Statut  Confirmé 
Série  MATHIHES 
Domaines  hepth 
Date  Mardi 13 Mars 2018 
Heure  14:30 
Institut  IHES 
Salle  Amphithéâtre Léon Motchane 
Nom de l'orateur  Sun 
Prenom de l'orateur  Zhe 
Addresse email de l'orateur  
Institution de l'orateur  YMSC, Tsinghua University & IHES 
Titre  Generalized Mcshane's identity via LandauGinzberg potential and triple ratios 
Résumé  (Joint work with Yi Huang) Goncharov and Shen introduced a LandauGinzberg potential on the FockGoncharov $A_{G,S}$ moduli space, where $G$ is a semisimple Lie group and $S$ is a ciliated surface. They used the potential to formulate a mirror symmetry via Geometric Satake Correspondence. This potential is the markoff equation for $A_{ PSL(2,R), S_{1,1} }$. When $S=S_{g,m}$, such potential can be written as a sum of rank $G*m$ partial potentials. We obtain a family of generalized Mcshane's identities by splitting these partial potentials for $A_{PSL(n,R),S_{g,m}}$ by certain pattern of cluster transformations with geometric meaning. We also find some interesting new phenomena in higher rank case, like triple ratio is bounded in mapping class group orbit. As applications, we find a generalized collar lemma which involves $\lambda 1 / \lambda 2$ length spectral, discreteness of that spectral etc. In further research, we would like to ask how can we integrate to obtain the generalized Mirzakhani's topological recursion with $\mathcal{W}_n$ constraint? 
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