Yong LinYau Mathematical Sciences Center, Tsinghua University, Beijing 100084, ChinaChong WangSchool of Mathematics, Renmin University of China, Beijing 100872, China; School of Mathematics and Statistics, Cangzhou Normal University, 061000 ChinaShing-Tung YauDepartment of Mathematics, Harvard University, Cambridge MA 02138, USA
CombinatoricsAlgebraic Topology and General Topologymathscidoc:2207.06004
In this paper, we give a necessary and sufficient condition that discrete Morse functions on a digraph can be extended to be Morse functions on its transitive closure, from this we can extend the Morse theory to digraphs by using quasi-isomorphism between path complex and discrete Morse complex, we also prove a general sufficient condition for digraphs that the Morse functions satisfying this necessary and sufficient condition.
Yong LinYau Mathematical Sciences Center, Tsinghua University, Beijing 100084, ChinaChong WangSchool of Mathematics, Renmin University of China, Beijing 100872, China; School of Mathematics and Statistics, Cangzhou Normal University, 061000 China
CombinatoricsAlgebraic Topology and General Topologymathscidoc:2207.06006
In this paper, we prove that discrete Morse functions on digraphs are flat Witten-Morse functions and Witten complexes of transitive digraphs approach to Morse complexes. We construct a chain complex consisting of the formal linear combinations of paths which are not only critical paths of the transitive closure but also allowed elementary paths of the digraph, and prove that the homology of the new chain complex is isomorphic to the path homology. On the basis of the above results, we give the Morse inequalities on digraphs.
Huimin ChangDepartment of Applied Mathematics, School of Education, The Open University of China, 100039 Beijing, ChinaYu ZhouYau Mathematical Sciences Center, Tsinghua University, 100084 Beijing, ChinaBin ZhuDepartment of Mathematical Sciences, Tsinghua University, 100084 Beijing, China
In this paper, we give a complete classification of cotorsion pairs in a cluster category C of type A^\infty_\infty via certain configurations of arcs, called τ-compact Ptolemy diagrams, in an infinite strip with marked points. As applications, we classify t-structures and functorially finite rigid subcategories in C, respectively. We also deduce Liu-Paquette's classification of cluster tilting subcategories of C and Ng's classification of torsion pairs in the cluster category of type A^\infty_\infty.
The CD inequalities are introduced to imply the gradient estimate of Laplace operator on graphs. This article is based on the unbounded Laplacians, and finally concludes some equivalent properties of the CD(K,1) and CD(K,n).
Let G=(V,E) be a locally finite graph. Firstly, using calculus of variations, including a direct method of variation and the mountain-pass theory, we get sequences of solutions to several local equations on G (the Schrödinger equation, the mean field equation, and the Yamabe equation). Secondly, we derive uniform estimates for those local solution sequences. Finally, we obtain global solutions by extracting convergent sequence of solutions. Our method can be described as a variational method from local to global.