During the Spring 2025 quarter, all of our meetings will be held in-person in Skye Hall 268.
Talks will run from 11a-12p PT, but feel free to show up early for socializing with the speaker.
Current organizers: Po-Ning Chen
Spring 2025 schedule
April 11, 2025:
Title: Special Lagrangians near adiabatic limits
Abstract: Constructing special Lagrangian submanifolds in compact Calabi-Yau manifolds is an important topic in both symplectic geometry and complex geometry, with potential applications to defining new invariants of Calabi-Yau manifolds. In this talk, I will describe a new gluing construction of special Lagrangian submanifolds in K3-fibered Calabi-Yau 3-folds, including examples diffeomorphic to S^3 and S^1 x S^2. This is joint work with Yu-Shen Lin.
April 18, 2025:
Filippo Mazzoli
Title: Constant mean curvature foliations of almost-Fuchsian manifolds
Abstract: Quasi-Fuchsian groups have been objects of extensive study since the 1890s. By naturally acting on the 3-dimensional hyperbolic space, they describe a wide class of complete, infinite volume, hyperbolic 3-manifolds, and their properties play a crucial role in Thurston's hyperbolization theorem and, more generally, in the study of the geometry and topology of 3-manifolds. Following Uhlenbeck, we say that a quasi-Fuchsian manifold is almost-Fuchsian if it contains an incompressible minimal surface with principal curvatures between -1 and 1. A conjecture by Thurston asserts that any almost-Fuchsian manifold admits a foliation by constant mean curvature (CMC) surfaces. In this talk, I will describe a result from an upcoming joint work with Nguyen, Seppi, and Schlenker, where we describe explicit conditions of the first and second fundamental forms of the minimal surface of an almost-Fuchsian manifold that guarantee the existence of a CMC foliation.
April 25, 2025:
Shane Rankin
Title: The Hard Lefschetz Theorem on Kähler Lie Algebroids
Abstract: Compact Kähler Manifolds are central objects of study in modern geometry. Several classical results use analytic and algebraic methods to constrain the topology of such an object - one such result being the Hard Lefschetz Theorem. In this talk, we'll discuss this theorem, and how and when it extends to a more general geometric setting: a Lie Algebroid. We'll discuss what these objects are and go over some basic examples of this theorem in context.
May 2, 2025:
Title: Generalized convex toric domains and symplectic embedding problems
Abstract: A convex toric domain $X_\Omega$ is a 4-dimensional subset of $\mathbb{R}^4$ that is the preimage of a bounded convex region $\Omega$ in the positive quadrant of the plane under the moment map. Here, we put no restrictions on $\Omega$ other than convexity. We consider how features of $\Omega$ such as the curviness of its boundary and the affine perimeter of the boundary impact symplectic packings problems. A main focus of the talk will be considering results about ellipsoidal embeddings into $X_\Omega$. The proof methods involve both ECH capacities and obstructions from exceptional classes. This is based on upcoming joint work with Dan Cristofaro-Gardiner and Dusa McDuff.
May 9, 2025:
Ruoyi Wang
The title is: Intrinsic metrics on bounded domains.
Abstract: We investigated the conditions for Bergman, Caratheodory, Kahler and Kobayashi metrics to be not pairwise distinct (up to scaling) and provided some characterization for strongly pseudoconvex domains using these intrinsic metrics.
May 16, 2025
Title: Can You Hear The Cone Points of an Orbifold Drum?
Abstract:
In 1966, Mark Kac famously asked "can you hear the shape of a drum?" More precisely, given a manifold, to what extent does the collection of eigenvalues of its Laplace-Beltrami operator determine the geometry of M? This is the "Inverse Isospectral Problem." Mathematicians have been actively working on the Inverse Isospectral Problem, and variations upon it, since the early 1900s and there are still many interesting open problems today. In this talk, I will start with a historical background of the Inverse Isospectral Problem, covering important examples and theorems. This will include discussion of planar domains and hyperbolic surfaces. I will then discuss work with a collaborator, Benjamin Linowitz, in which we studied whether one can "hear" the structure of the singular set of an orbifold. I will include many pictures, some videos, and discuss some open problems in the area.
May 23, 2025
Title: New Special Lagrangians in Calabi-Yau 3-Folds with Fibrations
Abstract: Special Lagrangian submanifolds, introduced by Harvey and Lawson, are an important class of minimal submanifolds in Calabi-Yau manifolds. In this talk, I will explain common constructions of special Lagrangians and then a gluing construction of a special Lagrangian in Calabi-Yau manifolds with K3-fibrations when the K3-fibres are collapsing. Furthermore, these special Lagrangians converge to an interval or loop of the base of the fibration at the collapsing limit. This phenomenon is similar to holomorphic curves collapsing to tropical curves in special Lagrangian fibrations and is only a tip of the iceberg of the DonaldsonScaduto conjecture. This is a joint work with Shih-Kai Chiu. This also generalizes to Calabi-Yau 3-folds with abelian fibrations in a joint work in progress with Sidharth Soundararajan.
May 30, 2025
Rolland Trapp
Title: Flat fully augmented links are determined by their complements
Abstract:
Gordon and Luecke proved that knots are determined by their complements, but the situation is different for links. For example, Dehn twists on embedded disks or annuli are homeomorphisms of a link complement that can frequently change link type. Mangum and Stanford refined the question slightly, showing there is an infinite class of links within which homeomorphic complements imply link equivalence. The main goal of this talk is to describe another infinite class of links, flat fully augmented links, with the same property. More precisely, we show that
Two flat fully augmented links have homeomorphic complements if and only if they are equivalent.
This is joint work with Christian Millichap and the talk will begin by introducing flat fully augmented links (flat FALs), a family of hyperbolic links with tractable geometry. Flat FAL complements contain reflection surfaces and thrice-punctured spheres, which are embedded totally geodesic surfaces. These surfaces, how they intersect, and their cusp behavior are the tools necessary to prove the theorem. Thus, geometry plays a crucial role in obtaining this topological result.
June 6, 2025
Title: Strong shortcuts and generating sets
Abstract: A group is strongly shortcut if it has a Cayley graph in which circles cannot embed at arbitrarily large scales with arbitrarily good bilipschitz constants. This can be shown to be a special case of the Gromov mesh condition implying simply connected asymptotic cones and polynomial Dehn function. Most classes of nonpositively curved groups are strongly shortcut, including CAT(0) groups, Helly groups, systolic groups and hierarchically hyperbolic groups. I will discuss various results on strongly shortcut groups, including recent joint work with Timothy Riley in which we showed that the strong shortcut property is not invariant under change of generating sets.