January 10 2025:
Po-Ning Chen
Title: Near horizon limit of wang-yau quasi-local mass
Abstract:
In this talk, we discuss the behavior of the Wang-Yau quasi-local mass on a family of surfaces approaching the apparent horizon (the near horizon limit). The vanishing of the norm of the mean curvature vector implies special properties for the Wang-Yau quasi-local energy and the optimal embedding equation. We utilize these features to prove the existence and uniqueness of the optimal embedding and investigate the minimization of the Wang-Yau quasi-local energy. In particular, we prove the continuity of the quasi-local mass in the near horizon limit.
January 17, 2025:
Title: Higher rank Teichmuller spaces, what are they, how many are there and where do they come from? (Part 1)
Abstract: These two talks will be about a somewhat recent paper joint with Bradlow, Garcia-Prada, Gothen and Oliveira, where we classify and parameterize all expected higher rank Teichmuller spaces using the theory of Higgs bundles. In this first talk I will define what I mean by a higher rank Teichmuller space and motivate their classification problem and give some examples and state the main classification theorem. In the second talk I will discuss the objects used in the classification (magical sl2-triples), Higgs bundles and the parameterizing objects.
January 24, 2025:
Title: Higher rank Teichmuller spaces, what are they, how many are there and where do they come from? (Part 2)
Abstract: These two talks will be about a somewhat recent paper joint with Bradlow, Garcia-Prada, Gothen and Oliveira, where we classify and parameterize all expected higher rank Teichmuller spaces using the theory of Higgs bundles. In this first talk I will define what I mean by a higher rank Teichmuller space and motivate their classification problem and give some examples and state the main classification theorem. In the second talk I will discuss the objects used in the classification (magical sl2-triples), Higgs bundles and the parameterizing objects.
January 31, 2025:
Title: Torus Actions in positive curvature
Abstract: I will present the structural components in the arguments of many recent papers involving group actions on positively curved spaces, explaining common strategies and obstacles to proving certain kinds of theorems about manifolds with group actions whose size is fixed relative to the dimension. Along the way I will sketch (and use as main example) how to prove the Hopf conjecture for manifolds with positive second intermediate Ricci curvature (and some fixed size torus action) (perhaps of dimension 0 mod 4). This is based on upcoming joint work with L. Kennard and L. Mouillé.
February 7, 2025:
Title: Non-uniqueness of mean curvature flow
Abstract: The smooth mean curvature flow often develops singularities, making weak solutions essential for extending the flow beyond singular times, as well as having applications for geometry and topology. Among various weak formulations, the level set flow method is notable for ensuring long-time existence and uniqueness. However, this comes at the cost of potential fattening, which reflects genuine non-uniqueness of the flow after singular times. With Xinrui Zhao, we show that even for flows starting from smooth, embedded, closed initial data, such non-uniqueness can occur. Our examples extend to higher dimensions, complementing the surface examples obtained by Ilmanen and White. Thus, we can't expect genuine uniqueness in general. Addressing this non-uniqueness issue is a difficult problem. With Alec Payne, we establish a generalized avoidance principle. We prove that level set flows satisfy this principle in the absence of non-uniqueness.
February 14, 2025
Title: Concave foliated flag structures and Hitchin representations in SL(3,R)
Abstract: In 1992 Hitchin discovered distinguished components of the PSL(d,R) character variety for closed surface groups pi_1S and asked for an interpretation of those components in terms of geometric structures. Soon after, Choi-Goldman identified the SL(3,R)-Hitchin component with the space of convex projective structures on S. In 2008, Guichard-Wienhard identified the PSL(4,R)-Hitchin component with foliated projective structures on the unit tangent bundle T^1S. The case d \ge 5 remains open, and compels one to move beyond projective geometry to flag geometry. In joint work with Alex Nolte, we obtain a new description of the SL(3,R)-Hitchin component in terms of concave foliated flag structures on T^1S.
February 21, 2025
Title: Obstructions to symplectic embeddings in dimension 6 and higher.
Abstract: McDuff and Siegel developed an embedding capacity for Liouville domains using Rational Symplectic Field Theory, which retains information when we take products with complex planes (stabilization) and is computable on 4-dimensional convex toric domains. The capacity comes from asymptotically cylindrical psuedoholomorphic curves in the completion of the domain. In the talk, we will introduce a restriction on the number of asymptotic ends and see that the altered computation formula gives new obstructions in stabilized embeddings. If there is time, we will discuss the construction of curves necessary for proving the computation formula.
February 28, 2025
Title: Anosov property of cyclic SO(2,3) Higgs bundles
Abstract: The concept of Anosov representations, introduced by F. Labourie, plays an important role in the study of higher Teichmüller theory. Given a Riemann surface X, by the celebrated non-Abelian Hodge correspondence, reductive representations from the fundamental group of X into a Lie group correspond to polystable Higgs bundles. In general, it is hard to check the Anosov property of a representation corresponding to a given Higgs bundle other than the known higher Teichmuller spaces or some trivial embeddings of known Anosov representations. Recently, S. Filip proved that some weight 3 variations of Hodge structure define Anosov representations. We extend his result and discover the Anosov property of representations corresponding to more general families of SO(2,3)-Higgs bundles.
March 7, 2025
Title: How to do Explicit Computations with Fukaya Categories of Surfaces
Abstract: Fukaya categories are interesting and subtle invariants of symplectic manifolds whose construction requires solving difficult differential equations and compactifying certain moduli spaces, which makes them difficult to compute explicitly. In the toy case of surfaces, however, everything can be made entirely combinatorial and explicit, allowing us to compute both in a particular fukaya categories and with families of related fukaya categories. In this talk we'll show explicitly how one might go about computing with fukaya categories of surfaces, with multiple examples. With any remaining time we'll discuss some applications to ongoing work with Peter Samuelson relating the hall algebra of the fukaya category of a surface to a certain skein algebra attached to that surface.