Academic Year 2020-2021

1. 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.
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2. 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.
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3. 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

4. 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.

5. 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.

6. 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.
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7. 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.
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8. 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.

9. 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.
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2019-2020 Schedule

2018-2019 Schedule