OLGA is an online workshop targeted at anyone interested in gauge theory, primarily early career researchers.

The scheduled events indicated below will take place on Zoom, and there will be a virtual break-out space hosted on the web app Gather, open for use throughout the day. Access information will be distributed to registered participants before the event via email. Registration for this event is now closed, please contact the organisers if you are unregistered but still want Zoom access.

OLGA seeks to increase both the visibility and participation of under-represented groups in the gauge theory community: we particularly encourage female early career researchers in gauge theory or a related area to join the workshop and to consider presenting a research poster, along with any others those who might not have considered doing so otherwise. Please get in touch with the event organisers if you feel unsure, or have further questions or comments.

Schedule 24th June (British Summer Time BST/Eastern Daylight Time EDT):

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15.00-16.00 BST
10.00-11.00 EDT

"What do you do?" poster presentations:
(authors must prepare a .pdf poster in advance to be displayed on Zoom)

John McCarthy (Imperial College London, poster)
Christopher Lang (University of Waterloo, poster)
Jaime Mendizabal (University College London, poster)
Guillermo Gallego (Universidad Complutense de Madrid, poster)
Andries Salm (University College London, poster)
Marcel Bigorajski (Georg-August University Gottingen, poster)
Matthew Magill (Uppsala University, poster)

16.00-16.15 BST
11.00-11.15 EDT

Break (short)

16.15-17.00 BST
11.15-12.00 EDT

Daniel Platt (Imperial College London):

Gluing constructions for instantons.

Abstract: Instantons are principal bundle connections with good curvature properties and many examples are constructed using gluing techniques. Gluing constructions are always done in three steps; first, an approximate solution is constructed; second, an estimate for the linearisation of the instanton operator is found; third, iteratively solving the linearised problem gives a solution to a non-linear equation which perturbs the approximate solution to a genuine solution. In the talk, I will showcase this three step process in three examples: Taubes's classical construction for ASD instantons; the construction of G2-manifolds by Joyce-Karigiannis; and my work in progress on constructing G2-instantons on the aforementioned G2-manifolds.

17.00-17.30 BST
12.00-12.30 EDT

Break

17.30-18.15 BST
12.30-13.15 EDT

Hector Papoulias (University Of Oxford)

Spin(7) Instantons and Hermitian-Yang-Mills Connections for the Stenzel Metric.

Abstract: The Spin(7) and SU(4) structures on a Calabi-Yau 4-fold give rise to certain first order PDEs defining special Yang-Mills connections: the Spin(7) instanton equations and the Hermitian Yang-Mills (HYM) equations respectively. The latter are stronger than the former. In 1998 C. Lewis proved that -over a compact base space- the existence of an HYM connection implies the converse statement. In this talk we demonstrate that the equivalence of the two gauge-theoretic problems fails to hold in generality. We do this by studying the invariant solutions on a highly symmetric noncompact Calabi-Yau 4-fold: the Stenzel manifold. We give a complete description of the moduli space of irreducible invariant Spin(7) instantons with structure group SO(3) and find that the HYM connections are properly embedded in it. This moduli space reveals an explicit example of a new removable singularity phenomenon: a sequence of Spin(7) instantons bubbles off near a Cayley submanifold and -after a topology change- converges to an HYM connection. This reveals a potential relationship between the two equation systems.

18.15-18.30 BST
13.15-13.30 EDT

Break (short)

18.30-19.15 BST
13.30-14.15 EDT

Mateo Galdeano (University Of Oxford)

G2 instantons in Physics: the heterotic G2 system

Abstract: The heterotic G2 system is the 7-dimensional analogue of the Strominger-Hull system. Solutions to this system describe compactifications of heterotic string theory with N=1 supersymmetry on a 7-dimensional manifold with a G2-structure. One of the main ingredients in this construction is a G2-instanton on the 7-dimensional manifold satisfying some additional constraints imposed by Physics. In this talk I will introduce the mathematical framework of the heterotic G2 system and motivate how higher-dimensional gauge theory helps in the task of solving the system. In particular I will discuss some of the existing solutions in the literature as well as presenting new solutions involving homogeneous 3-Sasakian manifolds. This talk is based on recent work with Xenia de la Ossa.

Schedule 25th June (British Summer Time BST/Eastern Daylight Time EDT):

15.00-15.45 BST
10.00-10.45 EDT

Laura Schaposnik (University of Illinois at Chicago):

On generalized hyperpolygons.

Abstract: In this talk we will introduce generalized hyperpolygons, which arise as Nakajima-type representations of a comet-shaped quiver, following recent work joint with Steven Rayan. After showing how to identify these representations with pairs of polygons, we shall associate to the data an explicit meromorphic Higgs bundle on a genus-g Riemann surface, where g is the number of loops in the comet. We shall see that, under certain assumptions on flag types, the moduli space of generalized hyperpolygons admits the structure of a completely integrable Hamiltonian system.

15.45-16.00 BST
10.45-11.00 EDT

Break (short)

16.00-17.00 BST
11.00-12.00 EDT

Xuemiao Chen (University of Maryland):

Singularities of Hermitian-Yang-Mills connections.

Abstract: After introducing some background about stable bundles and HYM connections, I will explain both the analytic and algebraic sides when studying singularities of HYM connections. It turns out that local algebraic invariants can be extracted to characterize the analytic side. In particular, the analytic tangent cone is an algebraic invariant. Based on joint works with Song Sun.

17.00-17.30 BST
12.00-12.30 EDT

Break

17.30-18.15 BST
12.30-13.15 EDT

Daniel Fadel (University of Peking):

Large mass monopoles on asymptotically conical 3-manifolds.

Abstract: I will consider the problem of the limiting behavior of sequences of monopoles of fixed charge and arbitrarily large masses (unbounded Yang-Mills-Higgs energies) on an asymptotically conical 3-manifold. I will start by reviewing joint work with Gonçalo Oliveira on the bubbling behavior of such sequences and then I will explain my ongoing project of further understanding the convergence problem outside the blow-up locus. Time permitting, I will mention some features of the corresponding problem in 7 dimensions and a few other future directions.

18.15-18.30 BST
13.15-13.30 EDT

Break (short)

18.30-19.15 BST
13.30-14.15 EDT

Alessandro Pigati (Courant Institute, New York University):

Yang-Mills-Higgs for Hermitian line bundles and codimension-two area.

Abstract: Minimizers (or more generally critical points) of gauge-invariant energies on a manifold are very often linked to other intrinsic geometric and topological objects. Inspired by earlier work by Jaffe-Taubes and Hong-Jost-Struwe (for the surface case), and by recent results on the scalar and vectorial Ginzburg-Landau energy, we show how the self-dual Yang-Mills-Higgs functional (for a Hermitian line bundle with structure group U(1), on any closed Riemannian manifold) provides an effective relaxation of the codimension-two area of submanifolds. Namely, zero sets of critical couples (section,connection) of rescalings of this functional converge to minimal, codimension-two submanifolds. If time allows, we will also mention some variational applications. This is based on a joint work with Daniel Stern and a more recent one with Daniel Stern and Davide Parise.

Organisers: Izar Alonso Lorenzo (University of Oxford, izar.alonsolorenzo@maths.ox.ac.uk ), Jakob Stein (University College London, jakob.stein.19@ucl.ac.uk)