Topology seminar

Location

Mondays at 2:30 in Vincent Hall 311

Talks in Spring 2020

• Jan 27 '20

  • Matthew Stoffregen (MIT)

  • Smooth 4-manifolds and the geometry of 3-manifolds

  One of the interests of low-dimensional topologists is understanding which smooth 4-manifolds can bound a given 3-manifold, or, as a special case, understanding the set of 3-manifolds up to so-called homology cobordism (to be defined in the talk). This question turns out to have applications to the study of triangulations of high-dimensional manifolds, and is a natural proving ground for Floer-theoretic techniques of studying 3-manifolds. In this talk, we will give some structure theorems about the homology cobordism group, and show that there are three-manifolds that are very far from having any of the seven non-hyperbolic Thurston geometries. This talk includes joint work with I. Dai, K. Hendricks, J. Hom, L. Truong, and I. Zemke.

• Mar 02 '20

  • George Shabat (Russian State University for the Humanities and Independent University of Moscow)

  • Dessins d'enfants applied to the geometry and arithmetic of moduli spaces of curves

  Dessins d'enfants theory, initiated by Alexander Grothendieck in the 1970's, establishes an equivalence between the category of certain graphs on topological surfaces and some arithmetic-geometric category (of Belyi pairs, i.e. (X, beta)'s, where X is a curve and beta: X ---> P^1 is a covering with 3 branch points). It turned out (Mumford-Penner-Kontsevich-...) that the decorated moduli spaces of curves M_{g,N} (C) x R_{>0}^N admit the orbifold cell decomposition in which the cells are parametrized by certain dessins d'enfants. In 1992 Kontsevich applied this construction to the proof of the famous Witten conjecture; the discrete version of this theory was suggested in [2].

  This discrete version relates Belyi pairs with the Deligne-Mumford stack \overline{M_{g,N}} over Spec Z and over finite fields; the corresponding counting problems were discussed in [1] and [3]. The talk will be devoted to the overview of the corresponding constructions, results and open problems; some examples will be presented.

  References

  [1] N. Amburg, E. Kreines and G. Shabat, Poincare polynomial for the moduli space \overline{M_{0,n}}(C) and the number of points in \overline{M_{0,n}}(F_q). arXiv: 1811.120976v1 [math.AG] 29 Nov 2018. [2] P. Norbury, Counting lattice points in the moduli space of curves. Math. Res. Lett. 17, 467-481 (2010). [3] G.Shabat, Counting Belyi pairs over finite fields. In 2016 MATRIX Annals (eds. David R. Wood, Jan de Gier, Cheryl E. Praeger, Terence Tao). MATRIX Book Series, Volume 1, pp. 305-322, 2018.