My personal expectation is to continue and to do some good research work in the interaction of analysis, geometry and probability theory, in connection with problems from statistical physics and other areas. I would like also to train some good students and help them to become good researchers in mathematics. About AMSS in future, I hope it can become more and more attractive and creative center of mathematics in the world. Wan: My research is primarily on Birch and Swinnerton-Dyer (BSD) conjecture and surrounding subjects. This relates special values of L-functions and arithmetic objects, which is one of the most important and deepest relations in mathematics. In near future I plan to finish up my previous work on such problems for modular forms, and try to generalize. Hopefully there will come up with new discoveries on this subject. For AMSS, it is a little hard to predict, as policies from government has been changing very frequently, which serve like a double-edged sword. But at the moment I certainly believe we can attract more and more talented young people and will gradually catch up with the world-class institutions in North American and European countries. Editors: Weijie Zhao (NSR) and Hao Tang (AMSS) Designer: Xiaoling Yu (NSR) Photo: Lin Wang (AMSS) © The Author(s) 2018. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png National Science Review Oxford University Press

# Academy of Mathematics and Systems Science, CAS: explore the sky and the earth with mathematics and systems science, and pursue truth with diligence and earnestness

, Volume Advance Article (3) – May 15, 2018
10 pages

Publisher
Oxford University Press
© The Author(s) 2018. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd.
ISSN
2095-5138
eISSN
2053-714X
D.O.I.
10.1093/nsr/nwy031
Publisher site
See Article on Publisher Site

### Abstract

View largeDownload slide View largeDownload slide The Academy of Mathematics and Systems Science (AMSS) of the Chinese Academy of Sciences (CAS) was founded in December 1998 with the integration of four institutes: the Institute of Mathematics (established 1952), the Institute of Applied Mathematics (established 1979), the Institute of Systems Science (established 1979) and the Institute of Computational Mathematics and Scientific/Engineering Computing (established 1995). Nowadays, besides the four institutes, the AMSS also houses six key laboratories and five centers (details shown in the next page). As a national comprehensive research center of mathematics and systems science, the goal of the AMSS is to become a world-renowned center for scientific research, talent training and scholarly exchanges in the field of mathematics and systems science, and an advisory center on national strategic issues. In the past 60 years, the AMSS has made outstanding contributions to the development of Chinese mathematics and systems science as well as to the national economy. During 2013–17, 3970 academic articles have been published by AMSS researchers. AMSS scientists have won more than 500 scientific awards, including one State Supreme Science and Technology Award (Wentsun Wu, 2001), four First Class Prizes of National Natural Science Award (the theory of functions of several complex variables over classical domains by Loo-Keng Hua, 1956; studies on characteristic classes and embedding classes by Wentsun Wu, 1956; studies on the Goldbach conjecture by Jingrun Chen, Yuan Wang and Chengdong Pan, 1982; symplectic geometric algorithm of Hamiltonian systems by Kang Feng et al., 1997), 30 Second Class Prizes of National Natural Science Award (consecutively getting this award from 2006 to 2017), as well as other important national and international prizes and awards. View largeDownload slide View largeDownload slide MAJOR RESEARCH CENTERS Center for Excellence in Mathematical Sciences The Center for Excellence in Mathematical Sciences (CEMS) of CAS was established on 12 December 2014 as a part of the CAS Pioneer Initiative (2015–30). CEMS aims at excellence in fundamental research and education of mathematical sciences by integrating the cutting-edge research in AMSS and the mathematical education in the University of Chinese Academy of Sciences. National Center for Mathematics and Interdisciplinary Sciences Based at AMSS, the National Center for Mathematics and Interdisciplinary Sciences (NCMIS) was founded on 24 November 2010 as a part of the CAS ‘Innovation 2020’ program. NCMIS aims to establish a national research platform for interdisciplinary cooperation between mathematics and other research fields, including information technology, economics and finance, advanced manufacturing, materials and environment, biology and medicine, physics and engineering. Hua Loo-Keng Center for Mathematical Sciences Hua Loo-Keng Center for Mathematical Sciences (HCMS) was set up on 20 March 2017. HCMS was supported by the project of National Natural Science Foundation of China, ‘Geometry, analysis, and computation on manifolds’. HCMS has 45 faculty members, including five Academicians of CAS. The main purpose of HCMS is to carry out high-level research on some of main directions of mathematical science through cooperation and exchange. Morningside Center of Mathematics The Morningside Center of Mathematics, CAS, was founded in 1996 with the endowment from CAS and Hong Kong Morningside Company. The Center is going to promote Chinese mathematics research by promoting mathematical communication with other countries and also Hong Kong, Macao, Taiwan and other areas. Taking the successful model of the Princeton Institute for Advanced Study as the reference, the Center is accomplishing an open, mobile and international research institution. View largeDownload slide View largeDownload slide ‘Never forget why you started, and your mission can be accomplished. In the past decades, the AMSS has gained plenty significant research outputs, and many talented researchers have been working or trained here. We obtained these achievements by standing on the shoulders of innumerable mathematical giants. Looking forward, we are confident to build the AMSS into a world-class research institute in the field of mathematics and systems science, and to make more contributions to scientific research and talents cultivation’ (Nanhua Xi, President of AMSS) RECENT RESEARCH HIGHLIGHTS Solutions of the optimal L2 extension problem and Demailly's strong openness conjecture Completed by: Xiangyu Zhou et al. Publications: Qi’an Guan and Xiangyu Zhou. A solution of an L2 extension problem with an optimal estimate and applications. Ann Math 2015; 181: 1139–208. Qi’an Guan and Xiangyu Zhou. A proof of Demailly's strong openness conjecture. Ann Math 2015; 182: 605–16. Several complex variables is a traditionally strong research area of the AMSS, which was initiated by Loo-Keng Hua and developed by Qikeng Lu. Xiangyu Zhou, a member of AMSS, has recently made important breakthroughs in this field with collaborators: (i) The solution of the optimal L2 extension problem. As applications, they completely solved the Suita conjecture and some open problems posed by Ohsawa et al. This work is commented on as ‘remarkable achievements’ by Ohsawa. (ii) The proof of Demailly's strong openness conjecture. As corollaries, they proved some conjectures and questions posed by Demailly, Ein, Lazarsfeld et al. and gave the new vanishing theorem for the pseudo-effective line bundle on compact Kähler manifolds. This work is commented on in the Math. Review as ‘among the greatest achievements’ in the intersection of ‘complex analysis and algebraic geometry in recent years’. Progress on the congruent number problem Completed by: Ye Tian Publications: Ye Tian. Congruent numbers with many prime factors. Proc Natl Acad Sci USA 2012; 109: 21256–8. Ye Tian. Congruent numbers and Heegner points. Camb J Math 2014; 2: 117–61. A positive integer is called ‘congruent’ if it is the area of a right-angled triangle, all of whose sides have rational length. The congruent number problem, which is the oldest unsolved major problem in number theory, is simply the question of deciding which positive integers are, or are not, congruent numbers. By discovering an induction method, together with relations between L-values and Heegner points, AMSS researcher Ye Tian established the following results on the congruent number problem: Theorem 1.For any given positive integer k, there are infinitely many square-free congruent numbers with exactly k prime divisors. Theorem 2.Let N ≡ 5, 6, 7 mod 8 be a square-free positive integer all except one of whose odd prime factors are ≡ 1 mod 8. Assume that the field $$\mathcal{Q}({\sqrt {- N}})$$ has no ideal class of exactly order 4. Then N is a congruent number, furthermore, the BSD conjecture (rank part) holds for the elliptic curve Ny2 = x3 − x. Proof of the fundamental conjectures of local theta correspondences Completed by: Binyong Sun et al. Publications: Wee Teck Gan and Binyong Sun. The Howe duality conjecture: quaternionic case. In: Jim Cogdell, Ju-Lee Kim and Chen-Bo Zhu (eds). Representation Theory, Number Theory, and Invariant Theory. Basel: Birkhäuser, 2017, 175–92. Binyong Sun and Chen-Bo Zhu. Conservation relations for local theta correspondence. J Amer Math Soc 2015; 28: 939–83. The classical invariant theory is one of the greatest achievements in the early study of classical groups. In the 1970s, Roger Howe initiated the theory of local theta correspondences, which is a profound extension of the classical invariant theory to the setting of infinite dimensional representations. There were the two most fundamental conjectures in the history of local theta correspondences, namely the Howe duality conjecture and the conservation relation conjecture of Kudla-Rallis. Jointly with collaborators, AMSS researcher Binyong Sun proved the conservation relation conjecture in full generality, and completed proof of the Howe duality conjecture by treating the last remaining case of quaternionic groups. View largeDownload slide Celebration of Professor Roger Howe's 70th birthday View largeDownload slide Celebration of Professor Roger Howe's 70th birthday Deep insights into electronic structure models and calculations Completed by: Aihui Zhou et al. Publications: Huajie Chen, Xiaoying Dai and Xingao Gong et al. Adaptive finite element approximations for Kohn-Sham models. Multiscale Model Sim 2014; 12: 1828–69. Xiaoying Dai, Xingao Gong and Aihui Zhou et al. A parallel orbital-updating approach for electronic structure calculations. arXiv 2014; arXiv:1405.0260. It is significant to understand the existing models and their approximations as well as to design a novel efficient and supercomputer-friendly algorithm for electronic structure calculations. In this field, AMSS researcher Aihui Zhou and his group made several important proofs and justifications. They proposed a mathematically rigorous proof of the Hohenberg-Kohn theorem, which is the foundation of density functional theory, and provided mathematical justification for the convergence of the existing widely used methods, as well as innovative methods. In particular, they obtained the uniform convergence and optimal complexity of the adaptive finite element approximations to linear eigenvalue problems and to Kohn-Sham equations. Besides these basic works, Zhou's group has also developed the package RealSPACES (Real Space Parallel Adaptive Calculation of Electronic Structure), which has been successfully applied to quantum chemistry and nanometer materials computations. View largeDownload slide An example of full potential calculation by RealSPACES: the calculated convergence curve for the ground state energy (a.u.) of C35 H44 O9 N12. Insert: the configuration of C35 H44 O9 N12. View largeDownload slide An example of full potential calculation by RealSPACES: the calculated convergence curve for the ground state energy (a.u.) of C35 H44 O9 N12. Insert: the configuration of C35 H44 O9 N12. Dimension-free matrix theory Completed by: Daizhan Cheng et al. Publications: Daizhan Cheng, Hongsheng Qi and Zhiqiang Li. Analysis and Control of Boolean Networks—a Semi-Tensor Product Approach. London: Springer-Verlag, 2011. Daizhan Cheng. On equivalence of matrices. arXiv 2016; arXiv: 1605.09523v3. The matrix theory is one of the most powerful mathematical tools widely applied in scientific research and engineering designs. However, the classical matrix theory faces a severe barrier of the dimension restriction. AMSS researcher Daizhan Cheng and colleagues proposed the dimension-free matrix theory to overcome this obstruction. This theory is based on the semi-tensor product (STP or M-product), which generalized and extended the conventional matrix product to two matrices of arbitrary dimensions. This theory also contains the V-product, which makes an arbitrary matrix a linear mapping to a vector space of arbitrary dimension, as well as M-addition and V-addition. It is a powerful matrix theory for modeling and analysing dimension-varying systems. New progress on the Hilbert's sixth problem Completed by: Feimin Huang et al. Publications: Feimin Huang, Yi Wang and Yong Wang et al. The limit of the Boltzmann equation to the Euler equations for Riemann problems. SIAM J Math Anal 2013; 45: 1741–811. Feimin Huang, Yi Wang and Tong Yang. Vanishing viscosity limit of the compressible Navier-Stokes equations for solutions to a Riemann problem. Arch Ration Mech Anal 2012; 203: 379–413. In 1900, David Hilbert delivered 23 mathematical problems, which have been influencing mathematical research in the world ever since. The justification of the compressible Euler limit of the Boltzmann equation is a part of Hilbert's sixth problem, ‘Mathematical treatment of the axioms of physics’. Cooperating with professor Tong Yang, from City University of Hong Kong, the AMSS group of Feimin Huang justified the limit from the Boltzmann equation to the compressible Euler equation in the setting of Riemann solutions by introducing two kinds of hyperbolic waves with different solution backgrounds to capture the extra masses carried by the hyperbolic approximation of the rarefaction wave and the diffusion approximation of contact discontinuity. Global value chain and trade in value-added measurement Completed by: Cuihong Yang et al. Publications: Cuihong Yang, Dietzenbacher E and Jiansuo Pei et al. Processing trade biases the measurement of vertical specialization in China. Econ Syst Res 2015; 27: 60–76. Xuemei Jiang, Quanrun Chen and Dabo Guan et al. Revisiting the global net carbon dioxide emission transfers by international trade. J Ind Ecol 2016; 20: 506–14. The expansion of global trade is characterized by increasing international fragmentation of production and traditional trade statistics on gross exports no longer gives accurate information about the actual value added obtained by the exporting countries/regions. The team led by Xikang Chen and Cuihong Yang proposed the idea to use value added by export/import instead of gross trade value to measure the trade volume and bilateral trade balances, and developed an input-occupancy-output model capturing processing trade for China (abbreviated as the DPN model) in order to illustrate the heterogeneity of processing trade. Cuihong Yang's team further developed the World Input-output Model capturing China's processing trade. They have submitted 14 policy reports to the Ministry of Commerce of China on this issue and gained international recognition in the past several years. View largeDownload slide China--US trade balance in gross volume, DVA content. View largeDownload slide China--US trade balance in gross volume, DVA content. Sharp stability for a multiscale method in solids Completed by: Pingbing Ming et al. Publications: Jianfeng Lu and Pingbing Ming. Convergence of a force-based hybrid method in three dimensions. Comm Pure Appl Math 2013; 66: 83–108. Jianfeng Lu and Pingbing Ming. Convergence of a force-based hybrid method with planar sharp interface. SIAM J Numer Anal 2014; 52: 2005–26. The quasicontinuum method (QC) is a successful multiscale method in solid mechanics coupling the atomic model and the continuum model, but its stability in high dimension largely remains an open problem. The AMSS group of Pingbing Ming proposed a new multiscale hybrid paradigm in the framework of QC, which couples the force fields at micro and macro scales by a smooth blending function, and proved that this method has sharp discrete H2 stability as well as the optimal convergence rate. And, for the force coupling method with rough blending function, the researchers have also proved its sharp discrete H2 stability in three dimensions. As a byproduct of the stability results, they proved that, for the Bravais lattice, the Lindmann stability criterion implies the Born stability criterion, which seems unjustified folklore in solid-state physics. View largeDownload slide A cartoon for QC with sharp interface. View largeDownload slide A cartoon for QC with sharp interface. INTERNATIONAL COOPERATION AMSS offers plenty of opportunities for researchers to communicate with scientists throughout the world. In 2016, around 80 person-times of AMSS researchers served on leader positions in important international academic conferences and organizations. The newly founded research center CEMS has 13 distinguished international visiting professors who are invited to visit CEMS for one to three months each year. These visiting professors have made significant contributions to the training and recruitment of young scholars in CEMS and AMSS. View largeDownload slide The 8th International Congress on Industrial and Applied Mathematics was held in Beijing in August 2015, and it was the first time to be held in Asia. Over 3400 scientists from over 70 countries and regions attended the Congress. View largeDownload slide The 8th International Congress on Industrial and Applied Mathematics was held in Beijing in August 2015, and it was the first time to be held in Asia. Over 3400 scientists from over 70 countries and regions attended the Congress. ACADEMIC LEADERS Prof. Xiangyu Zhou, Academician of CAS, is a world leading expert in several complex variables and complex geometry. Zhou and his coauthor solved the extended future tube conjecture, the optimal L2 extension problem and Suita conjecture, and proved Demailly's strong openness conjecture. Zhou has received many significant awards including Tan Kah Kee Science Award (2016) and the State Natural Science Award of China (2004). Prof. Yuefei Wang, deputy director of CEMS, vice director of NCMIS and a member of HCMS, has devoted in the research of complex dynamical systems, non-Archimedean dynamical systems, conformal geometry and SLE. His group has focused on and made contributions to the study of dynamics of transcendental holomorphic maps in the complex field, p-adic fields and Berkovich spaces, minimal decompositions of p-adic rational maps, etc. Prof. Fuzhou Gong's research area is stochastic analysis and its application. His major contributions are the proof of Poincare inequalities for the weighted first order Sobolev spaces on loop spaces and the proof of Log-Sobove inequalities with the neat and explicit expressed potentials on loop spaces. He also pays attention to the applications of stochastic analysis on many fields. He has received several influential mathematical awards and national funds. Prof. Lei Guo's major research area is systems and control science. He has made fundamental contributions to the theory of adaptive control, adaptive filtering, feedback capability, flocks synchronization and PID control, etc. Prof. Guo was elected Academician of CAS, Fellow of IEEE, Fellow of TWAS, Fellow of IFAC and Foreign Member of Royal Swedish Academy of Engineering Science. Prof. Yaxiang Yuan works on numerical methods for non-linear optimization. He has made outstanding contributions to trust region algorithms, quasi-Newton methods and non-linear conjugate gradient methods and subspace methods. Yuan was elected as Academician of CAS, Fellow of SIAM and Fellow of TWAS. He has won numerous awards and is currently the president of the Chinese Mathematical Society. Professor Shouyang Wang, Institute of Systems Science, AMSS Prof. Shouyang Wang, director of Center for Forecasting Science, is a leading specialist in systems engineering and economic forecasting. He has made outstanding achievements in the areas of decision analysis, financial risk management, economic analysis and forecasting, etc. Besides academic research work, he has also submitted more than 180 policy research reports to the State Council. Many of his policy suggestions have been adopted by the government. Professor Xiao-Shan Gao, Institute of Systems Science, AMSS Prof. Xiao-Shan Gao's research area is computer mathematics. He established the theories of differential sparse resultant and differential Chow form, introduced complete and highly efficient algorithms of geometric constraint solving, solved basic problems in parallel robotics and computer vision, and established the Area Method for automated generation of short and readable proofs for geometric theorems. Gao is vice president of the Chinese Mathematics Society and vice president of the China Society of Industrial and Applied Mathematics. INTERVIEW View largeDownload slide Zhiming Chen, Director of Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS View largeDownload slide Zhiming Chen, Director of Institute of Computational Mathematics and Scientific/Engineering Computing, AMSS NSR: What is the importance of mathematics and systems science research? How about China's support on this field? Chen: Mathematics plays a fundamental and decisive role in modern science and high-end technology. I think our government has recognized the importance of mathematics and the support to mathematics has been increased substantially in the last years. There is still much room of improvement in the funding structure and policy for mathematics, especially in the areas where mathematics interacts with the other scientific disciplines. View largeDownload slide Jinhu Lü, Professor at Institute of Systems Science, AMSS View largeDownload slide Jinhu Lü, Professor at Institute of Systems Science, AMSS Lü: Mathematics and systems science are critical to the development of a country and profoundly affect the comprehensive national strength. Developed countries tend to maintain the leading position of mathematics and systems science as their major strategic need. China pays high attention to the development of mathematics and systems science. National Natural Science Foundation of China, CAS, Ministry of Science and Technology and other departments have set up the Special Funds or Major Project of Mathematical and Systems Science to support the original research in these fields and yielded fruitful results. View largeDownload slide Xiangdong Li, Professor at Institute of Applied Mathematics, AMSS View largeDownload slide Xiangdong Li, Professor at Institute of Applied Mathematics, AMSS Li: AMSS is a national research institute in mathematics in China. It plays an important role in the fundamental research of mathematics as well as its applications in China. The Chinese government gives us very strong support, not only in finance, but also provides us with very good environment in research. View largeDownload slide Xin Wan, Professor at Institute of Mathematics, AMSS View largeDownload slide Xin Wan, Professor at Institute of Mathematics, AMSS Wan: I think mathematical science is the leading subject for the study of both natural science and social science. On one hand, it studies problems motivated from physics, biology, economics, etc., and in turn provides powerful tools for these study. On the other hand, mathematics itself has some parts with independent interest and principles, which, although may not see immediate applications at the moment, might be of crucial importance in the future. In my opinion, the development of mathematics normally goes ahead of the human knowledge system, and usually provides directions for other disciplines. Nowadays, there have been aggressive programs initiated by the Chinese government supporting developments of science including mathematics, like the ‘Recruitment Program for Global Experts’ (and some others). These not only provide plenty amount of grants, nice environments for scientists, but also come with life benefits so as to help us better focus on work. Observably there have been more and more good mathematicians attracted by these programs to come back work in China. NSR: Would you please describe the research environment of AMSS? Chen: AMSS encourages excellence and innovation, emphasizes originality in research that has long-lasting impact. These are the most attractive characteristics of the research environment in AMSS. Lü: AMSS provides a relaxed, free and open research environment to encourage us to explore freely and devote ourselves into research, including: (i) for different disciplines and different directions, instead of the past ‘one size fits all’ evaluation method, AMSS establishes the diversified international evaluation system; (ii) for young researchers, AMSS gives them much more freedom, so that they can concentrate on doing much more important scientific problems; (iii) AMSS gives us great support in manpower, material resources and other aspects, so that researchers can concentrate on devoting themselves into research for long-term in their areas. Li: AMSS provides us very good environment to do mathematical research work. We have the freedom to do our research in mathematics based on our own interest. AMSS also provides each researcher a basic grant to perform research activity. Wan: Besides the national level support, AMSS also provides additional amount of grants as complement. Inviting people has been pretty easy for me. If I want to organize a conference I can also apply for some special funding for this. AMSS is the leading institution in China on number theory, which is my field of study. I can discuss my mathematics with a lot of colleagues. I have benefited a lot from such communications for my own research. Mostly during summer, the department invites some famous mathematicians to visit for longer period. This helps us to keep connection with most recent developments. Administrative burden like things related to reimbursement has not influenced me much (such burden is complained by a few friends working in other universities). NSR: What is the most predominant characteristic of AMSS? Chen: AMSS is the melting pot of mathematics. People working in every aspects of the theory and application of mathematics find their places in AMSS. Lü: (i) AMSS has a glorious history and impressive achievements. The free academic exploration, painstaking research scientific spirits have been formed in its long history. (ii) AMSS is a high-level research platform in which mathematics and other disciplines work together. AMSS has a group of masters in mathematics and related disciplines. It is promising to make a major breakthrough on a number of key scientific questions. Li: We have very good researchers in different fields of mathematics. So we can discuss various topics in mathematics and we may do some collaboration with our colleagues. Moreover, we have many excellent visitors from China and abroad to exchange ideas and to do collaboration in mathematical research. We are also strongly supported to go abroad to attend conferences and to visit some leading mathematical institutes and universities in the world. This provides us a good opportunity to do research collaboration with many good mathematicians in the world. Wan: I think the most important characteristic of AMSS that attracted me is the flexibility, compared to universities. The system is to some extent similar to the CNRS in France or IAS in the US. I have more freedom arranging my time on research, on teaching time and teaching subjects, and on visits. Personally I like this freedom and I think it indeed helps bringing my ability of research into full play. NSR: What are your future expectations for your research and the development of AMSS? Chen: I expect that AMSS will be one of the very top research institutions in mathematics worldwide in the coming years. I hope I can contribute to the success of AMSS in a substantial way. Lü: My research fields include complex networks, non-linear circuits and systems, and big data. Combining theory and application, I will try to obtain a series of original results by overcoming a number of bottleneck problems. I hope that AMSS can achieve a series of essential breakthroughs in a number of major key scientific issues, which will lead the rapid development of related fields. As a result, AMSS will do much more contributions to our country. Li: My personal expectation is to continue and to do some good research work in the interaction of analysis, geometry and probability theory, in connection with problems from statistical physics and other areas. I would like also to train some good students and help them to become good researchers in mathematics. About AMSS in future, I hope it can become more and more attractive and creative center of mathematics in the world. Wan: My research is primarily on Birch and Swinnerton-Dyer (BSD) conjecture and surrounding subjects. This relates special values of L-functions and arithmetic objects, which is one of the most important and deepest relations in mathematics. In near future I plan to finish up my previous work on such problems for modular forms, and try to generalize. Hopefully there will come up with new discoveries on this subject. For AMSS, it is a little hard to predict, as policies from government has been changing very frequently, which serve like a double-edged sword. But at the moment I certainly believe we can attract more and more talented young people and will gradually catch up with the world-class institutions in North American and European countries. Editors: Weijie Zhao (NSR) and Hao Tang (AMSS) Designer: Xiaoling Yu (NSR) Photo: Lin Wang (AMSS) © The Author(s) 2018. Published by Oxford University Press on behalf of China Science Publishing & Media Ltd. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/about_us/legal/notices)

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National Science ReviewOxford University Press

Published: May 15, 2018

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