本课题主要从两个大的方面来研究和发展地图理论。一方面是正则、半正则凯莱地图的存在性证明和分类等，主要研究课题包括：某些有限群正则凯莱地图的存在性证明；在正则凯莱地图存在的前提下，分类这些地图；某些有限群t-正则凯莱地图存在性证明；循环群、交换群等有限群半正则凯莱地图的计算。另一方面是对正则地图的外对称性研究，主要研究内容包括：构造具有极大指数群的(n,m)-型正则地图；证明具有极大指数群的(3,m)-型有限正则地图的存在性，这里6｜(m-a),其中a是不取1和-1的整数；构造(k,k)- 型自对偶且自Petrie对偶的正则地图；验证是否存在偶数度正则、自对偶且自Petrie对偶的正规凯莱地图。

Our research for map theories is divided into two main aspects. The first one is the proof of existence and classification of regular and half-regular Cayley maps.We will consider the following problems: Prove the existence of regular Cayley maps for some finite groups; classify all the regular Cayley maps under the condition of existence; prove or disprove the existence of regular t-balanced Cayley maps of some finite groups;enumerate the half-regular Cayley maps of cyclic and abelian finite groups.The second is about the external symmetries of regular maps. The problems are as follows:Construct the (n,m)-type regular maps with largest exponent groups; prove the existence of (3,m)-type regular maps with largest exponent groups, where 6｜(m-a) and the integer a cann't be 1 and -1;construct (k,k)-type self-dual and self-Petrie-dual regular maps;prove or disprove the existence of self-dual and self-Petrie-dual regular normal Cayley maps of even valency.