Xiaolong Hans Han

Welcome!

I'm currently a postdoc at Yau Mathematical Sciences Center, Tsinghua University. I am working with Prof. Yunhui Wu.

 Previously I was a PhD student at University of Illinois at Urbana-Champaign, under the advisory of Prof. Nathan Dunfield, and graduated in May 2021. 

I was a program associate for Random and Arithmetic Structures in Topology at MSRI, in fall 2020. 

Research Interests

I am interested in hyperbolic geometry, low-dimensional manifolds, and their connections to analytic invariants etc. One theme I am interested in is effective geometrization, in which we describe quantitatively the geometric invariants of a hyperbolic manifold (with dimension at least 3) in terms of its topological invariants. In one of my papers, I bound the L^2 norm (geometric complexity) of a cohomology in terms of its Thurston norm (topological complexity) for non-compact hyperbolic 3-manifolds. I used tools from Hodge theory and minimal surfaces. 

  In my recent paper, I study the connection between the Thurston norm, best Lipschitz circle-valued maps, and   maximal stretch laminations, building on the recent work of Daskalopoulos and Uhlenbeck, and Farre, Landesberg, and Minsky. I show that the distance between a level set and its translation is the reciprocal of the Lipschitz constant, bounded by the topological entropy of the pseudo-Anosov monodromy if M fibers. For infinitely many examples constructed by Rudd, the entropy is bounded from below by one-third the length of the circumference.