PATCH Seminar

All are welcome to attend PATCH, which is a daylong event for faculty and students in the Philadelphia area with interests in geometry and topology. Speakers are Michael Landry, Saint Louis University, and Hiro Lee Tanaka, Texas State University. View the schedule here.

Morning Background Talks (Park 278)

• 10–11 a.m. “A stack of broken lines: Like BG, but for Morse Theory,” Tanaka
  • Abstract: I will discuss some needed background for the afternoon talk.
• 11–11:30 a.m. Break
• 11:30 a.m.–12:30 p.m. “Objects associated with 3-manifolds fibering over the circle,” Landry
  • Abstract: A fundamental example in low-dimensional topology is a closed oriented 3-manifold fibering over the circle. Thurston's study of this example led to the celebrated Nielsen-Thurston classification of surface homeomorphisms and the Thurston norm on homology. I will introduce these concepts before further developing some of the rich structure present in the example, touching on flows, foliations, and homeomorphisms of surfaces with infinitely generated fundamental group. I will mention joint work with Minsky and Taylor that fits into the story.

Afternoon Talks (Park 336)

• 2:30–3:30 p.m. “A stack of broken lines: Like BG, but for Morse Theory,” Tanaka
  • Abstract:  A G-bundle over X is a family of copies of G, with one copy for every element of X. Families like this arise when studying nice functions on manifolds (i.e., in Morse theory) -- where instead of families of groups, families of broken lines live on moduli of gradient trajectories. And just like G-bundles are classified by an object called BG, it turns out you can write down the object that classifies families of broken lines -- this object is the stack of broken lines. The amazing fact is that this (geometric) object has an incredibly deep connection to the (algebraic) idea of associativity, and I'll try to explain why this is true. If time allows (which it might not) I'll try to explain why this object is expected to play a central role in enriching Morse theory and various Floer theories over stable homotopy theory. This is joint work with Jacob Lurie.
• 3:30–4 p.m. Break
• 4-5 p.m. “Toward a dynamical theory of Thurston's norm,” Landry
  • Abstract: One might hope to generalize the picture described in the morning talk to the setting of 3-manifolds that do not necessarily fiber over the circle. I will give some of the history of this endeavor, mentioning three conjectures of Mosher from the 1990s. Then I will describe joint work with Tsang that aims to make progress on these conjectures using modern objects called veering branched surfaces.

Mathematics