Recent progress of various Volume Conjectures for links as well as 3-manifolds
报告题目(Title)：Recent progress of various Volume Conjectures for links as well as 3-manifolds
报告人(Speaker)：陈庆陶 教授 （NYU Abu Dhabi）
地点(Place)： 腾讯会议 ID：831 587 387
时间(Time)：2021.4.22 (周四下午) 1:30-3:00
邀请人(Inviter)：彦文娇
报告摘要
The original Volume Conjecture of Kashaev-Murakami-Murakami predicts a precise relation between the asymptotics of the colored Jones polynomials of a knot in S^3 and the hyperbolic volume of its complement. I will discuss two different directions that lead to generalizations of this conjecture.
The first direction concerns different quantum invariants of knots, arising from the colored SU(n) (with the colored Jones polynomial corresponding to the case n=2). I will first display subtle relations between congruence relations, cyclotomic expansions and the original Volume Conjecture for the colored Jones polynomials of knots. I will then generalize this point of view to the colored SU(n) invariant of knots. Certain congruence relations for the colored SU(n) invariants, discovered in joint work with K. Liu, P. Peng and S. Zhu, lead us to formulate cyclotomic expansions and a Volume Conjecture for these colored SU(n) invariants. If time permits, I will briefly discuss similar ideas for the Superpolynomials that arise in HOMFLY-PT homology.
Another direction for generalization involves the Witten-Reshetikhin-Turaev and the (modified) Turaev-Viro quantum invariants of 3-manifolds. In a joint work with T. Yang, I formulated a Volume Conjecture for the asymptotics of these 3-manifolds invariants evaluated at certain roots of unity, and numerically checked it for many examples. Interestingly, this conjecture uses roots of unity that are different from the one usually considered in literature. These 3-manifolds invariants are only polynomially large at usual root of unity as the level of the representation approaches infinity, which is predicted by Witten's Asymptotic Expansion Conjecture. True understanding of this new phenomenon requires new physical and geometric interpretations that go beyond the usual quantum Chern-Simons theory.
Currently these new Volume Conjectures have been proved for many examples by various groups. However, like original Volume Conjecture, a complete proof for general cases is still an open problem in this area. In a recent joint work with J. Murakami, I proved the asymptotic behavior of the quantum 6j-symbol evaluated at unusual root of unity, which could explain the Volume Conjectures for the asymptotics of the Turaev-Viro invariants in general.
主讲人简介
纽约大学阿布扎比分校理学院助理教授，2004年毕业于复旦大学数学系，2009年获美国加州大学伯克利分校数学系博士学位。陈博士曾在多个国际知名学术机构任职, 如在位于瑞士的苏黎世联邦理工学院数学系任博士后研究员(Postdoctoral Researcher)，资深研究员(Senior Researcher)。他在数学物理中的整体微分几何，量子拓扑等研究方向取得了一批原创结果，在国际一流期刊上发表学术论文十余篇。他与杨田合作的论文获得2019年度世界华人数学家联盟(ICCM)“最佳论文奖Best Paper Award”，他是2000年国际物理奥林匹克(IPHO)中国国家集训队队员，获得1999年全国中学生物理竞赛上海赛区一等奖第一名并实验第一名。
