Waterloo Differential Geometry Working Seminar

Academic Year 2020-2021

1. August 18

   Spencer Whitehead - Symmetric Instantons (Abstract)

   Ansatze to the anti-self-dual instanton equations allow one to easily generate approximately 5/8ths of the solutions at any given charge, but to find the remaining instantons is a more difficult problem. An approach developed by Sutcliffe at the beginning of the 21st century is to impose symmetry conditions on the gauge potential of an instanton. In the case of a SU(2) instanton, we will see how these symmetry conditions can be used with the ADHM construction to turn the problem into one of the representation theory of the symmetry group, and how one can use this theory to construct instantons having the symmetry of the dodecahedron and rhombitruncated icosidodecahedron of lower charges than the corresponding JNR instanton. We will then see how the theory is affected when the structure group is changed from SU(2) to SU(N), and the symmetry group is changed from a discrete subgroup of SO(3) to a discrete subgroup of SO(4). Finally, we will look at how in this setting, the equivariant index theorem can be used to derive a class of 'integrality theorems' restricting the instanton solutions.

2. August 4

   Caleb Suan - Monogenic Forms and Decomposition of Spinor-Valued Forms (Abstract)

   In this talk, we follow the text by Delanghe, Sommen, and Souček and demonstrate a splitting of the space of spinor-valued forms. We describe some important operators and their identities as well as properties of the exterior covariant derivative when restricted to certain subspaces of spinor-valued forms. Time permitting, we will define monogenic forms and some of their homological properties.

   Nicholas Kayban - Riemannian Submersions and the O'Neill Tensors (Abstract)

   In an introductory Riemannian geometry course, one typically encounters the Euler, Gauss, and Codazzi equations, which relate the curvature of a submanifold to the curvature of the ambient manifold via the second fundamental form. The O'Neill equations are analogous equations for the case of a Riemannian submersion. In this talk we define Riemannian submersions and discuss the Fubini Study metric on $CP^n$ as an example. We also consider a vector bundle $E$ over a Riemannian manifold $(M,g)$ where the $E$ is endowed with a Riemannian metric induced from a fibre metric on $E$, a connection on $E$, and the Riemannian metric $g$ on $M$, such that the projection is a Riemannian submersion. The O'Neill tensors are defined, and we state the fundamental equations. We determine the O'Neill tensors of the Fubini-Study metric and the Riemannian metric on $E$. The O'Neill tensors are then applied to show that the sectional curvature of the Fubini-Study metric is bounded between 1 and 4.

3. July 28

   Max Chemtov - Subalgebras of so(7) Induced by the Coassociative 4-Form on R^7 (Abstract)

   Consider the 2-forms on R^7 resulting from evaluating the coassociative 4-form on two vectors. It turns out that, when such vectors are restricted to an associative 3-plane P, these 2-forms along with those in Λ^2(P) give a subalgebra of so(7) isomorphic to so(3) ♁ so(3). After giving some brief context, we will discuss this fact and characterize the intersections of such subalgebras for different choices of P.

   Daren Cheng - The second variation of area of minimal surfaces in four-manifolds (Part 2) (Abstract)

   I'll continue talking about the work of Micallef and Wolfson on minimal surfaces in oriented Riemannian 4-manifolds. I'll start by recalling from my previous talk the averaged 2nd variation formula of area, and the index lower bound that follows from it under a condition on the scalar and Weyl part of the ambient curvature. I'll then explain how this curvature condition is implied by a few other more classical ones, for instance the condition that the sectional curvatures are strictly pointwise 1/4-pinched. Finally I'll talk about some applications of the index lower bound.

4. July 21

   Spiro Karigiannis - Metric compatible connections in dimension 3 (Abstract)

   Let $(M, g)$ be an oriented Riemannian manifold. If $D$ is a $g$-compatible connection on $TM$, then the difference $D - \nabla$, where $\nabla$ is the Levi-Civita connection of $g$, is uniquely determined by the torsion $T$ of $D$. The Ricci curvature $F_{ij}$ of $D$ is in general not symmetric. Its skew part can be expressed in terms of the torsion $T$ and its covariant derivative $\nabla T$. In dimension $3$, we can further exploit the fact that $\Lambda^2 T^* M \cong T^* M$ via the Hodge star to express the torsion as a $2$-tensor on $M$. Moreover, in dimension $3$, even if $T \neq 0$, the curvature $4$-tensor $F_{ijkl}$ of $D$ is still completely determined by the Ricci tensor $F_{ij}$. I will explain these various facts and briefly discuss why I am interested in such objects.

5. July 14

   Daren Cheng - The second variation of area of minimal surfaces in four-manifolds (Abstract) (Zoom Info)

   I'll talk about the work of Micallef-Wolfson (Math. Ann. '93) on closed, immersed, oriented minimal surfaces in oriented Riemannian 4-manifolds, focusing on the result where they obtain a topological lower bound on the index plus nullity of such minimal surfaces under a condition involving the scalar curvature and Weyl tensor of the ambient space. The main ingredient is an "averaged" version of the second variation formula of the area functional, which I'll derive at the beginning. I'll then explain how the aforementioned curvature condition arises from this formula and mention a few situations where it is known to hold. Finally I'll describe how the index+nullity lower bound follows with the help of the Riemann-Roch theorem, applied to the normal bundle.
   
   Meeting ID: 958 7361 8652
   Passcode: 577854

6. July 7

   Aidan Patterson - Local Normal Forms for Hamiltonian Actions of Poisson-Lie Groups (Abstract) (Zoom Info)

   In this talk, we'll discuss what it means for a Poisson-Lie group to act Hamiltonianly on a symplectic manifold, the bridge between classical and Poisson-Lie Hamiltonian actions, and the analogue to the classical local normal form theory for Hamiltonian actions of Poisson-Lie groups on symplectic manifolds. We will restrict our attention to compact Poisson-Lie groups, where these constructions can be made explicit enough to provide a class of computable (in theory) examples. This talk is based on a project supervised by Dr. Harada and Dr. Lane at McMaster University.
   
   Meeting ID: 846 7793 5521
   Passcode: 780946

7. June 30

   Shengda Hu - Curvature of generalized holomorphic bundles (Abstract) (Zoom Info)

   We continue with the discussion on generalized connections on a Riemannian manifold. We will discuss properties of curvatures on generalized holomorphic vector bundles over a generalized Kähler manifold and generalized analogues of classical notions.
   
   Meeting ID: 846 7793 5521
   Passcode: 780946

8. June 23

   Shengda Hu - Curvature for connections in generalized geometry (Abstract) (Zoom Info)

   We continue with the discussion on generalized connections on a Riemannian manifold, discuss the curvature identities and generalize holomorphic bundle over a generalized Kähler manifold.
   
   Meeting ID: 958 7361 8652
   Passcode: 577854

9. June 16

   Spiro Karigiannis - Decomposition of curvature tensor for metrics with torsion, continued (Abstract)

   Last time we reviewed the classical decomposition of the Riemann curvature tensor into scalar, traceless Ricci, and Weyl curvature. This time we will examine special features in dimensions 3 and 4. Then I will consider the more general case of a metric compatible connection with torsion, and see how this decomposition generalizes.

10. June 2

    Spiro Karigiannis - Decomposition of curvature tensor for metrics with torsion (Abstract)

    I will first review the classical decomposition of the Riemann curvature tensor into scalar, traceless Ricci, and Weyl curvature, with an emphasis on special features in dimensions 3 and 4. Then I will consider the more general case of a metric compatible connection with torsion, and see how this decomposition generalizes.

11. May 12

    Shengda Hu - Some computations for connections in generalized geometry (Abstract)

    We look at generalized connections on a Riemannian manifold. We will consider curvature in generalized geometry and look to extend classical computations to the generalized situation.

12. April 7

    Da Rong Cheng - Non-minimizing solutions to the Ginzburg-Landau equations (Abstract)

    Abstract: I'll talk about the very recent paper by Ákos Nagy and Gonçalo Oliveira, where they use two different methods, one variational and the other perturbative, to construct new examples of non-minimizing critical points of the Ginzburg-Landau functional on Hermitian line bundles over closed Riemannian manifolds. I'll begin with some necessary background on the functional and its Euler-Lagrange equations. Then I'll focus on the second method in the Nagy-Oliveira paper, which uses the Lyapunov-Schmidt reduction and applies in particular to closed manifolds of any dimension with trivial first cohomology.
    Reference  

13. March 24

    Christopher Lang - On the charge density and asymptotic tail of a monopole (Abstract)

    We follow Harland and Nogradi's 2016 paper [1] where they define an abelian magnetic charge density for non-abelian monopoles. This agrees asymptotically with the conventional charge distributions but is smooth inside of the monopole. We then show the relationship between this charge density and the tail of a monopole, as given by Hurtubise. Finally, we show how this charge density can be obtained from the Nahm data of the monopole directly.
    Reference  

14. March 10

    Anton Iliashenko - Harmonic Mappings (Abstract)

    For any smooth mapping between Riemannian manifolds, one can associate a variety of invariantly defined functionals. Consider the energy functionals E, which are of great geometrical and physical interest. We will examine the extremals of E, interpreted as the zeroes of the Euler-Lagrange equation associated with E. Special cases of these extremal mappings include geodesics, harmonic maps, etc. The talk is based on the first chapter of “Harmonic Mappings of Riemannian Manifolds” by James Eells, Jr. and J. H. Sampson

15. March 3

    Spiro Karigiannis - Variational characterization of instanton-submanifolds (Abstract)

    There are two classes of calibrated submanifolds. The "instantons" are those whose tangent spaces are invariant under a vector cross product. In the U(m) and G2 cases, we show that these submanifolds can be characterized as being critical with respect to variations of the ambient metric in the direction of closed forms. The Spin7 case seems to be different. This is a continuation of the last talk I gave, based on forthcoming joint work with Daren Cheng and Jesse Madnick.

16. February 24

    Christopher Lang - The Many Faces of Monopoles (Abstract)

    In this talk, we introduce the four ways of looking at monopoles: solutions of the Bogomolny equations, Nahm data, spectral curves, and rational maps. We then discuss the relationships between these equivalent descriptions and some of the advantages and disadvantages of using them.

17. February 10

    Ragini Singhal - Six dimensional nearly Kähler manifolds of Cohomogeneity one (Abstract)

    We will discuss a paper by Podesta-Spiro where the authors consider six-dimensional strict nearly Kähler manifolds acted on by a compact, cohomogeneity one automorphism group G. We will see how they classify the compact manifolds of this class up to G-diffeomorphisms.
    Reference  

18. February 3

    Caleb Suan - Pinching Estimates and Eigenvalues of Curvature Operators (Abstract)

    In this talk, we will go through various results from the paper “Curvature Operators: Pinching Estimates and Geometric Examples” by Bourguignon and Karcher, which relate bounds on sectional curvatures of manifolds to eigenvalues of curvature operators.
    Reference  

19. January 20

    Spiro Karigiannis - Isometric immersions which are minimal with respect to special variations of ambient metric (Abstract)

    Let L be a fixed compact oriented submanifold of a manifold M. Consider the volume functional of L with respect to variations of an ambient Riemannian metric on M. It is easy to show that with respect to general variations, there are no critical points. However, if (M, g) has additional extra structure, then with respect to a particular special class of metric variations, there can be critical points. I will discuss a result of Arezzo-Sun in this context concerning complex submanifolds of a Kahler manifold. In ongoing work with Daren Cheng and Jesse Madnick, we have generalized this result to both the non-integrable setting and to G₂-structures. Surprisingly, the Spin(7) analogue is false.

20. January 13

    Ragini Singhal - New G₂-holonomy cones and exotic nearly Kähler structures on S⁶ and S³ × S³ (Abstract)

    In this influential paper by Lorenzo Foscolo and Mark Haskins the authors prove the existence of the first complete inhomogeneous nearly Kähler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kähler structure on the 6-sphere and on the product of a pair of 3-spheres. They also conjecture that these are the only simply connected (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions. In this talk we will discuss a basic introduction and techniques used in the paper. We will talk about some geometric structures and their relationship and also see the various steps involved in proving the result.
    Reference