Topology Reading Group (TRG) 2022

Fall 2022 Meeting Time and Place:

Tuesdays 3pm-4pm, Boston College Maloney Hall 560.

For Spring 2022 TRG info, head over to Fraser’s site, here; and for Fall 2021 TRG info, head over to Braeden’s site, here.

Together with Mujie Wang, I am organizing this year’s (Fall 2022 and Spring 2023) Topology Reading Group.

Topology Reading Group (TRG) is an informal seminar for graduate students and post-docs at Boston College and nearby universities. Participants can volunteer to speak about anything topology/geometry/dynamics-related that they’d like to.

If you’d like to speak, or to attend, please email either:

  • Kevin, at: kevin (dot) yeh (at) bc (dot) edu (that’s me!)
  • Mujie, at: mujiew (at) bc (dot) edu

Talks:

September 6th: Kevin Yeh

  • Title: An Introduction to the Classification of 4-Manifolds Part I
  • Abstract: The classification of 4-manifolds differs substantially from that of other dimensions. One such distinction is the crucial role the intersection form plays in their classification. In Kevin’s portion of the talk, we will introduce this algebraic object, together with several of its numerical invariants. We will end this portion with some classification results obtained via the intersection form in the topological category.

September 13th: Ali Naseri Sadr

  • Title: An Introduction to the Classification of 4-Manifolds Part II
  • Abstract: Ali will continue the thread by focusing on the smooth category; in particular the theorems of Freedman and Donaldson. We will see how the existence of a smooth structure on a 4-manifold can substantially restrict which intersection forms it can carry. Finally, we will finish by computing the intersection forms of certain manifolds using characteristic classes. Working knowledge in topological and smooth manifolds, and their homology and cohomology, will be assumed.

September 20th: Qingfeng Lyu

  • Title: An Introduction to the Classification of 3-Manifolds
  • Abstract: In this talk, we will ambitiously tell the story of how people developed a classification of 3 manifolds in the past century (in line with the past talks). We hope to mention prime decomposition, torus decomposition, hyperbolization and geometrization. Pure story. No actual proof included. All are welcome.

September 27th: Ali Naseri Sadr

  • Title: Construction of an Exotic R^4
  • Abstract: I will introduce instantons and show you how we can use the moduli space of instantons to prove 2E_8 is not smooth; Then we will use this to construct an exotic R^4 embedded in the connected sum of three copies of S^2 “cross” S^2.

October 4th: Joe Boninger

  • Title: Morse Theory and the Calculus of Variations
  • Abstract: We’ll give a rapid introduction to Morse theory, then discuss its relationship with the calculus of variations. This combination uses topology and differential geometry to study the path spaces of smooth manifolds. We’ll state the Morse Index Theorem, and give some fun applications if time permits. (One application is Bott Periodicity, which could be discussed in a future talk.) No background in differential geometry is needed.

October 11th: NO MEETING.

  • Title:N/A
  • Abstract:N/A

October 18th: Laura Seaberg

  • Title: We Have Invariants Too, Guys
  • Abstract: What is a helpful notion of ‘isomorphic’ dynamical systems, and how can we tell different dynamical systems apart? This talk will introduce the notion of topological entropy, an invariant of a dynamical system which roughly describes the growth rate of the number of distinguishable orbits. We will explore some cute examples. Only prerequisites are some basic topology/analysis and a can-do attitude!

October 25th: Matthew Zevenbergen

  • Title: The Gromov Norm
  • Abstract: We will introduce the Gromov norm, which is defined using homology to give a measure of the complexity of the fundamental class of an oriented manifold. We’ll do some computations with closed surfaces. Finally, we’ll discuss some applications of the Gromov norm to hyperbolic geometry, including Mostow Rigidity. Notes here.

November 1st: Joaquin Ignacio Lema Perez

  • Title: Counting Simple Closed Geodesics in Surfaces
  • Abstract: If S is a hyperbolic closed surface (genus g>1), there are dynamic reasons to expect tons of closed geodesics. Concretely, Hubbard and Selberg proved that if N_T is the number of closed geodesics of length less or equal to T, then N_T grows like e^T/T. One can tweak the question and ask what happens with the growth of the number of simple closed geodesics (i.e. those with no self-intersection). In her amazing Ph.D. thesis, Maryam Mirzakhani uses some good old ergodic theory to prove that this number grows like L^(6g-6) (you guessed it, that’s the dimension of Teichmuller space!). We are not going to prove any of that. Instead, we will prove a baby case: if S is a flat torus and N_T is the number of simple closed geodesics of length less or equal to T, then this number grows like L^2. If time permits, we will mention a “recipe” that if followed, allows us to prove Mirzakhani’s theorem.

November 8th: Braeden Reinoso

  • Title: Taut Foliations, L-Spaces, and Symplectic Fillings
  • Abstract:I’ll sketch a proof of the well-known fact that Heegaard Floer L-spaces don’t admit taut foliations. This is one-third of the famous “L-space conjecture.” The proof itself is very short once you have all the ingredients–it’s mostly just an excuse to introduce all the concepts in the title and see how they interact. This hopefully will also serve as a good way to see how some of the different flavors of Heegaard Floer homology interact, if you haven’t seen something like that before. If I have time, I’ll also discuss a short generalization of the proof relating to the Thurston norm.

November 15th: Jacob Caudell

  • Title: (tentative) Lens Spaces in 3 and 4 Dimensions
  • Abstract: From the cut-and-paste perspective, lens spaces—the 3-manifolds with finite cyclic fundamental group, the irreducible 3-manifolds admitting genus one Heegaard splittings—are the simplest non-trivial 3-manifolds. In his resolution of the lens space realization problem, Greene conjectured that the only lens spaces bounding simply-connected 4-manifolds with b_2 = 1 are those which are realized by Dehn surgery on a knot in the 3-sphere, i.e. those bounding a 4-manifold admitting a handle decomposition consisting of a single 0-handle and a single 2-handle. Ballinger recently produced an infinite family of counterexamples to this conjecture by a construction that can straightforwardly, if not tediously, produce infinitely more infinite families of lens spaces that bound 4-manifolds with b_2 = 1 but are not surgery on a knot in the 3-sphere. In this talk, we will give an overview of lens spaces and the 4-manifolds they bound and present Ballinger’s construction and some related constructions.

November 22nd: THANKSGIVING, NO MEETING

  • Title:N/A
  • Abstract:N/A

December 6th: Fraser Binns

  • Title:TBD
  • Abstract:TBD

December 13th: Ali Naseri Sadr

  • Title:TBD
  • Abstract:TBD