Research

Do you want to know the profound connections behind different branches of mathematics such as geometry and number theory?

Do you want to know how to depict knots in mathematical language?

Do you want to know therelationshipbetween the Yang-Baxter equation and quantum groups?

Do you want to know what quantum is and where it comes from?

Have you ever heard of God playing dice?

Do you believe that quantum communication is not afraid of“eavesdropping”?

Today wewilltake you into the team of algebra and its application, you can experience the beauty of mathematics,

we will show you the great charm of quantum information science,

we welcome you to join us in the future with your love for mathematics and your inquisitive spirit.

Basic information

Team Leader：

FAN Zhaobing,Professor, doctoral supervisor, currently Dean of School of Mathematical Sciences, Harbin Engineering University. He used to be theVice Dean of the School of Sciences and the School of Mathematical Sciences of Harbin Engineering University, and the Director of the International Exchange and Cooperation Office. He was awarded a doctor's degree in mathematics at Kansas State University, and successively worked as a postdoctoral and visiting assistant professor at the State University of New York Buffalo and Kansas State University. He is mainly engaged in geometric representation theory and quantum communication theory research, such as the geometric realization of quantum subgroups, Kazhdan Lusztig theory, canonical basis theory, Schur Weyl duality theory, Quantum key distribution and quantum teleportation. He has published academic papers in different journalsincluding Mem Amer Math Soc., Comm. Math Phys, Trans Amer Math Soc., Int. Math Res. Not. IMRN, etc.. He is theprincipal investigator ofa number of fund projects including the Outstanding Youth Fund of Heilongjiang Province, the National Natural Science Fund, the Natural Science Fund of Heilongjiang Province, and Fundfor the Returned Overseas Chinese Scholars.

Team members:

YANG Gao (Associate Professor), XIAO Husileng(Associate Professor), QIN Chao (Associate Professor), KIM SUNHO (Associate Professor), YAO Hongmei (Associate Professor), MA Haitao (Lecturer), WANG Qi (Lecturer), and BI Yingjin (Lecturer)。

Team introduction: The team is committed to the research of cutting-edge problems in algebra, including geometric representation theory, algebraic representation theory and quantum information. The main research content of geometric representation theory is Hall algebra, geometric realization of quantum groups, Springer theory, etc. Algebraic representation theory mainly studies Lie algebras, Lie superalgebras, W-algebras, etc. Quantum information mainly studies quantum resource theory, quantum communication and quantum network. In the past five years, the team has published more than 70 SCI papers and trained 9 doctoral students and more than 30 master students. The team has a good foundation for international cooperation, and the team members have established good cooperative relations with many well-known mathematicians, including: Tomoyuki Arakawa, Soek-Jin Kang, WANGWeiqiang, LIHaisheng and others. Team members have published many papers in Mem. Amer. Math. Soc.,Comm. Math. Phys.,Adv. Math.,Trans. Amer. Math. Soc., Math. Z., Transform. Groups,Phys. Rev.A and other journals, hosted the National Fund project, Outstanding Youth Fund of Heilongjiang Province, Excellent Youth FundofHeilongjiang Province, National Youth Fund, National Defense Key Laboratory Fund project, etc., and won the first prize of Natural science in universities in Heilongjiang Province.

Main research directions

1. Geometric Representation Theory

At the beginning of 1980s, as a group of well-known experts and scholars at home and abroad introduced geometric tools into representation theory, geometricrepresentation theory emerged as an important research field. Geometric representation theory is a method to study algebra with geometric tools, which has greatly promoted the development of representation theory and has a great influence on the development of representation theory.

G.Lusztig

Quantum group is one of the most popular and advanced research fields in mathematics, and it is closely related to branches of mathematics and physics. In the field of mathematics, quantum groups are closely related to Lie groups, Lie algebras, special functions, Knot theory, Low-dimensional topology, operator algebra, non-commutative geometry, combinatorial theory, etc. It is also closely related to quantum backscattering theory, integrable model theory, quantum fiel

At present, the team closely following the international frontier in the field of geometric representation theory. It mainly studying Schur-Weyl duality theory, Kazhdan-Lusztig theory, Springer theory, mainly solving important problems including the Schur-Weyl duality of I-quantum groups and geometric realization (including abnormal layer method and K-theory method, etc.), Springer correspondence of symmetric Spaces of orthogonal groups, and so on.

2. Generalized Kac-Moody Algebra

To prove the Moonshine Cconjecture, Borcherds introduced the generalized Kac-Moody algebra. This is abroaderclass of infinite dimensional Lie algebras, which is characterized by allowing imaginary single roots to appear in sets of single roots, which greatly enriches Lie theory. Our team focuses on quantized generalized Kac-Moody algebras and their representation theory, canonical and crystal-based theory, categorization problems, and geometric realization.

3. Representation Theory of Superalgebraic

The theory of Lie superalgebras originated from mathematical physics, which was developed by physicists in the study of supersymmetry problems in quantum field theory. At present, the study of nonmodular Lie superalgebras has been rich in theories and is still widely studied by mathematicians. The study of modular Lie superalgebras is relatively slow, and the classification of finite dimensional single modular Lie superalgebras is still unsolved. Therefore, modular Lie superalgebras have become an interesting research topic and is of great significance. The team focuses on the representation and structure theory of W- (super) algebras, and the structure and representation of modular Lie superalgebras.

4. Quantum resource theory

Quantum resource theory regards quantum entanglement as a resource, systematically describes the transformation between two states, the entanglement measure of quantum states, and the purification of quantum states. Specifically, it includes the quantification of various quantum resources from quantum entanglement to quantum coherence, and the study of static resources such as the conversion between different quantum resources through quantum extension, and the corresponding extension of dynamic resources. At the same time, it conducts theoretical research with auxiliary quantum states as catalyst to realize more efficient transformation between quantum states.

5. Quantum communication

Quantum communication is the transmission of information using the basic principles of quantum mechanics.

Quantum communication has absolute security and high efficiency in theory. Quantum communication includes quantum key distribution, quantum secure direct communication, quantum secret sharing, quantum dense coding and quantum teleportation.

Quantum key distributionuses channels to establish random key (QKD) on both sides. The team mainly studied the maximum transmission distance of quantum keys in different media, and designed a new cross-media quantum key distribution protocol.

Quantum teleportationis the transmission of any unknown photon information without the need to transfer the particle itself (first proposed by C.H.Bennett, an academician of the American Academy of Sciences). The team mainly studies catalytic quantum teleportation, high-dimensional quantum teleportation and multi-degree of freedom quantum teleportation.

6. Quantum networks

Quantum networks are many processes involved in quantum evolution. It includes quantum computing networks, fault-tolerant quantum networks, quantum storage networks, entanglement distribution networks, preparation and measurement networks, and trusted relay networks. The team is concerned about the measurement of resources in quantum networks, mainly using the relative entropy of quantum network hypothesis test to measure the entanglement distillation of quantum networks and to study how many resources need to be consumed to simulate a given quantum network. The theory is applied to the design of key distribution protocols for quantum networks in different media.

Contact us

Interested students are welcome to join the research team!

Doctoral students and visiting scholars are welcome to join us.

The team enrolls 3-5 doctoral students and 5-7graduateseveryyear.

Undergraduate students who want to join the team to study in advance, please contact the counselor LIU Dongping（Wechat:liudongping234567）.

FAN Zhaobing  Professor: fanzhaobing@hrbeu.edu.cn

YANG Gao  Associate Professor : yanggao_izumi@foxmail.com

XIAO Husileng Associate Professor : hsl1523@163.com

KIM SUNHO  Associate Professor : kimsunho81@hrbeu.edu.cn

YAO Hongmei  Associate Professor : hongmeiyao@163.com

QIN Chao  Associate Professor : qinch23@mail.sysu.edu.cn

MA Haitao  Lecturer : hmamath@163.com

WANG Qi  Lecturer : Wang777@ustc.edu.cn

BI Yingjin  Lecturer : bbjingjin@126.com