Organizers: Anneke Bart, John Kalliongis, Michael Landry
Time: 4-5pm central, more or less alternating Tuesdays
Location: 120 Ritter Hall
| Date | Speaker | Title and abstract |
|---|---|---|
| September 3 | Michael Landry, SLU | Simultaneous universal circles In low-dimensional topology it is common to study spaces by studying associated geometric and dynamical structures. Three examples are codimension-1 foliations, flows, and actions of the fundamental group on the circle. I will give some background on these things before describing a construction that brings all three together, called a simultaneous universal circle. This is joint work with Minsky and Taylor. |
| September 17 | John Kalliongis, SLU | Planar Crystallographic Groups: Mapping Classes Abstract. A planar crystallographic group π is a uniform discrete group of isometries on the complex plane. There are seventeen of these groups up to isomorphism. Via the universal covering map, the orbifold Mπ = C/π is endowed with an affine structure and with a flat structure. The mapping class group mcg(π) is the group of components of the affine self-diffeomorphism group Aff(Mπ). We will discuss mcg(π) for these groups. |
| October 1 | Minh Lam Nguyen, WashU |
Spectral invariants and positive scalar curvature on 4-dimensional cobordism Abstract: Information about sectional curvature and Ricci curvature tends to make the underlying manifold "rigid" topologically. This is not the case for scalar curvature, e.g., obstruction to existence of positive scalar curvature (psc) is often via some topological invariants. In this talk, we use the Chern-Simons-Dirac functional to define an R-filtration on monopole Floer homology HM(Y) of a rational homology 3-sphere Y. We define a numerical quantity ρ (spectral invariant) that measures the non-triviality of HM(Y). It turns out that ρ is an invariant of Y with a geometric structure. Using ρ, we give an obstruction to psc on ribbon homology cobordsim between 2 rational homology spheres. |
| October 15 | Federico Salmoiraghi, Queens University |
Application of convex surfaces theory to Anosov flows Anosov flows are an important class of dynamical systems due to their ergodic and geometric properties. Even though they represent examples of chaotic dynamics, they enjoy the remarkable property of being stable under small perturbations. In this talk, I will explain how, perhaps surprisingly, Anosov flows are related to both integrable plane fields (foliations) and totally non-integrable plane fields (contact structures). The latter represents a less-studied approach that has the potential to make new connections to other branches of mathematics, such as symplectic geometry and Hamiltonian dynamics. As main application, I will show how convex surface theory introduced by Giroux in the 90s in the context of contact structures, gives a general framework for cut-and-paste techniques on Anosov flows. |
| October 29 | Andy Miller, University of Oklahoma |
Moduli for planar crystallographic groups and the isometry realization problem There are 17 isomorphism classes of planar crystallographic groups. By definition, each one admits embeddings into the isometry group of the (Euclidean) complex plane whose image is uniform and discrete. Two of these embeddings determine the same 'modulus' if they are conjugate by a similarity. For a given group, the space of moduli can be viewed as a connected manifold with dimension zero, one or two. We will discuss these moduli spaces with a focus on looking at specific examples that illustrate basic principles. Moduli play a role in determining if a subgroup of the outer automorphism group of a planar crystallographic group G can be realized by an isomorphic group of isometries of the closed 2-orbifold obtained as the quotient of the complex plane by G. (The outer automorphism group of G can be identified with the mapping class group of the orbifold). This is joint work with John Kalliongis. |
| November 12 | Stacey Harris, SLU |
Work on Causal Boundary for Generic Spacetimes Boundary-constructions can give an eye to global structures--on topological spaces, on manifolds with geometry. Spacetimes have a fundamental structure in between topology and geometry: causality, the relation that exists between event p and event q, when the physics at p can effect on the physics at q. The Causal Boundary construction is a universal construction on spacetimes--universal in a categorical sense, of creating the minimal causally-complete object containing the original spacetime. But it's not an easy construction to perform, and all published accounts are on spacetimes with high degree of symmetry (such as spherically symmetric or permitting a global isometric R-action) or having a highly specific algebraic structure to the geometry (such as warped product geometry). But this is highly unsatisfactory from a physical standpoint, as physically important models that, on physical grounds, ought to be nearly identical in causal structure (such as uncharged vs. charged vacuum, spherically symmetric, static black holes), have vastly different Causal Boundaries. It's widely believed this is an artifact of physically unrealistic exact symmetries. But no one knows. I present a sketch of physical measurements--things locally observed by each of a global field of observers--that allow one to conclude that the Causal Boundary is that of the Schwarzschild black hole model. These conditions thus suffice for what amounts to something like a generic class of spacetimes. |
| November 19 (note irregular date) | Junzhi Huang, Yale |
Depth-one foliations, pseudo-Anosov flows and universal circles In 3-dimension topology, the study of foliations, flows and \pi_1-actions on 1-manifolds are closely related. Given a 3-manifold M, one can construct a \pi_1(M)-action on a circle from either a cooriented taut foliation or a pseudo-Anosov flow in M by works of Thurston, Calegari-Dunfield and Fenley. When the foliation is depth-one and the pseudo-Anosov flow is transverse to the foliation and has no perfect fits, we show that the circle actions from both settings are topologically conjugate. Moreover, the two circles admit extra structures that are compatible in the most natural sense. |
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