Differential geometry and symplectic topology seminar (Spring 2022)

Abstracts

• Zhenhua Liu
  • Calibrated surfaces appear naturally in many geometrical contexts, like holomorphic varieties or special Lagrangians. They are also area-minimizing, thus serving as the prime examples in geometric measure theory. However, just recall that a sequence of smooth holomorphic curves can converge to one with self-intersections and branch points. Singularities appear naturally when one considers moduli spaces of calibrated surfaces. Except for the 2-d case, where all area-minimizing surfaces are branched minimal immersions, we know very little beyond the codimension two Hausdorff dimension bound. For example, can the singular set of a 3-dimensional calibrated surface be the Cantor set? In this talk, we will discuss some recent constructions, and answer this question affirmatively. In fact, we can construct 3-d surfaces calibrated by smooth 3-forms on smooth compact Riemannian manifolds, so that the singular sets can be prescribed to be any closed subset of a closed interval, or even any closed subset of any finite combinatorial graph that contains at least some neighborhood of each vertex. Thus, any real number between 0 and 1 can be realized as the dimension of the singular set of a 3-d calibrated surface, and we can have a Cantor set worth of singularity. In general dimensions, we provide a sharp answer to a conjecture by Almgren.
• Jiayin Pan
  • We start with a simple phenomenon in an open manifold M with nonnegative Ricci curvature: the set of all minimal representing geodesic loops of pi_1(M,p) may not be contained in any bounded sets of M. We will talk about how this escape phenomenon is related to the group structure of fundamental groups and the metric structure of asymptotic cones. Part of this talk is joint work with Guofang
• Andrew Sageman-Furnas
  • Which data determine an immersed surface in Euclidean three-space up to rigid motion? A generic surface is locally determined by only a metric and mean curvature function. However, there are exceptions. These may arise in a family, like the isometric family of vanishing mean curvature surfaces transforming a catenoid into a helicoid, or as two surfaces called a Bonnet pair.
  • For compact surfaces, Lawson and Tribuzy proved in 1981 that a metric and non-constant mean curvature function determine at most one immersion with genus zero, but at most two compact immersions (compact Bonnet pairs) for higher genus. This led them to ask if compact Bonnet pairs exist.
  • In this talk, we discuss our recent construction of the first examples of compact Bonnet pairs. It uses a local classification by Kamberov, Pedit, and Pinkall in terms of special surfaces that are called isothermic. Moreover, we describe how a structure-preserving discrete theory for isothermic surfaces and Bonnet pairs led to this discovery.
  • The smooth theory is joint work with Alexander Bobenko and Tim Hoffmann and the discrete theory is joint work with Tim Hoffmann and Max Wardetzky.
• Peng Zhou
  • Just as one can compute the cohomology of a smooth manifold using Morse theory with different Morse functions, one can compute the partially wrapped Fukaya category of a Weinstein pair using microlocal sheaf theory with different Lagrangian skeletons. In the setup of mirror symmetry for toric GIT quotient, we give an explicit interpolation between various Lagrangian skeleton using the notion of windows subcategory in VGIT. This is based on works https://arxiv.org/abs/2011.03719 and https://arxiv.org/abs/2011.06114 (with Jesse Huang).
• Zhichao Wang
  • Recently, X. Zhou proved that the Almgren-Pitts min-max solution has multiplicity one for bumpy metrics (Multiplicity One Theorem). In this talk, we exhibit the first set of examples of non-bumpy metrics on the $(n+1)$-sphere ($2\leq n\leq 6$) in which the varifold associated with the two-parameter min-max construction must be a multiplicity-two minimal $n$-sphere. This is proved by a new area-and-separation estimate for certain minimal hypersurfaces with Morse index two inspired by an early work of Colding-Minicozzi. This is a joint work with X. Zhou.
• Alex Waldron
  • I’ll describe some recent work on 2D harmonic map flow, in which I show that a familiar bound on the blowup rate at a finite-time singularity is sufficient for continuity of the body map. This is relevant to a conjecture of Topping.
• Gleb Smirnov
  • Symplectomorphism groups may be considered intermediate objects between Lie groups of isometries and full diffeomorphisms. A central question about the relationship between diffeomorphisms and symplectomorphisms is to describe the smoothly trivial symplectic mapping class group, that is, the group of isotopy classes of symplectomorphisms that are smoothly isotopic to the identity. After a short introduction to symplectic mapping class groups, I will explain how to use Seiberg-Witten theory to get information about them. In particular, I will prove that the smoothly trivial symplectic mapping class groups of many K3 surfaces are infinitely generated, thus extending a recent result of Sheridan and Smith.
• Laura Starkston
  • We will discuss how to present the symplectic topology of the complement of a rational cuspidal curve in a closed symplectic 4-manifold. Even when we know abstractly that the complement of a divisor has a Stein structure, we can ask what is an explicit Stein handlebody diagram or Lefschetz fibration for that complement? This is a difficult question in general. Here we discuss a scenario where we can find explicit Stein handlebody diagrams. This is joint work with Marco Golla.
• Christos Mantoulidis
  • The p-widths of a closed Riemannian manifold are a nonlinear analogue of the spectrum of its Laplace–Beltrami operator, which was defined by Gromov in the 1980s and corresponds to areas of a certain min-max sequence of hypersurfaces. By a recent theorem of Liokumovich–Marques–Neves, the p-widths obey a Weyl law, just like the eigenvalues do. However, even though eigenvalues are explicitly computable for many manifolds, there had previously not been any >= 2-dimensional manifold for which all the p-widths are known. In recent joint work with Otis Chodosh, we found all p-widths on the round 2-sphere and thus the previously unknown Liokumovich–Marques–Neves Weyl law constant in dimension 2.
• Mark McLean
  • Consider a smooth submersion from a symplectic manifold P to the complex line with symplectic fibers. Then we prove that the cohomology of P over the integers is additively isomorphic to the cohomology of the fiber times the base. More generally, we prove such an isomorphism holds with respect to any complex oriented cohomology theory, such as complex cobordism. These results are new even in the special case of smooth projective morphisms to the complex line. To prove our result we use Morava K theories, which are generalized cohomology theories approximating cohomology over the integers modulo each prime power and which admit virtual fundamental classes. Our proof also contains a new construction of a global Kuranishi chart for the moduli space of curves. This is joint work with Abouzaid and Smith.
• Jonathan Zhu
  • Waists and widths measure the size of a manifold with respect to measures of families of submanifolds. We’ll discuss related area estimates for minimal submanifolds, as well as applications to quantitative symplectic camels.
• Martin Lesourd
  • The study of Positive Scalar Curvature (PSC) is rich in open problems. I’ll discuss two of these: ‘‘what topologies are compatible with PSC?’’, and ‘‘what is the geometry of PSC?’’. Both of these are related to the Positive Mass Theorem, which I’ll also discuss. The basic set of results I’ll present concerns the removal of the compactness assumption that has been more or less ubiquitous in studies on the topology and geometry of PSC. Without compactness, many of the classic techniques pioneered in the 1970s-80s don’t seem to work. Spearheaded by a flurry of papers on PSC by M.Gromov, a certain variant of the classic geometric technique of Schoen-Yau 1979 has come to the fore and allowed generalizations of several classic results which I’ll discuss. This is based on joint work over 4 papers, 2 with R.Unger and S-T. Yau, and 2 with R.Unger and D.Lee.