Counting incompressible surfaces in 3-manifolds

Time:Tues.,10:15-11:15, Dec.6, 2022

Venue: Venue / 地点 Zoom ID: 405 416 0815, PW: 111111


Speaker: Speaker / 主讲人 Nathan Dunfield, UIUC


Counting embedded curves on a hyperbolic surface as a function of their length has been much studied by Mirzakhani and others. I will discuss analogous questions about counting incompressible surfaces in a hyperbolic 3-manifold, with the key difference that now the surfaces themselves have more intrinsic topology. As there are only finitely many incompressible surfaces of bounded Euler characteristic up to isotopy in a hyperbolic 3-manifold, it makes sense to ask how the number of isotopy classes grows as a function of the Euler characteristic. Using Haken’s normal surface theory and facts about branched surfaces, we can characterize not just the rate of growth but show it is (essentially) a quasi-polynomial. Moreover, our method allows for explicit computations in reasonably complicated examples. This is joint work with Stavros Garoufalidis and Hyam Rubinstein. Based on


Nathan Dunfield has been at University of Illinois at Urbana-Champaign since 2007. Previously, He spent four years at Harvard and four years at Caltech after getting my PhD from the University of Chicago sometime back in the 20th century. In 2013, He became a Fellow of the American Mathematical Society. His research area is the topology and geometry of 3-manifolds.


DATEDecember 6, 2022
Related News
    • 0

      Contact 3-manifolds and contact surgery

      AbstractContact structures on 3-manifolds are given by a hyperplane distribution in the tangent bundle satisfying a condition called "complete non-integrability". Contact structures fall into one of two classes: tight or overtwisted. Ozsvath and Szabo introduced invariants of contact structures using Heegaard Floer homology. In this talk, I will survey some recent results about the tightness an...

    • 1

      Counting sheaves on K3 surfaces in 3 folds and 4 folds

      AbstractWe discuss the enumerative geometry of moduli spaces of sheaves with 2 dimensional support on K3 surfaces in K3-fibered threefolds and 4 folds. The reason to study such specific geometric setups is that these often provide computable invariants which govern deep information about geometry of moduli spaces of solutions to N=2 Super Yang-Mills theory on K3 surfaces, when the base surface ...