Organizers: Anneke Bart, John Kalliongis, Michael Landry
Time: 4-5pm central, more or less alternating Tuesdays
Spring 2025 Location: 202 Ritter Hall
| Date | Speaker | Title and abstract |
|---|---|---|
| January 21 | Michael Landry, SLU | The mother of all 3-manifolds I will discuss a proof of the famous result of Lickorish and Wallace that every closed, orientable, connected 3-manifold can be obtained by performing Dehn surgery on a link in the 3-sphere. |
| February 4 | Fernando Al Assal, UW Madison | Asymptotically geodesic surfaces in hyperbolic 3-manifolds Abstract: Let M be a hyperbolic 3-manifold. We say a sequence of distinct (non-commensurable) essential closed surfaces in M is asymptotically geodesic if their principal curvatures go uniformly to zero. When M is closed, these sequences exist abundantly by the Kahn-Markovic surface subgroup theorem, and we will discuss the fact that such surfaces are always asymptotically dense, even though they might not equidistribute. We will also talk about the fact that such sequences do not exist when M is geometrically finite of infinite volume. Finally, time permitting, we will discuss some partial answers to the question: does the existence of asymptotically geodesic surfaces in a negatively-curved 3-manifold imply the manifold is hyperbolic? This is joint work with Ben Lowe. |
| February 18 | Charles Ouyang, WashU | Postponed due to snow |
| March 4 | Calin Belean, SLU | An Introduction To Symplectic Manifolds Abstract: Symplectic manifolds have been an object of study in mathematical physics from classical mechanics to string theory. In this talk we will introduce the symplectic form, Hamiltonians, and Lagrangian submanifolds with examples. We will end with connections to modern research. This talk is suitable for anyone with an understanding of manifolds and differential forms. |
| March 7, 4pm in Ritter 236 (note irregular date and location) | Nathalie Rieger, Yale | From Lorentzian manifolds to signature-type change Hartle and Hawking (1983) suggested that signature-type change could have profound conceptual implications, leading to the development of the "no-boundary" proposal for the universe’s initial conditions. In this framework, singularity-free universes have no distinct beginning yet possess an origin of time. Mathematically, this corresponds to manifolds with a smooth transition from a Riemannian to a Lorentzian signature, where time emerges at the interface. Motivated by the "no-boundary" proposal, I explore this topic and introduce elements of a novel framework for signature changing manifolds, characterized by a smooth yet degenerate metric. Additionally, I extend certain Lorentzian techniques and results to the context of signature change. |
| March 18 | Aliakbar Daemi, WashU |
The mapping class group action on the odd character variety is faithful An important source of (possibly singular) symplectic manifolds is provided by the space of all representations of the fundamental group of a Riemann surface into a Lie group G modulo conjugation. For instance, using the group SO(3) and following this construction, one obtains a smooth symplectic manifold, called the odd character variety. Any diffeomorphism of the underlying Riemann surface determines a symplectomorphism of any of these representation varieties including the odd character variety. This gives rise to a homomorphism from (a finite extension of) the mapping class group of the surface to the symplectic mapping class group of the odd character variety. Dostoglou and Salamon asked whether this homomomorphism is injective when the genus is at least 2. In this talk, I will discuss how we can answer this question positively for any genus, generalizing an earlier work of Ivan Smith who answered the question for the genus 2 case. The proof uses tools from instanton homology, the Atiyah--Floer Conjecture, and some classical facts about the mapping class groups of Riemann surfaces. This talk is based on joint work with Chris Scaduto. |
| April 1 | Charles Ouyang, WashU | New Minimal Lagrangians in ℂℙ2 Abstract: Minimal Lagrangian tori in ℂℙ2 are the expected local model for particular point singularities of Calabi-Yau 3-folds, and numerous examples have been constructed. In stark contrast, very little is known about higher genus examples, with the only ones to date due to Haskins-Kapouleas and only in odd genus. Using loop group methods, we construct new examples of minimal Lagrangian surfaces of genus (k-1)(k-2)/2 for large k. In particular, we construct the first examples of such surfaces with even genus. This is joint work with Sebastian Heller and Franz Pedit. |
| April 15 | Michael Sullivan, SIU Carbondale |
A Survey of Smale Flows We study flows on 3-manifolds (mostly the 3-sphere) that are structurally stable with one dimensional invariant sets. If there are a finite number of periodic orbits, such flows are called nonsingular Morse-Smale flows and are classified up to topological flow equivalence by a theorem of Masaaki Wada (1989). Such flows with infinitely many periodic orbits are called nonsingular Smale flows. The attractors and repellers are still isolated periodic orbits, but there are chaotic saddle sets. These are studied using template theory initiated by Joan Birman and the late Robert Williams in the 1980s. We study how the templates, attactors and repellers fit together. Work by recent PhD students is included. |
| April 29 | Ari Stern, WashU |
| 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. |