兰州大学数学与统计学院

数学院高兴教授

研究方向

代数图论，罗巴(Rota-Baxter)代数，算子代数，rough path，regularity structure and Stochastic PDE，密码学，数据处理.

个人简历

        高兴，男，博士，基础数学教授，博士生导师，兰州大学萃英学者。在兰州大学数学与统计学院获得学士和博士学位。2015年8月至2016年8月间，在美国罗格斯大学交流访问。主要从事Rota-Baxter代数和代数图论等领域的研究，在Journal of Algebra、 Journal of Pure and Applied Algebra、Discrete Applied Math.等国际期刊上发表学术论文近六十篇。主持数学天元基金、青年科学基金、面上项目、甘肃省自然科学基金项目, 曾参与国家自然科学基金项目2项和甘肃省自然科学基金项目1项。已经指导毕业博士研究生1名，硕士研究生4名。
        罗巴(Rota-Baxter)代数简介： 罗巴代数在1960 年由美国概率论学家G.巴克斯特首先定义。很快得到Atkinson 等著名分析学家和Cartier,罗塔(Rota)等著名组合学家的重视。而在20世纪八十年代，李代数上的罗巴算子又以经典杨-巴克斯特方程的算子形式由前苏联理论物理学家独立发现。新世纪的开始标志着罗巴代数引人注目的复兴。2000年，罗巴代数作为一个基本结构出现在Connes (菲尔兹奖得主)和Kreimer 有关量子场论重整化的开创性工作中。同年Augiar 的工作将罗巴代数与结合杨-巴克斯特方程和Loday 的dendriform 代数联系起来。同年郭锂教授和Keigher把自由交换罗巴代数以混合洗牌(mixable shuffle)乘法构造出来，开始了交换罗巴代数的系统研究。而同年M.Hoffman定义拟洗牌(quasi-shuffle)乘法来研究数论中的多元zeta 值（MZV），成为罗巴代数在MZV 中应用的联接点。在此之后，出现了许多罗巴代数方面的文章，联系到量子场论（QFT），杨-巴克斯特方程，数论，operad，Hopf 代数，交换代数，组合和微分方程边值问题等。郭锂教授应邀为美国数学会会刊(Notice AMS)的WHAT IS …专栏撰文，把罗巴代数介绍给数学界,此文的发表，不仅说明罗巴代数和曾在此专栏介绍的其它领域一样，已成为公认的新兴学科，也认同郭锂教授在罗巴代数领域的领军地位。完成罗巴代数的首部专著《An Introduction to Rota-Baxter Algebra》，由美国International Press和中国高等教育出版社出版。

教学及指导学生情况

已经指导毕业博士研究生1名，硕士研究生4名

发表论文及专著

---------------------------------------------------accepted---------------------------------------------------
[61] Y. Y. Zhang, H. H. Zhang and X. Gao*, Free Ω-Rota-Baxter systems and Gröbner-Shirshov bases, Journal of Algebra and its Applications, 
          Online 2024.1.24  doi.org/10.1142/S0219498825501622
[60] Y. Y. Zhang, X. Gao* and D. Manchon*, Free Rota-Baxter family algebras and Free (tri)dendriform family algebras, accepted by Algebras and Representation Theory, 2023
          Online 2023.2.9 DOI: 10.1007/s10468-022-10198-3   数学学科一流期刊
[59] X. Gao,  L. Guo*, W. Y. Sit, S. Zheng, Rota-Baxter type operators, rewriting systems  and Gröbner-Shirshov bases,  arXiv:1412.8055.   
---------------------------------------------------2024-----------------------------------------------------  
[58] H. H. Zhang, X. Gao and L. Guo*, Compatible structures of nonsymmetric operads, Manin products and Koszul duality, Applied Categorical Structures, 32(1), 2024, Article 2.
          DOI: 10.1007/s10485-023-09760-x  Published 2024.1.10
---------------------------------------------------2023-----------------------------------------------------  
[57] X. Gao, L. Guo*, Y. J. Liu and Z. C. Zhu, Operated groups, differential groups and Rota-Baxter groups with an emphasis on the free objects, Communications in Algebra, 51(11) (2023), 4481-4500. 
          DOI: 10.1080/00927872.2023.2212775   Online: 2023.5.24
[56] Z. C. Zhu, H. H. Zhang and X. Gao*, Free weighted (modified) differential algebras, free (modified) Rota-Baxter algebras and Gröbner-Shirshov bases, International Journal of Algebra and Computation, 33(4) (2023), 787-817.   Pub: 2023.6.30 
          DOI: 10.1142/S0218196723500352
[55] H. H. Zhang, X. Gao* and L. Guo*, Operator identities on Lie algebras, rewriting systems and Gröbner-Shirshov bases, Journal of Algebra, 620 (2023), 585-629. 数学学科一流期刊
          DOI: 10.1016/j.jalgebra.2023.01.001   2023年04月15日
[54] X. M. Wang, L. Foissy* and X. Gao, The dual of infinitesimal unitary Hopf algebras and planar rooted forests, International Electronic Journal of Algebra, 33(33) (2023), 178-204. 
          DOI: 10.24330/ieja.1220707  Pub: 2023.9.1
---------------------------------------------------2022-----------------------------------------------------  
[53] Y. Zhang and X. Gao*, Weighted infinitesimal bialgebras, 数学年刊A辑, 43(2) (2022), 153-190.
          DOI: 10.16205/j.cnki.cama.2022.0011
[52] C. M. Bai, X. Gao, L. Guo and Y. Zhang, Operator forms of the nonhomogeneous associative classical Yang-Baxter equation, Advances in Theoretical and Mathematical Physics, 26(7) (2022), 1965-2009.                                                                                   
          DOI: 10.4310/ATMP.2022.v26.n7.a1  Pub. 2023.8.30
[51] Y. Zhang, X. Gao* and L. Guo*, Hopf algebra of multidecorated rooted forests, free matching Rota–Baxter algebras and Gröbner-Shirshov bases, Pacific Journal of Mathematics, 317(2) (2022), 441-475.  Pub 2022.7.14
          DOI: 10.2140/pjm.2022.317.441
[50] Y. Zhang, Z. C. Zhu, J. W. Zheng and X. Gao*, Weighted infinitesimal unitary Bialgebras and weighted associative Yang–Baxter equations, Advances in Mathematics (China), 51(6)  (2022), 1011-1028.
          DOI: 1000-0917(2022)06-1011-18   11月
[49] J. W. Wang, Z. C. Zhu and X. Gao*, New Operated Polynomial Identities and Gröbner-Shirshov Bases, Mathematics, 10(6) (2022), 961-975.  
          DOI: 10.3390/math10060961    3月
[48] X. Gao, H. H. Zhang and L. Guo*, Rota’s Program on Algebraic Operators, Rewriting Systems and Gröbner–Shirshov Bases, Advances in Mathematics (China), 51(1) (2022). 
          DOI: 10.11845/sxjz.2021003a   1月
[47] X. S. Peng, Y. Zhang, X. Gao and Y. F. Luo*, Universal enveloping of (modified) λ-differential Lie algebras, Linear and Multilinear Algebra, 70(6)  (2022), 1102-1127.
          DOI: 10.1080/03081087.2020.1753641   Pub 2022.4.13    数学学科一流期刊

---------------------------------------------------2021-----------------------------------------------------  
[46] Y. Zhang, X. Gao and Y. F. Luo*, Weighted infinitesimal unitary bialgebras of rooted forests, symmetric cocycles and pre-Lie algebras, Journal of Algebraic Combinatorics, 53 (2021), 771–803.
          DOI: 10.1007/s10801-020-00942-7  Pub 2021.5.1  数学学科一流期刊
[45] X. Gao and L. Guo* and Y. Zhang, Commutative matching Rota-Baxter operators, shuffle products with decorations and matching Zinbiel algebras, Journal of Algebra, 586 (2021), 402-432.
          DOI: 10.1016/j.jalgebra.2021.06.032  Pub 2021.11.15   数学学科一流期刊
[44] T. J. Zhang, X. Gao* and L. Guo*, Reynolds algebras and their free objects from bracketed words and rooted trees, Journal of Pure and Applied Algebra, 225(12) (2021), 106766.  
          DOI: 10.1016/j.jpaa.2021.106766  Pub 2021.5.4 数学学科一流期刊
[43] X. S. Peng, Y. Zhang*, X. Gao and Y. F. Luo, Dendriform-Nijenhuis bialgebras and DN-associative Yang-Baxter equations, Journal of Algebra, 575 (2021), 78-126.
          DOI: 10.1016/j.jalgebra.2021.01.042   Pub 2021.7.1  数学学科一流期刊
---------------------------------------------------2020-----------------------------------------------------  
[42] X. S. Peng, Y. Zhang*, X. Gao and Y. F. Luo, Left counital Hopf algebras on bi-decorated planar rooted forests and Rota-Baxter systems, Bulletin of the Belgian Mathematical  Society Simon Stevin, 27(2) (2020), 219-243.  Pub 2020.7.10
          DOI: 10.36045/bbms/1594346416 
[41] Y. Zhang, X. Gao and L. Guo*, Matching Rota-Baxter algebras, matching dendriform algebras and matching pre-Lie algebras, Journal of Algebra, 552 (2020), 134-170. 
          DOI: 10.1016/j.jalgebra.2020.02.011  Pub 2020.6.15  数学学科一流期刊
[40] Y. Zhang and X. Gao*, Hopf algebras of planar binary trees: an operated algebra approach, Journal of Algebraic Combinatorics, 51 (2020), 567-588.
          DOI: 10.1007/s10801-019-00885-8  Online 2019.5.20  Pub 2020.6  数学学科一流期刊
[39] X. M. Wang*, S. J. Xu and X. Gao, A Hopf algebra on subgraphs of a graph, Journal of Algebra and Its Applications, 19(9) (2020), 2050164.
          DOI: 10.1142/S0219498820501649   
[38] Y. Y. Zhang, X. Gao and D. Manchon*, Free (tri)dendriform family algebras, Journal of Algebra, 547 (2020), 456-493.  
          DOI: 10.1016/j.jalgebra.2019.11.027   Pub 2020.4.1 数学学科一流期刊
       ---------------------------------------------------2019-----------------------------------------------------  
[37] Y. Zhang*, D. Chen, X. Gao and Y. F. Luo, Weighted infinitesimal unitary bialgebras on rooted forests and weighted cocycles, Pacific Journal of Mathematics, 302(2) 2019, 741-766.
          DOI: 10.2140/pjm.2019.302.741    Pub 2019.11.27
[36] M. Rosenkranz*, X. Gao and L. Guo, An algebraic study of multivariable integration and linear substitution, Journal of Algebra and Its Applications, 18(11) (2019), 1950207. 
          DOI: 10.1142/S0219498819502074  Pub 2019.8.19
[35] Y. Y. Zhang* and X. Gao, Free Rota–Baxter family algebras and (tri)dendriform family  algebras, Pacific Journal of Mathematics, 301 (2019), 741-766. 
          DOI: 10.2140/pjm.2019.301.741
[34] L. Qiao, X. Gao and L. Guo*, Rota-Baxter modules toward derived functors, Algebras and Representation Theory, 22 (2019), 321-343.   数学学科一流期刊
          DOI: 10.1007/s10468-018-9769-5  Online:2018.2.22
[33] X. G. Zhang, X. Gao and L. Guo*, Modified Rota–Baxter Algebras, Shuffle Products and Hopf Algebras, Bull. Malays. Math. Sci. Soc. 42(6) (2019), 3047-3072. 
          DOI: 10.1007/s40840-018-0648-3   Online 2018.6.16  Pub 2019.11   
[32] X. Gao* and X. Wang, Infinitesimal unitary Hopf algebras and planar rooted  forests, Journal of Algebraic Combinatorics, 49(4) (2019), 437-460. 
          DOI:10.1007/s10801-018-0830-6  Pub 2019.6  数学学科一流期刊
[31] X. G. Zhang, X. Gao and L. Guo*, Free modified Rota-Baxter algebras and Hopf algebra, International Electronic Journal of Algebra, 25 (2019), 12-34. 
          DOI: 10.24330/ieja.504101  Pub 2019.8.1
-----------------------------------------------------2018-----------------------------------------------------
[30] X. Gao, P. Lei and T. J. Zhang*, Left counital Hopf algebras on free Nijenhuis algebras, Communications in Algebra, 46(11) (2018), 4868-4883.   
          DOI: 10.1080/00927872.2018.1459641      Pub 2018.4.23
[29] J. Zhang and X. Gao*, Free operated monoids and rewriting systems, Semigroup Forum, 97 (2018), 435-456.
          DOI: 10.1007/s00233-018-9939-0   Pub 2018.4.17
[28] X. Gao* and T. J. Zhang, Averaging algebras, rewriting systems and Gröbner–Shirshov bases, Journal of Algebra and Its Applications, 17(7) 2018, 1850130.
          DOI:  10.1142/S021949881850130X   Pub 2018.6.13
[27] X. Gao, L. Guo and M. Rosenkranz*, On rings of differential Rota-Baxter operators, International Journal of Algebra and Computation, 28(1) (2018), 1-36. 
          DOI: 10.1142/S0218196718500017 
-----------------------------------------------------2017---------------------------------------------------
[26] X. Gao and W. F. Keigher*, Interlacing of Hurwitz series, Communications in Algebra，45(5) (2017), 2163-2185.
          DOI: 10.1080/00927872.2016.1226885
[25] X. Gao*, W. F. Keigher and M. Rosenkranz, Divided powers and compositions in integro-differential algebras, Journal of Pure and Applied Algebra, 221(10) (2017), 2525-2556.
          DOI: 10.1016/j.jpaa.2017.01.001   Pub 2017.10  数学学科一流期刊
[24] X. Gao and L. Guo*, Rota's Classification Problem, rewriting systems and   Gröbner-Shirshov bases, Journal of Algebra, 470 (2017), 219-253.   
          DOI: 10.1016/j.jalgebra.2016.09.006   Pub 2017.1.15    数学学科一流期刊
[23] 张天杰，高兴，郭锂，自由Rota-Baxter代数的因子分解，Factorizations in Free Rota-Baxter Algebras, 数学学报, 5 (2017), 789-798.
          DOI: 10.12386/A2017sxxb0067
-----------------------------------------------------2016-----------------------------------------------------
[22] T. J Zhang, X. Gao and Li Guo*，Hopf algebras of rooted forests, cocyles and free Rota-Baxter algebras，Journal of Mathematical Physics, 57 (2016), 101701.  
          DOI: 10.1063/1.4963727
[21] X. Gao*, Q. Li, J.W. Wang and W.Z. Wang, On 2-Extendable Quasi-abelian Cayley Graphs, Bulletin of the Malaysian Mathematical Sciences Society， 39 (2016), 43-57.
          DOI: 10.1007/S40840-015-0287-X     Pub 2016.11.18 
 ----------------------------------------------------2015-----------------------------------------------------
[20] X. Gao Huazhong Lv and Yifei Hao, The Laplacian and signless Laplacian spectrum of semi-Cayley graphs over abelian groups, J. Appl. Math. Comput., 2015: 1-13.  
[19] 高兴*, 吕华众，王维忠，The extendability of Bi-Cayley graphs of dihedral groups, 浙江大学学报, 42 (2015), 526-528. 
[18] X. Gao，L. Guo* and M. Rosenkranz, Free integro-differential algebras and  Gröbner–Shirshov bases, Journal of Algebra, 442 (2015), 354-396. 
          DOI: 10.1016/j.jalgebra.2014.10.016    Pub 2015.11.15   数学学科一流期刊
-----------------------------------------------------2014-----------------------------------------------------
[17] 樊馨蔓，杨东，高兴，单演半群的Г图，浙江大学学报, 41(3)  ( 2014), 258-260.
[16] X. Gao and L. Guo*, Constructions of free commutative integro-differential algebras,  Lecture Notes in Comput. Sci., 8372 (2014), 1-22. 
[15] W.Z Wang*, Y.F. Luo and X. Gao, On incidence energy of some graphs, Ars Combinatorics, 114 (2014), 427-436. 
[14] X. Gao，L. Guo* and S. H. Zheng, Construction of free commutative integro-differential algebras by the method of Gröbner-Shirshov bases, Journal Algebra and Its Application, 13 (2014), 1350160.  
          DOI: 10.48550/arXiv.1302.0041  Pub 2014.8.1
-----------------------------------------------------2013-----------------------------------------------------
[13] X. X. Fan, Y. F. Luo and X. Gao, Tricyclic graphs with exactly two main eigenvalues,  Cent. Eur. J. Math. 11(10) 2013, 1800-1816.  
-----------------------------------------------------2012-----------------------------------------------------
[12] 樊馨蔓，高兴，杨东，完全单周期半群的Cayley图的可迁性，兰州大学学报, 48(6) ，2012, 102-104.
[11] X. Gao *，Y.F. Luo and W.W. Liu, Kirchhoff index in line, subdivision and total graphs of a regular graph, Discrete Applied Mathematics, 160 (2012) , 560-565.
-----------------------------------------------------2011-----------------------------------------------------
[10] X. X Fan*, X. Gao and Y. F. Luo, Spectral characterations of a specific class of trees,   Ars Combinatorics, 102 (2011), 147-159.  
[9] Y. F. Hao*, X. Gao and Y. F. Luo, On Cayley graphs of symmetric inverse semigroups, Ars Combinatorics, 100 (2011), 307-319.  
[8] Y. F. Hao*, X. Gao and Y. F. Luo, On the Cayley graphs of Brandt semigroups, Communications in Algebra, 39 (2011), 2874-2883.  
[7] X. Gao *, W. W Liu and Y. F. Luo, On Cayley graphs of normal bands, Ars Combinatorics, 100 (2011), 409-419.
[6] X. Gao*, W. W. Liu and Y. F. Luo, On the extendability of certain semi-Cayley graphs of finite abelian groups, Discrete Mathematics, 311 (2011), 1978-1987.   
[5] X. Gao *, Y. F. Luo and W. W. Liu, Resistance distances and the Kirchhoff index in Cayley  graphs, Discrete Applied Mathematics 159 (2011), 2050-2057.  
-----------------------------------------------------2010-----------------------------------------------------
[4] X. Gao * and Y. F. Luo, The zero-divisor graph of a completely 0-simple semigroup, Southeast Asian Bulletin of Mathematics 34 (2010), 657-662.
[3] Y. Dong and X. Gao*, D-Saturated Property of the Cayley graphs of Semigroups,  Semigroup  Forum 80 (2010), 174-180. 
[2] X. Gao* and Y. F. Luo, The spectrum of semi-Cayley graphs over abelian groups, Linear  Algebra and its Applications 432 (2010), 2974-2983.  
-----------------------------------------------------2009-----------------------------------------------------
[1] Y. F. Luo and X. Gao *, On the extendability of Bi-Cayley graphs, Discrete Mathematics, 309 (2009), 5943-5949.

项目成果

国家自然科学基金（面上项目），2021--2024，主持;
国家自然科学基金（青年基金），2013--2015，主持;
甘肃省自然科学基金，2013--2015，主持;
国家自然科学基金（天元基金），2011.1--2011.12，主持;
中央高校基本科研业务费，2011--2012， 主持;
中央高校基本科研业务费，2013.1--2014.6， 主持.
国家自然科学基金面上项目(11371177)，2014-2017，参与

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