On October 28, the opening ceremony of Shandong National Center for Applied Mathematics, one of the first 13 national centers for applied mathematics announced by the Ministry of Science and Technology, was held. Led by the national major strategy, the Center focuses on key industry problems, and carries out the research on interdisciplinary theory and application featuring uncertainty and nonlinear expectation. It is aimed at building a platform for exchange of mathematical sciences and mathematical applications to help relevant industries solve problems and make breakthroughs in bottlenecks. What are the research areas of the Center? What are the previous academic and research achievements? We will take you to experience the research strength of Shandong National Center for Applied Mathematics from the perspective of little stories…

Cryptological mathematics safeguard cybersecurity

“Password cracking is very important. Without cracking, it is difficult to have standardized password applications, and it is also hard to strengthen the commercial password system.” Wang Xiaoyun, Academician of the Chinese Academy of Sciences and Dean of the School of Cyber Science and Technology (Research Institute), Shandong University, said. Wang Xiaoyun studied at Shandong University for bachelor’s, master’s and doctoral degrees. On the eve of graduating with a master’s degree, she changed her research direction from “analytic theory of numbers” to the emerging “cryptography”. Since then, she has been tightly bound to cryptological mathematics and has made outstanding achievements.

For many years, MD5 and SHA-1 algorithms based on the Hash function promulgated by the National Institute of Standards and Technology (NIST) are the two most advanced algorithms recognized internationally, and the latter is even regarded as the cornerstone of computational security systems, known as the “White House Password”. They are widely used in finance, securities and other e-commerce fields. At the Annual International Cryptology Conference held in Santa Barbara, California, USA in 2014, Professor Wang Xiaoyun read out the results of MD4, MD5, HAVAL-128 and RIPEMD, the four internationally renowned cryptographic algorithms, cracked by her along with her research team, making a stir in the audience. Her research results announced the collapse of the impregnable MD5, the world standard for cryptography, and caused an uproar in the cryptographic community. At the RSA Conference 2005 held in the United States, the paper written by Professor Wang Xiaoyun, et al. proved that SHA-1 was also cracked theoretically, which was another breakthrough in the field of international cryptography after deciphering MD5, and it just took two months.

After cracking two major international cryptographic algorithms, Professor Wang Xiaoyun gave up participating in the design of a new international standard cryptographic algorithm solicited by the USA from the world. Instead, she started to design a domestic standard for cryptographic algorithms. Nowadays, SM3, designed by her, has been a guardian in many industries, and has been widely used in finance, state grid, transportation and other important economic fields of China.

Financial mathematics, from inception to development

In 2020, the Mathematics and Computer Science Prize was awarded to Academician Peng Shige, Professor of Shandong University for his pioneering contributions in the fields of backward stochastic differential equations, nonlinear Feynman-Kac equations and nonlinear expectations. These three research projects have set the trend in financial risk control with their applications and roles in the financial sector. Peng Shige is also regarded as a pioneer of financial mathematics in China.

In the 1990s, Peng Shige and Professor Pardoux created “backward stochastic differential equations” with stochastic optimal control, and obtained nonlinear Feynman-Kac equations. Inspired by his academic peers, his revealed that his research results could be applied to finance. In 1993, Peng Shige used “backward stochastic differential equations” and calculated that there were some serious problems in China’s futures trading overseas. Investors have a greater than 70% probability of losing per trade and less than 30% probability of winning, which will inevitably lead to a significant loss of Chinese capital. He immediately wrote two letters, of one which was forwarded by the President of Shandong University to the leadership of Shandong Province; the other was submitted to the National Natural Science Foundation of China. In these two letters, he stated his basic views on the international futures and options markets, as well as the huge risks that China currently faces in overseas futures trading. He also suggested that research on risk analysis and control of the international futures market be carried out without delay, and the training of financial professionals enhanced. When receiving the letter, Shandong Province immediately stopped overseas futures trading, and the National Natural Science Foundation of China also quickly submitted Peng Shige’s letter to the CPC Central Leading Group for Financial and Economic Affairs. As a result, corresponding measures were taken to avoid the massive loss of China’s financial assets.

In the economic and financial fields, the future needs to be predicted before making decisions, and this is where linear mathematical expectations do not work. For this reason, Peng Shige pioneered the field of nonlinear mathematical expectations. Peng Shige found through the study of the financial crisis that swept the world that, financial theory is rooted in the modern probability theory, but the probability theory itself suffers from model uncertainty. Model uncertainty is the root cause of the financial crisis. The financial risk measurement tools commonly used in the world are not dynamic or sensitive to extreme risks. The probability of these extreme risks is very low, but when they do occur, they will often be devastating. In response to this problem, Peng Shige creatively introduced the concept of “G-Brownian motion” into stochastic control theory and established a set of “nonlinear mathematical expectations” that corresponds to model uncertainty, which has become the basic tool for the dynamic measurement of financial risks in modern China. At present, the results of this theory can be used to solve the prices of risky financial products under more general and complex circumstances. This theory is recognized as a fundamental tool for studying the pricing of derivative securities in the financial market and is also praised by experts as a “powerful and beautiful tool”.

Medical mathematics: Solving medical problems with mathematical models

Currently, mathematics plays an increasingly important role in biomedical studies and provides a broad space for the development of applied research on mathematics.

Li Guojun, Professor of the School of Mathematics, Shandong University, has been engaged in bioinformatics since 2004. He proved Chvátal’s conjecture and several open puzzles in the graph theory; he ended the debate on several difficult approximability problems in the algorithm theory community, achieved a series of results in biological data mining and software development, and published more than ten papers in top professional journals as the first or corresponding author. In 2019, Professor Li Guojun and his research team again published the latest results on transcriptome reconstruction in Genome Biology, a top international journal. They developed TransLiG, a de novo transcriptome assembler. The test assessment showed that TransLiG clearly outperformed the most popular splicing algorithm and solved a key bottleneck in analyzing RNA-seq data.

At present, although China’s precision medical research on cancer based on omics data has achieved some theoretical and applied results, there still has not been a fundamental change in the overall technology. One of the main bottlenecks is the lack of basic research on mathematical models and algorithms, resulting in the processing and analysis of multi-omics big data far from the ideal accuracy. For example, how to discover individual relevance from large scale and multi-layered omics data to predict the specificity recorded in cancer by regulatory modules. On this basis, Liu Bingqiang, Professor of the School of School of Mathematics, and his team declared the project “Mathematical Models and Algorithms in Precision Medicine of Breast Cancer”. This project will target bottlenecks in precision medicine of breast cancer, create a new mathematical theory and methodology for multi-omics data analysis of tumors, and a build a multi-omics atlas and regulatory network for breast cancer in the Chinese population. It will also systematically clarify features of the resistance transfer factor in breast cancer, discover new targets and biomarkers at the network level, and establish a precision prediction model for drug resistance and prognosis to promote the fundamental technological change in precision medicine of breast cancer.

Marine mathematics: Solve field problems according to the actual situation

Rui Hongxing, Professor of the School of Mathematics, has been engaged in the research on the numerical solution to partial differential equations, numerical methods for oil and water resources and applications for many years. Engaged in basic algorithms and theories, he is committed to solving hot and difficult problems in this research field, and he has also undertaken many research projects for state-owned enterprises and research institutions according to the actual situation. His research results of the “Theory and Application of Numerical Simulation Methods for Oil and Water Resources” were awarded with the Second Prize of Natural Sciences Awards of the Ministry of Education in 1997, and the “Research on Integrated Prevention and Control of Seawater Intrusion” was awarded with the Third Prize for Science and Technology Progress of the Ministry of Water Resources. In recent years, he has been in active cooperation with the Petroleum Exploration and Production Research Institute of SINOPEC to explore new practical methods and technologies for numerical simulation of complex reservoirs. With regard to the research on the methods and theories of numerical simulation of oil and water resources, Professor Rui Hongxing constructed a finite element algorithm featuring second-order accuracy in 2002 after long-term studies, gave a complete theoretical analysis and solved the problem that the feature line algorithm has only had first-order accuracy for the time step since it was proposed, which significantly improved the computational accuracy. He proposed a finite element algorithm characterized by conservation in 2010, which completely overcame the drawback that the feature line algorithm could not keep the conservation of mass. In 2017, he proved the superconvergence of the MAC algorithm for the Stokes equation, solved the classical problem that has existed for more than 50 years since its discovery in actual computing in the 1960s, and developed the MAC algorithm.

It’s mentioned in the plan of the Shandong National Center for Applied Mathematics that the Center will focus on key issues, such as financial risk, cryptographic computing, ocean dynamic system simulation, precision medicine big data, intelligent control system, etc., based on key bottlenecks in industries, such as finance, information security, energy, life and health, artificial intelligence, etc. On the basis of nonlinear expectations, it will carry out the research on interdisciplinary issues related to uncertainty by means of financial mathematics, cryptological mathematics, marine mathematics, life mathematics, intelligent mathematics and communication mathematics. With the great support from the state and local provinces, advantages of mathematics and outstanding leading talents in a variety of fields, the story of Shandong National Center for Applied Mathematics has just begun…