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Seminars (MAM)

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Event When Speaker Title Presentation Material
MAM 20th July 2015
14:00 to 15:00
Geometric invariant theory for graded unipotent group actions and applications
Mumford's geometric invariant theory (GIT) provides a method for constructing (projective completions of) quotient varieties for linear actions of complex reductive groups on affine and projective varieties, and has many applications (for example in the construction of moduli spaces in algebraic geometry). Mumford's GIT can be extended to actions of linear algebraic groups which are not necessarily reductive, but many of the nice properties belonging to reductive GIT no longer hold for non-reductive actions. The aim of this talk is to describe conditions under which these properties still hold for suitable non-reductive actions, and to discuss some applications
MAM 20th July 2015
16:00 to 17:00
ALG and ALH spaces
gravitational instanton is a complete hyper-Kahler Riemannian 4-manifold with curvature tensor in L^2. I will try to review what we know about gravitational instantons with slower than cubic volume growth at infinity.
MAM 21st July 2015
11:00 to 12:30
Physics of Moduli Space Dynamics of Solitons
MAM 21st July 2015
15:30 to 16:30
Decay and Moduli of Yang Mills Instantons
The inverse square law of electromagnetism tells us that electromagnetic fields decay quadratically. Determination of moduli of instantons on ALF spaces (such as Taub NUT) and of calorons, require similar decay information. In this talk, I will discuss instanton decay and moduli on ALF spaces and on R\times T^3. This reports on joint work with Sergey Cherkis and Andres Larrain and joint work with Savdeep Sethi and Travis Maxfield.
MAM 22nd July 2015
11:00 to 12:30
Analysis on singular spaces
I will give a brief overview of the use of manifolds with corners in analysis on singular and non-compact manifolds. I will describe applications to Hodge theory, index theory, and analytic torsion on these spaces.
MAM 22nd July 2015
15:30 to 17:00
New results on euclidean monopole metrics
We discuss a partial compactification of the moduli space, M_k, of SU(2) magnetic monopoles on R^3, wherein monopoles of charge k decompose into widely separated `monopole clusters' of lower charge which go off to infinity at comparable rates. The hyperkahler metric on M_k has a complete asymptotic expansion, the leading terms of which generalize the asymptotic metric discovered by Bielawski, Gibbons and Manton in the case that the monopoles are all widely separated. This is part of a larger work in progress to compactify the M_k as manifolds with corners.
MAM 23rd July 2015
11:00 to 12:30
Hitchin's self-duality equation and limiting configurations
We discuss a partial `differential geometric´ compactification of Hitchin's Higgs moduli space by so-called limiting configurations. The first part of the talk will address their existence. The second talk deals with convergence near infinity.

The content of this talk is based on arXiv:1405.5765 and 1502.01692 [math.DG] joint with R. Mazzeo and H. Weiss.
Video for this talk will not be available due to technical errors. We pologise for the inconvenience

MAM 23rd July 2015
14:00 to 15:00
Ruling out non-collapsed singularities in Riemannian 4-manifolds via the symplectic geometry of their twistor spaces
The twistor space of a Riemannian 4-manifold carries a natural closed 2-form. Asking that it be symplectic gives an interesting curvature inequality (which includes, for example, anti-self-dual Einstein metrics of non-zero scalar curvature). I will explain how the theory of J-holomorphic curves in the twistor space can be used to rule out certain types of degeneration in families of manifolds satisfying the curvature inequality. In particular, this shows that anti-self-dual Einstein metrics of negative scalar curvature cannot develop non-collapsed singularities. If there is time, I will end with speculation about other Riemannian uses for these symplectic structures and various conjectures concerning them.
Video for this talk will not be available due to technical errors. We pologise for the inconvenience
MAM 23rd July 2015
16:00 to 17:00
The renormalized volume of hyperbolic 3 manifolds
Video for this talk will not be available due to technical errors. We pologise for the inconvenience
MAM 24th July 2015
11:00 to 12:30
Instanton and Bow Moduli Spaces
Video for this talk will not be available due to technical errors. We pologise for the inconvenience
MAM 24th July 2015
14:00 to 15:00
A Vasy Analysis of quantum N-body type problems
I will discuss aspects of analysis in N-body scattering theory and its relationship to other related problems, such as higher rank symmetric spaces, which have a similar structure to certain kinds of moduli spaces
Video for this talk will not be available due to technical errors. We pologise for the inconvenience
MAMW01 27th July 2015
10:00 to 11:00
Folded hyperKähler metrics
Nigel Hitchin recently introduced folded hyperKähler metrics in his study of the limit of higher Teichmüller spaces. I will explain a construction of such metrics.
MAMW01 27th July 2015
11:30 to 12:30
H Auvray An analytic construction of dihedral ALF gravitational instantons
Gravitational instantons are 4-dimensional complete non-compact hyperkähler manifolds with some curvature decay at infinity. The asymptotic geometry of these spaces plays an important role in a conjectural classification; for example, instantons of euclidean, i.e. quartic, large ball volume growth, are completely classified by Kronheimer, whereas the cubic regime, i.e. the {\it ALF (Asymptotically Locally Flat)} case, is not fully understood yet. More precisely, ALF instantons with {\it cyclic topology at infinity} are classified by Minerbe; by contrast, a classification in the {\it dihedral} case at infinity is still unknown. A wide, conjecturally exhaustive, range of dihedral ALF instantons were constructed by Cherkis-Kapustin, adopting the moduli space point of view, and studied explicitly by Cherkis-Hitchin. I shall explain in this talk another construction of such spaces, based on the resolution of a Monge-Ampère equation in ALF geometry.
MAMW01 27th July 2015
14:30 to 15:30
P Boalch Non-perturbative hyperkahler manifolds
Following Kronheimer's construction of the ALE spaces from the ADE affine Dynkin graphs, and Kronheimer-Nakajima's subsequent extension of the ADHM construction, a large class of hyperkahler manifolds attached to graphs emerged, known as "quiver varieties". Nakajima has shown they play a central role in representation theory. If the underlying graph is of a special type it turns out that the corresponding quiver varieties have natural partial compactifications, which also admit complete hyperkahler metrics. They arise as spaces of solutions to Hitchin's equations on Riemann surfaces, with wild boundary conditions. (They were constructed in work with Biquard published in 2004). The class of graphs for which this works are known as "supernova" graphs and includes all the complete multipartite graphs. In particular the square and the triangle are supernova graphs, and so some gravitational instantons arise in this way. In this talk I will review this story focussing on specific examples and recent developments such as the algebraic construction of the underlying holomorphic symplectic manifolds (the "wild character varieties").
MAMW01 27th July 2015
16:00 to 17:00
Kähler metrics and Chern forms on the moduli space of punctured Riemann surfaces
I will review known properties of the WP and TZ metrics on the moduli space of punctured Riemann surfaces, discuss Chern forms of the associated line bundles and some open problems. This is a joint work with J. Park and L.P. Teo.
MAMW01 28th July 2015
09:00 to 10:00
On the geometry of some Hyperkaehler manifolds
I will discuss the geometry of some hyperkaehler manifolds : the QALE geometry of the Hilbert scheme of n-points in the complex plane or the QAC geometry of the cotangent bundle of Grassmannian.
MAMW01 28th July 2015
10:00 to 11:00
X Zhu Nodal degeneration of hyperbolic metrics and application to Weil-Petersson metric on the moduli space
This is joint work with Richard Melrose. We analyze the behavior of the Laplacian on the fibres of a Lefschetz fibration and use it to describe the behavior of the constant curvature metric on a Riemann surface of genus $>1$ undergoing nodal degeneration. We apply this to deduce the asymptotics of the Weil-Petersson metric on the moduli space $\mathcal{M}_g$.
MAMW01 28th July 2015
11:30 to 12:30
Asymptotics of hyperboilic, Weil-Peterssen and Takhtajan-Zograf metrics
This will be a continuation of the talk by Xuwen Zhu on our joint work concerning the regularity of the fibre hyperbolic metrics up to the singular fibres for Lefschetz fibrations. In particular this applies to the universal curve over moduli space. I will discuss the marked case with the moduli space $\mathcal{M}_{g,n}$ of surfaces of genus $g$ with $n$ ordered distinct points in the stable range, $2g+n\ge3.$ As in the unmarked case the description of the regularity of the fibre hyperbolic metrics, up to the divisors forming the `boundary' of the Knudsen-Deligne-Mumford compactification, implies boundary regularity for the Weil-Peterssen metric. In this case it also leads to an asymptotic description of the Takhtajan-Zograf metric which contributes to the Chern form of the determinant bundle for $\bar\partial$ on the fibres of the universal curve.
MAMW01 28th July 2015
14:30 to 15:30
Renormalized volume on the Teichmuller space of punctured Riemann surfaces
We define and study the renormalized volume for geometrically finite hyperbolic 3-manifolds that may have rank-1 cusps. We prove a variation formula, and show that for certain families of convex co-compact hyperbolic metrics degenerating to a geometrically finite hyperbolic metric with rank-1 cusps, the renormalized volume converges to the renormalized volume of the limiting metric.
MAMW01 28th July 2015
16:00 to 17:00
LD Saper Perverse sheaves on compactifications of locally symmetric spaces
Perverse sheave have important applications to representation theory, number theory, and algebraic geometry. I will discuss work in progress to understand the category of perverse sheaves on the Baily-Borel compactification of a Hermitian locally symmetric space; the method is to first work on the less singular reductive Borel-Serre compactification and then push down. Along the way I will introduce these various compactifications and give examples.
MAMW01 29th July 2015
09:00 to 10:00
R Bielawski Asymptotics and compactification of monopole moduli space
I shall give several (equivalent) descriptions of asymptotic metrics on various regions of SU(2)-monopole moduli spaces, as well as of their gluing which provides an asymptotic description of SU(2)-monopole metric. I shall also describe the common compactification of the monopole moduli space and of the above asymptotic approximation.
MAMW01 29th July 2015
10:00 to 11:00
Coulomb branches of 3-dimensional $\mathcal N=4$ gauge theories
Let $M$ be a quaternionic representation of a compact Lie group $G$. Physicists study the Coulomb branch of the 3-dimensional supersymmetric gauge theory associated with $(G,M)$, which is a hyper-Kaehler manifold, but have no rigorous mathematical definition. When $M$ is of a form $N\oplus N^*$, we introduce a variant of the affine Grassmannian Steinberg variety, define convolution product on its equivariant Borel-Moore homology group, and show that it is commutative. We propose that it gives a mathematical definition of the coordinate ring of the Coulomb branch. If time permits, we will discuss examples arising from quiver gauge theories.
MAMW01 29th July 2015
11:30 to 12:30
ALG and the SU($\infty$) Toda equation
The purpose of the talk is to pose a question about 4-dimensional moduli spaces of Higgs bundles: what are the minimal assumptions one needs to prove that the moduli space is ALG?
MAMW01 29th July 2015
16:00 to 17:00
'Breakout' Session
Format: 2 x (10 min introduction + 5-10min discussion) M. Singer - Spectral Curves: What are they good for? B. Schroers - Monopole Clouds: What are they?

Problems: For those who have a problem that they can clearly state in less than 3mins and explain why it is interesting in less than (a further) 2mins
MAMW01 30th July 2015
09:00 to 10:00
T Hausel Hyperkähler toy models
Motivated by visualizing hyperkähler moduli spaces, we present here examples, typically four dimensional, which can serve as toy models understanding their shape.
MAMW01 30th July 2015
10:00 to 11:00
Quantization of integrable systems of periodic monopoles
I will review what we learned over the recent years on the quantization of the periodic monopole moduli spaces appearing as the phase spaces of algebraic integrable systems in connection with the supersymmetric gauge theories (Seiberg-Witten integrable systems).
MAMW01 30th July 2015
11:30 to 12:30
Schiffer variations and Abelian differentials
Given a Riemann surface and an Abelian differential, we consider Cech style deformations based at zeros of the differential. Deformations are given in terms of slit mappings, degenerate Schwarz Christoffel mappings. We describe the associated deformation cocycles valued in vector fields.

Schiffer gave an exact formula for the change in the double pole Green's function corresponding to his conformal gluing deformation. We follow his approach and develop the second order variation formula for the double pole Green's function. Consequences are second order variation formulas for Abelian differentials and for the Riemann period matrix. The second variation of the period matrix is in the style of Rauch's celebrated formula and is given in terms of the 2-jet of the corresponding differentials at the base point zero.

Applications may include the Teichmuller geodesic flow on the space of Abelian differentials and the curvature of the Siegel upper half space metric on the image of Teichmuller space by the period matrix mapping.
MAMW01 30th July 2015
14:30 to 15:30
The charge density of a monopole and its asymptotic tail
Co-author: Daniel Nogradi

Magnetic monopoles are finite-energy solutions of the Yang-Mills-Higgs equations in non-abelian gauge theory. From afar, they resemble sources of magnetic charge in Maxwell electromagnetism (hence the name). It is a long-standing problem to identify a smooth magnetic charge density which induces the asymptotic magnetic field of a monopole. In this talk I will present a novel solution to this problem. I define a charge density by summing the squared norms of an L^2-orthonormal basis for the kernel of a Dirac operator associated with the monopole -- this is the analog for monopoles of the Bergman kernel in Kaehler geometry. I will show that the expansion of its induced magnetic field agrees with the asymptotic field of the monopole, to all orders in 1/r. I will also discuss the explicit evaluation of this asymptotic field.

MAMW01 30th July 2015
16:00 to 17:00
The elliptic genus - a view from conformal field theory
The elliptic genus of a compact Calabi-Yau manifold Y is a weak Jacobi form of weight zero. It is a topological invariant of Y which allows a reinterpretation as part of the partition function of so-called associated conformal field theories. As such, the coefficients of the elliptic genus count generic dimensions of spaces of states across the moduli space of such theories. On the other hand, if the theories have extended supersymmetry, then this conformal field theoretic view on the elliptic genus implies a novel decomposition of the underlying virtual bundle. This phenomenon occurs, for example, if Y is a hyperKaehler manifold. The talk will review the elliptic genus from this perspective in terms of geometric data.
MAMW01 31st July 2015
09:00 to 10:00
Polynomial Pick forms for affine spheres, real projective polygons, and surface group representations in PSL(3,R).
Abstract: (Joint work with David Dumas.) Discrete surface group representations into PSL(3, R) correspond geometrically to convex real projective structures on surfaces; in turn, these may be studied by considering the affine spheres (an interpretation of the Hitchin system of equations in this case) which project to the convex hull of their universal covers. As a sequence of convex projective structures leaves all compacta in its deformation space, a subclass of the limits is described by polynomial cubic differentials on affine spheres which are conformally the complex plane. We show that those particular affine spheres project to polygons; along the way, a strong estimate on asymptotics is found, which translates to a version of Stokes data. We begin by describing the basic objects and context and conclude with a sketc h of some of the useful technique and an application.
MAMW01 31st July 2015
10:00 to 11:00
Mass in Kaehler Geometry
Given an ALE (asymptotically locally Euclidean) Riemannian manifold, one can define a real number called its mass that measures an important feature of the asymptotic geometry. In this lecture, I will describe a new result that offers a reinterpretation of the mass of ALE Kaehler manifolds. In the AE (asymptotically Euclidean) case, this not only implies the positive mass theorem for Kaehler manifolds, but also yields a Penrose-type inequality for the mass.
MAMW01 31st July 2015
11:30 to 12:30
Coulomb Branch and the Moduli Space of Instantons
The moduli space of k G instantons on C^2, where G is a classical gauge group, has a well known HyperKahler quotient formulation known as the ADHM construction. The extension to exceptional groups is an open problem. In string theory this is realized using a system of branes, and the moduli space of instantons is identified with the Higgs branch of a particular supersymmetric gauge theory with 8 supercharges. A less known, and less studied aspect of moduli spaces of instantons is that they can be realized as the Coulomb branch of a supersymmetric gauge theory in 2+1 dimensions. Recent developments on the understanding of the Coulomb branch gives us a nice solution to the problem where G is an exceptional group, thus allowing a systematic study of these moduli spaces. I will discuss these developments, and present the corresponding quivers, and the Coulomb branch Hilbert Series - the main tool which lead to the recent progress.
MAM 3rd August 2015
14:45 to 15:45
A Dancer Symplectic and hyperkahler implosion
We describe implosion constructions in symplectic and hyperkahler geometry. We show how the latter case may be approached via Nahm moduli spaces, and also describe quasi-Hamiltonian analogues of implosion.
MAM 3rd August 2015
16:00 to 17:00
L Kamenova On Kobayashi's conjecture for K3 surfaces and hyperk\"ahler manifolds"
The Kobayashi pseudometric on a complex manifold M is the maximal pseudometric such that any holomorphic map from the Poincare disk to M is distance-decreasing. Kobayashi conjectured that this pseudometric vanishes on Calabi-Yau manifolds. Using ergodicity of complex structures, we prove this conjecture for any hyperkahler manifold that admits a deformation with a Lagrangian fibration, and its Picard rank is not maximal. For hyperkahler manifolds with maximal Picard rank we need an extra assumption, the SYZ conjecture. We shall discuss the proof of Kobayashi's conjecture for K3 surfaces and for certain hyperkahler manifolds. These results are joint with S. Lu and M. Verbitsky.
MAM 4th August 2015
11:00 to 12:00
Higgs bundles, spectral data, and fiber products of curves
I will discuss some interesting relations among Higgs bundles, from the point of view of spectral data, that result from isogenies among low dimensional Lie groups.
MAM 5th August 2015
11:00 to 12:00
L Fredrickson A construction of limiting solutions of Hitchin's equations
I'll describe a construction of solutions of Hitchin's equations on a compact Riemann surface near the "ends" of the SL(n,C)-Hitchin moduli space. This construction generalizes Mazzeo-Swoboda-Weiss-Witt's construction of SL(2,C) solutions of Hitchin's equations where the Higgs field is "simple." In the generalized construction, moduli spaces of irregular connections arise. This is ongoing work.
MAM 5th August 2015
16:00 to 17:00
(Open) Problem Discussion Session
MAM 6th August 2015
11:00 to 12:00
J Hurtubise Monopoles on circle bundles
In this talk, I will consider monopoles with Dirac type singularities on a circle bundle over a Riemann surface, and show that they correspond to holomorphic objects on the surface. When the bundle is trivial, one can show that the correspondence is bijective; in general, this is not yet proven, but the class of holomorphic objects is more interesting.
MAM 7th August 2015
11:00 to 12:00
Topology and Compactifications of Moduli Spaces
MAM 7th August 2015
14:00 to 15:00
J Lotay The moduli space of hyperKaehler metrics on 4-manifolds with boundary
MAM 11th August 2015
14:00 to 15:00
R Maldonado Moduli space of periodic monopoles and Hitchin equations on a cylinder
The Nahm transform relates periodic monopoles to periodic solutions of the Hitchin equations. I will identify the moduli in the 2-monopole case and construct the asymptotic metric. The symmetries of the system allow us to identify two geodesic submanifolds. I will describe their topology and the resulting monopole scattering through symmetric geodesics in the centre of the moduli space.
MAM 12th August 2015
14:00 to 15:00
HyperKaehler metrics with circle action
MAM 12th August 2015
15:30 to 16:30
Asymptotically Conic Calabi-Yau Manifolds
MAM 13th August 2015
15:00 to 16:00
Two predictions from physics relevant to metric and analytic aspects of moduli spaces
University of Cambridge Research Councils UK
    Clay Mathematics Institute London Mathematical Society NM Rothschild and Sons