Algebraic and Analytic Methods in Gauge Theory (4-6 September 2024)

All talks will take place in the "Topos" in the Department of Mathematics (Directions to the Topos).

Wednesday 4 September

Time		Title
13:00-13:15	Arrival and Coffee in the Topos
13:15-14:15	Eloise Hamilton (Cambridge)	Moduli spaces for not necessarily semistable Higgs bundles
14:15-14:30	Coffee and discussion
14:30-15:30	Fatemeh Rezaee (Cambridge)	Smoothings of stable maps
15:30-16:00	Coffee and discussion
16:00-17:00	Oscar Garcia-Prada (ICMAT)	Kähler-Yang-Mills equations and gravitating vortices

Thursday 5 September

Time		Title
09:30-09:45	Arrival and Coffee in the Topos
09:45-10:45	Peter Gothen (Porto)	The conformal limit and projective structures
10:45-11:00	Coffee and discussion
11:00-12:00	Ana Peon-Nieto (Santiago de Compostela)	Classification of very stable Higgs bundles
12:00-15:15	Lunch and discussion
15:15-16:15	Hartmut Weiß (Kiel)	Singular Solutions to Hitchin’s Equation and Harmonic Maps to the Conformal 3-Sphere
16:15-16:30	Coffee and discussion
16:30-17:30	Jaime Mendizabal (UCL)	Constructing monopole moduli spaces with the b and scattering calculuses

Friday 6 September

Time		Title
09:15-09:30	Arrival and Coffee in the Topos
09:30-10:30	Stuart Hall (Newcastle)	Rigidity of 'odd' complex Grassmannians and some other symmetric spaces
10:30-10:45	Coffee and discussion
10:45-11:45	Nuno Romao (IHES)	Vortex moduli in toric fibre bundles
11:45-12:00	Coffee and discussion
12:00-13:00	Andrew Dancer (Oxford)	Implosion and contraction constructions in hyperkähler geometry
13:00-	Lunch and discussion

Abstracts

13:15, 4 September Eloise Hamilton (Cambridge)
Title. Moduli spaces for not necessarily semistable Higgs bundles
Abstract. The stack of Higgs bundles is a geometric object that parametrises all Higgs bundles. Amongst these are a special class of Higgs bundles, the semistable ones, corresponding to Higgs bundles that give rise to solutions to Hitchin's equations. A moduli space for semistable Higgs bundles can be constructed algebraically using Geometric Invariant Theory (GIT), or analytically using gauge-theoretic methods. This talk is guided by the following question: can moduli spaces also be constructed for Higgs bundles that are not necessarily semistable? The answer I will give is that this can be done algebraically using Non-Reductive GIT, after stratifying the stack of Higgs bundles. The question of whether or not these moduli spaces can be constructed analytically is still open.

14:30, 4 September Fatemeh Rezaee (Cambridge)
Title. Smoothings of stable maps
Abstract. To construct a compact moduli space smaller than the entire space of stable maps, it is natural to ask which stable maps appear as limits of a one-parameter family of maps with smooth domain. In this talk, based on joint work with Mohan Swaminathan, I will partially answer this question by giving (separate) sufficient and necessary conditions for a stable map to be smoothable.

16:00, 4 September Oscar Garcia-Prada (ICMAT)
Title. Kähler-Yang-Mills equations and gravitating vortices
Abstract. I will introduce the Kähler-Yang-Mills equations on a holomorphic vector bundle over a compact complex manifold. These equations, inspired by the Donaldson-Uhlenbeck-Yau theorem for bundles, and the Yau–Tian–Donaldson conjecture for constant scalar curvature Kähler metrics, intertwine the curvature of a Hermitian Yang–Mills connection on the bundle and the scalar curvature of a Kähler metric on the manifold. After this, I will consider special symmetric solutions on a compact Riemann surface known as gravitating vortices.

09:45, 5 September Peter Gothen (Porto)
Title. The conformal limit and projective structures
Abstract. The non-abelian Hodge correspondence maps a polystable \(SL(2,R)\)-Higgs bundle on a compact Riemann surface \(X\) of genus \(g>1\) to a connection which, in some cases, is the holonomy of a branched hyperbolic structure. On the other hand, Gaiotto's conformal limit maps the same bundle to a partial oper, i.e., to a connection whose holonomy is that of a branched complex projective structure compatible with \(X\). I shall explain how these are both instances of the same phenomenon: the family of connections appearing in the conformal limit can be understood as a family of complex projective structures, deforming the hyperbolic ones into the ones compatible with \(X\). Moreover, when the Higgs bundle has zero Toledo invariant, this deformation is optimal, inducing a geodesic in the Teichmüller metric.
This is joint work with Pedro M. Silva.
Partially supported by CMUP (UIDB/00144/2020 and UIDP/00144/2020) funded by FCT (Portugal) with national funds.

11:00, 5 September Ana Peon-Nieto (Santiago de Compostela)
Title. Classification of very stable Higgs bundles
Abstract. The seminal work of Hausel and Hitchin provides examples of dual BBB and BAA branes on the moduli space of Higgs bundles, the latter being given by upward flows from very stable regular nilpotent Higgs bundles. I will speak about ongoing work on the classification of components of the global nilpotent cone into very stable and wobbly, which are respectively those components containing or not a very stable Higgs bundle. I will show that, essentially, the only very stable Hodge bundles correspond to the regular (by Hausel—Hitchin's work) and subregular nilpotent orbits. Time permitting, I will analyse some invariants (the virtual equivariant multiplicities defined by Hausel and Hitchin) involved in the geometry of the dual BBB branes of very stable subregular upward flows, which moreover provide an obstruction to the existence of very stable points in a given component.

15:15, 5 September Hartmut Weiß (Kiel)
Title. Singular Solutions to Hitchin’s Equation and Harmonic Maps to the Conformal 3-Sphere
Abstract. I will report on ongoing work with Sebastian Heller and Lothar Schiemanowski on solutions to Hitchin’s equation which are singular along a system of simple closed curves on the underlying Riemann surface. These are constructed by gluing methods. We use them to construct harmonic maps to the conformal 3-sphere with a specified behaviour when passing through the sphere at infinity. Earlier examples of such were obtained by Heller and Heller.

16:30, 5 September Jaime Mendizabal (UCL)
Title. Constructing monopole moduli spaces with the b and scattering calculuses
Abstract. For the construction of the moduli spaces of framed monopoles from a differential-geometric point of view we must study PDEs on a non-closed manifold, which involves a careful analysis of the asymptotics. We will explain some analytic techniques which can be used to take into account the asymptotic behaviour of monopoles along different subbundles and obtain a hyper-Kähler structure on the moduli space, as well as regularity and decay properties of the monopoles themselves.

09:30, 6 September Stuart Hall (Newcastle)
Title. Rigidity of 'odd' complex Grassmannians and some other symmetric spaces
Abstract. I will discuss recent work with Paul Schwahn and Uwe Semmelmann where we demonstrate that the canonical Einstein metric on complex Grassmannians of \(k\)-planes in \(\mathbb{C}^n\) is rigid (isolated in the space of all Einstein metrics) if \(n\) is odd. Time permitting, I will discuss outstanding rigidity questions for other symmetric spaces having \(SU_n\) as the isometry group.

10:45, 6 September Nuno Romao (IHES)
Title. Vortex moduli in toric fibre bundles
Abstract. I will describe moduli spaces of abelian vortices, targeted in any compact Kähler toric manifold \(X\), on a compact Kähler manifold of arbitrary complex dimension. A key step for this result is the proof of a 20-year-old conjecture by J. Baptista (established by him and other authors for \(X=\mathbb{P}^n\)) concerning an identification of moduli between linear and nonlinear vortices, which I shall explain. This is joint work with Marcel Bökstedt.

12:00, 6 September Andrew Dancer (Oxford)
Title. Implosion and contraction constructions in hyperkahler geometry
Abstract. We discuss the implosion and contraction constructions in real symplectic geometry, and how they can be extended to the hyperkähler situation. In this talk we emphasise links with non-reductive geometric invariant theory and the Moore-Tachikawa category.

Last updated 3 September, 2024