Hung Tran: The curvature of the second kind and a conjecture of Nishikawa

• Hung Tran (Texas Tech)

The Riemannian curvature tensor can be seen as acting on either the space of skew-symmetric or symmetric maps giving rise to different operators. The classical theory focuses mostly on the former while the latter only received sporadic attention. In this talk, we'll investigate manifolds for which the curvature of the second kind satisfies certain positivity conditions. It is somewhat surprising that it is closely related to the notion of isotropic curvature which has far-reaching applications in geometry and topology. Our highlighted result is the resolution of a differentiable sphere conjecture proposed by Nishikawa in 1986. This is joint work with Matthew Gursky and Xiaodong Cao.

Mauricio Che: Ends of spaces with lower curvature bounds

• Mauricio Che (Durham University)

The ends of a space are the connected components of its ideal boundary. Under certain curvature conditions, it is possible to give uniform bounds for the number of ends of Riemannian manifolds. In this talk I will recall previous work by Z.-D. Liu in this direction and show a generalization of this result in the setting of metric measure spaces satisfying the curvature dimension condition CD(0,N) outside a compact set. This is joint work with Jesús Núñez-Zimbrón. Preprint: https://arxiv.org/abs/2108.10659

Cordelia Webb: Equivariant Minimal Surfaces in Complex Hyperbolic Space

• Cordelia Webb (University of York)

The moduli space of equivariant minimal surfaces in complex hyperbolic space over a compact surface X, acquires an analytic structure via an embedding into the direct product of the Teichmuller space of X and the character variety of the fundamental group of X. This latter space is homeomorphic to the moduli space of PU(n, 1)-Higgs bundles. Moreover, an equivariant minimal surface admits a harmonic sequence of maps. We consider how these different perspectives can be used to further explore the structure of the moduli space of such equivariant minimal surfaces.

Ravil Gabdurakhmanov: Calderon's problem for the connection Laplacian

• Ravil Gabdurakhmanov (University of Leeds)

We consider a vector bundle equipped with a connection over a compact Riemannian manifold of dimension greater than two with boundary. We show that when all geometric data are real-analytic the topology and geometry of such a vector bundle are determined uniquely, up to a gauge equivalence, by the Cauchy data of the connection Laplacian. This result has applications to Calderon's problem for some non-linear equations.

Carolyn Gordon: (Infinitesimal) Maximal Symmetry and Homogeneous Expanding Ricci Solitons

• Carolyn Gordon (Dartmouth College)

We introduce notions of maximal symmetry and infinitesimal maximal symmetry of left-invariant Riemannian metrics on Lie groups. We (i) explore questions of existence and (ii) ask whether those left-invariant metrics with the ``nicest'' curvature properties exhibit special symmetry properties. Concerning (ii), we focus on simply-connected solvable Lie groups. In this setting, we find that left-invariant Einstein metrics are always maximally symmetric; in contrast, left-invariant Ricci solitons are always infinitesimally maximally symmetric but some are not maximally symmetric. This is joint work with Michael Jablonski.