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。
---------------------------------------------------accepted--------------------------------------------------- [56] 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. DOI: 10.1007/s10468-022-10198-3 [55] X. Gao, L. Guo*, W. Y. Sit, S. Zheng, Rota-Baxter type operators, rewriting systems and Gröbner-Shirshov bases, arXiv:1412.8055. [54] H. H. Zhang, X. Gao and L. Guo*, Compatible structures of nonsymmetric operads, Manin products and Koszul duality, accepted by Applied Categorical Structures. [53] C. M. Bai, X. Gao, L. Guo and Y. Zhang, Operator forms of the nonhomogeneous associative classical Yang-Baxter equation, accepted by Advances in Theoretical and Mathematical Physics. ---------------------------------------------------2023----------------------------------------------------- [52] 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. DOI: 10.1016/j.jalgebra.2023.01.001 [51] 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 (2023), 178-204. DOI: 10.24330/ieja.1220707 ---------------------------------------------------2022----------------------------------------------------- [50] 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. 7月14日 [49] 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 1月 [48] 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 [47] 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 [46] 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 ---------------------------------------------------2021----------------------------------------------------- [45] 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 [44] 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 11月 [43] T. J. Zhang, X. Gao* and L. Guo*, Reynolds algebras and their free objects from bracketed words and rooted trees, J. Pure Appl. Algebra，225 (2021), 106766. Doi: 10.1016/j.jpaa.2021.106766 12月 [42] 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 6月 ---------------------------------------------------2020----------------------------------------------------- [41] 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 (2020), 219-243. Doi: 10.36045/bbms/1594346416 7月 [40] 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 6月 [39 ] 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 [38] X. M. Wang*, S. J. Xu and X. Gao, A Hopf algebra on subgraphs of a graph, Journal of Algebra and Its Applications, 19(09) (2020), 2050164. Doi: 10.1142/S0219498820501649 [37] 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, 4月 -----------------------------------------------------2019----------------------------------------------------- [36] 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 [35] 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 1月 [34] 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 [33] L. Qiao, X. Gao and L. Guo*, Rota-Baxter modules toward derived functors, Algebras and Representation Theory 22 (2019), 321-343. Doi: 10.1007/s10468-018-9769-5 [32] X. G. Zhang, X. Gao and L. Guo*, Modified Rota–Baxter Algebras, Shuffle Products and Hopf Algebras, Bull. Malays. Math. Sci. Soc. 42 (2019), 3047-3072. Doi: 10.1007/s40840-018-0648-3 [31] X. Gao* and X. Wang, Infinitesimal unitary Hopf algebras and planar rooted forests, Journal of Algebraic Combinatorics, 49 (2019), 437-460. Doi:10.1007/s10801-018-0830-6 [30] 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 -----------------------------------------------------2018----------------------------------------------------- [29] 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 [28] J. Zhang and X. Gao*, Free operated monoids and rewriting systems, Semigroup Forum, 97 (2018), 435-456. Doi: 10.1007/s00233-018-9939-0 4月 [27] X. Gao* and T. J. Zhang, Averaging algebras, rewriting systems and Gröbner–Shirshov bases, Journal of Algebra and Its Applications, 16(2) 2018, 1850130. Doi: 10.1142/S021949881850130X [26] 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--------------------------------------------------- [25] X. Gao and W. F. Keigher*, Interlacing of Hurwitz series，Communications in Algebra，45(5) (2017), 2163-2185. [24] X. Gao*, W. F. Keigher and M. Rosenkranz, Divided powers and compositions in integro-differential algebras, J. Pure Appl. Algebra, 221 (2017), 2525-2556. [23] X. Gao and L. Guo*, Rota's Classification Problem, rewriting systems and Gröbner-Shirshov bases, Journal of Algebra, 470(2017), 219-253. -----------------------------------------------------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. [21] X. Gao*, Q. Li, J.W. Wang and W.Z. Wang, On 2-Extendable Quasi-abelian Cayley Graphs, Bull. Malays. Math. Sci. Soc. 39 (2016), S43-S57. -----------------------------------------------------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 and M. Rosenkranz, Free integro-differential algebras and Gröbner–Shirshov bases, Journal of Algebra, 442 (2015), 354-396. -----------------------------------------------------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, J. Algebra and Its Application, 13 (2014), 1350160. -----------------------------------------------------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 fifnite 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.
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
作者:高兴
