Events

Instantons over manifolds with special holonomy are a distinguished class of connections on principal bundles. In their seminal paper from 1998 Donaldson and Thomas envisioned invariants of special holonomy manifolds based on the moduli space of such instantons. However, a rigorous construction of these invariants faces formidable challenges arising from the possible non-compactness of this moduli space. One source of non-compactness comes from the formation of non-removable singularities, that is, a sequence of instantons might converge to an instanton that is singular along a subset of the underlying manifold. In this talk, I will focus on instantons over 6-manifolds with SU(3)-structures that have isolated singularities of conical type. In particular, I will discuss their moduli space, its structure, and its virtual dimension. This talk is based on joint work with Yuanqi Wang.

MC 5417

The classification of matrices by their principal minors has been an area of interest to many since at least the 1980s. In this talk we expand on this work by outlining a proof that, if two orthogonal projection matrices have the same principal minors, then the coordinatewise absolute value of their images coincide. Generalizing the ideas of the proof gives rise to a stratification of the Grassmannian which is coarser than the well-known matroid stratification of Gelfand-Goresky-MacPherson-Serganova. We conclude this talk with some applications of this theorem to geometry.

MC 5403

We will prove the existence and uniqueness of a Euclidean SU(2)-monopole with a given charge, center, S^1 symmetry group, and axis of symmetry. This will be proven using a theorem of Hitchin regarding the equivalence of SU(2)-monopoles and the cohomological data obtained from spectral curves.

MC 5417

We will begin our discussion of the first jump inversion theorem for structures by covering some necessary background such as defining the jump of a structure and effective bi-interpretability.

MC 5301

In joint work-in-progress with Rahim Moosa, we explore the notion of internality in continuous logic and show that, whenever this phenomenon arises in a metric structure, a definable binding group arises together with it. This requires navigating some of the difficulties of what definability means in the continuous sense.

MC 5403

We discuss contact strata in the arc space of the flag variety. In the preseminar period, we briefly introduce arc spaces and discuss their relevance in birational geometry.

MC 5403