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高兴

兰州大学数学与统计学院     教授   高兴

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
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---------------------------------------------------

[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.   

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
荣誉、获奖
社会工作
其它信息

作者:高兴