Events

Carlos Valero, McGill University

The Calderón problem refers to the question of whether one can determine the Riemannian metric on a manifold with boundary from its "Dirichlet-to-Neumann (DN) map", which maps a function on the boundary to the normal derivative of its harmonic extension. In this talk, we define the analogue of the DN map for the spinor Laplacian twisted by a unitary connection and show that it is a pseudodifferential operator of order 1, whose symbol determines the Taylor series of the metric and connection at the boundary. We go on to show that if all the data are real-analytic, then the spinor DN map determines the connection modulo gauge.

MC 5417

Ákos Nagy, BEIT Canada

Motivated by recent theoretical and experimental developments in the physics of hyperbolic crystals, I will introduce the noncommutative Bloch transform for Fuchsian groups which I will call the hyperbolic Bloch transform (HBT). The HBT transforms wave functions on the hyperbolic plane to sections of irreducible, flat, Hermitian vector bundles over the orbit space and transforms the hyperbolic Laplacian into the covariant Laplacian. I will prove that the HBT is injective and “asymptotically unitary”. If time permits, I will talk about potential applications to hyperbolic band theory. This is a joint work with Steve Rayan (arXiv:2208.02749).

MC 5417

Guillermo Gallego, Universidad Complutense de Madrid

Multiplicative Higgs bundles are a natural analogue of Higgs bundles on Riemann surfaces, where the Higgs field now takes values on the adjoint group bundle, instead of the adjoint Lie algebra bundle. In the work of Charbonneau and Hurtubise, they have been related to singular monopoles over the product of a circle with the Riemann surface.

In this talk we study the natural action of an involution of the group on the moduli space of multiplicative Higgs bundles, also from the point of view of monopoles. This provides a "multiplicative analogue" of the theory of Higgs bundles for real groups.

MC 5403

Andy Royston, Penn State University

The extended Bogomolny equations are a system of PDE's for a connection and a triplet of Higgs fields on a three-dimensional space. They are a hybrid of the Bogomolny equations and the Nahm equations. After reviewing how these latter systems arise in the study of magnetic monopoles, I will present an energy functional for which solutions of the extended Bogomolny equations are minimizers in a fixed topological class. In this construction, the connection and Higgs triplet are defined on all of R^3 and couple to additional dynamical fields localized on a two-plane that are analogous to jumping data in the Nahm equations. Solutions can therefore be interpreted as finite-energy BPS solitons in a three-dimensional theory with a planar defect. This talk is based on work done in collaboration with Sophia Domokos.

MC 5417

Charles Cifarelli, CIRGET & Stony Brook

I will present recent joint work with V. Apostolov on a new construction of complete steady gradient Kähler-Ricci solitons on C^n, using the theory of hamiltonian 2 forms, introduced by Apostolov-Calderbank-Gauduchon-Tønnesen-Friedman, as an Ansatz. The metrics come in families of two types with distinct geometric behavior, which we call Cao type and Taub-NUT type. In particular, the Cao type and Taub-NUT type families have a volume growth rate of r^n and r^{2n-1}, respectively. Moreover, each Taub-NUT type family contains a codimension 1 subfamily of complete Ricci-flat metrics.

MC 5417

François Greer, Michigan State University

An old question going back to S.T. Yau asks whether there are finitely many diffeomorphism types for smooth projective Calabi-Yau manifolds of a given dimension. The answer is affirmative for dimensions one and two (elliptic curves and K3 surfaces). It has recently been settled for Calabi-Yau threefolds admitting elliptic fibrations. We discuss the case of CY3’s admitting abelian surface or K3 fibrations. 

MC 5417

Alex Waldron, University of Wisconsin-Madison

Given a principal bundle over a compact Riemannian 4-manifold or special-holonomy manifold, it is natural to ask whether a uniform gap exists between the instanton energy and that of any non-minimal Yang-Mills connection. This question is quite open in general, although positive results exist in the literature. We'll review several of these gap theorems and strengthen them to statements of the following type: the space of all connections below a certain energy deformation retracts (under Yang-Mills flow) onto the space of instantons. As applications, we recover a theorem of Taubes on path-connectedness of instanton moduli spaces on the 4-sphere, and obtain a method to construct instantons on quaternion-Kähler manifolds with positive scalar curvature.

MC 5417