Geometry Seminar

Spring 2021

Thursdays (and sometimes Tuesdays) 3:40-4:30pm, in Ryerson 358 (currently on zoom)

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• 4/22/21
• 4/29/21
• 5/13/21
• 5/20/21
• 5/27/21
• 6/17/21

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• contacts

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• previous quarters
• W2021
• F2020
• S2020
• W2020
• F2019
• S2019
• W2019
• F2018
• S2018
• W2018
• F2017
• S2017
• W2017
• F2016
• S2016
• W2016
• F2015
• S2015
• W2015

• • Thursday April 22 at 3:40PM-5:00PM in Zoom
  • Laurent Bartholdi, University of Göttingen
  • Dimension series and homotopy groups of spheres
  • Abstract: The lower central series of a group G is defined by 𝛾₁ = G and 𝛾_n = [G,𝛾_n-1]. The "dimension series", introduced by Magnus, is defined using the integral group algebra: 𝛿_n = {g: g is in the n^th power of the augmentation ideal}. It has been, for the last 80 years, a fundamental problem of group theory to relate these two series. One always has 𝛿_n ≥ 𝛾_n, and a conjecture by Magnus, with false proofs by Cohn, Losey, etc., claims that they coincide; but Rips constructed an example with 𝛿₄ / 𝛾₄ cyclic of order 2. On the positive side, Sjogren showed that 𝛿_n / 𝛾_n is always a torsion group, of exponent bounded by a function of n. Furthermore, it was believed (and falsely proven by Gupta) that only 2-torsion may occur. In joint work with Roman Mikhailov, we prove however that every torsion abelian group occurs as a quotient 𝛿_n / 𝛾_n; this proves that Sjogren's result is essentially optimal. Even more interestingly, we show that this problem is intimately connected to the homotopy groups 𝛑_n(S^m) of spheres; more precisely, the quotient 𝛿_n / 𝛾_n is related to the difference between homotopy and homology. We may explicitly produce p-torsion elements starting from the order-p element in the homotopy group 𝛑_2p(S²) due to Serre.
• • Thursday April 29 at 4:10PM-5:00PM in Zoom
  • Joshua Greene, Boston College
  • Peg Problems
  • Abstract: Toeplitz asked in 1911 whether every Jordan curve in the Euclidean plane contains the vertices of a square. The problem remains open, but it has given rise to many interesting variations and partial results. I will describe some of these and the proof of a result which is best possible when the curve is smooth: for any four points on the circle and for any smooth Jordan curve in the Euclidean plane, there exists an orientation-preserving similarity which carries the four points onto the curve. The proof involves symplectic geometry in a surprising way. Joint work with Andrew Lobb.
• • Thursday May 13 at 4:10PM-5PM in Zoom
  • Kasia Jankiewicz, University of Chicago
  • Background on growth rates and hyperbolic group
  • Abstract: This is a preliminary talk for the talk by Koji Fujiwara. We will provide background on the growth rates of groups and hyperbolic groups.
• • Thursday May 13 at 5:10PM-6PM in Zoom
  • Koji Fujiwara, Kyoto University
  • The rates of growth in a hyperbolic group
  • Abstract: I will discuss the set of rates of growth of a hyperbolic group with respect to all its finite generating sets. It turns out that the set is well-ordered, and that every real number can be the rate of growth of at most finitely many generating sets up to automorphism of the group. This is a joint work with Sela. If time permits, I will discuss other families of groups.
• • Thursday May 20 at 4:10PM-5:00PM in Zoom
  • Danny Calegari, University of Chicago
  • An introduction to slice knots
  • Abstract: A knot K in the 3-sphere is (smoothly) slice if it bounds a smooth embedded disk in the 4-ball; more generally, the *slice genus* is the least genus of a smooth embedded surface in the 4-ball K bounds. The signature of a slice knot is zero, and its Alexander polynomial factors as a product f(t)f(t^-1). Detecting whether a knot is slice is difficult; Conway’s knot was recently shown to be slice after 50 years by Lisa Piccirillo, using Rasmussen’s s-invariant. We shall define this invariant, and how it gives a bound on slice genus. This is an expository talk.
• • Thursday May 27 at 4:10PM-5:00PM in Zoom
  • Ciprian Manolescu, Stanford University
  • Khovanov homology and the search for exotic 4-spheres
  • Abstract: A well-known strategy to disprove the smooth 4D Poincare conjecture is to find a knot that bounds a disk in a homotopy 4-ball but not in the standard 4-ball. Freedman, Gompf, Morrison and Walker suggested that Rasmussen’s invariant from Khovanov homology could be useful for this purpose. I will describe three recent results about this strategy: that it fails for Gluck twists (joint work with Marengon, Sarkar and Willis); that an analogue works for other 4-manifolds (joint work with Marengon and Piccirillo); and that 0-surgery homeomorphisms provide a large class of potential examples (joint work with Piccirillo).
• • Thursday June 17 at 3:00-4:00pm in Zoom
  • Bruno Martelli, Università di Pisa
  • Hyperbolic 5-manifolds that fiber over the circle
  • Abstract: We show that the existence of hyperbolic manifolds fibering over the circle is not a phenomenon confined to dimension 3 by exhibiting some examples in dimension 5. More generally, there are hyperbolic manifolds with perfect circle-valued Morse functions in all dimensions n ≤ 5, a fact that leads us naturally to ask whether this may hold for any n. One consequence of this result is the existence of hyperbolic groups with finite-type subgroups that are not hyperbolic. The main tool is Bestvina - Brady theory applied to some hyperbolic n-manifolds that decompose very nicely into right-angled polytopes, enriched with the combinatorial game recently introduced by Jankiewicz, Norin and Wise. These are joint works with Battista, Italiano, and Migliorini.

Due to the high number of requests, we are no longer accepting speakers via self-invitations.

For questions, contact

• Benson Farb, farb at math.uchicago.edu
• Shmuel Weinberger, shmuel at math.uchicago.edu
• Danny Calegari, dannyc at math.uchicago.edu
• Kasia Jankiewicz, kasia at math.uchicago.edu
• Justin Lanier, jlanier at math.uchicago.edu
• Kevin Schreve, kschreve at math.uchicago.edu