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Autumn 2019

Leeds Geometry Seminar

Organiser: Gerasim Kokarev

Time and venue:
Wednesday 3pm, Roger Stevens LT10. Coffee/Tea served in the Common Room after 4pm.
Programme:

25 September 2019
Rene Garcia (Leeds)
Geometry of the space of asymmetric vortex-antivortex pairs
Abstract. In this talk, I will describe the geometry of the space of vortex-antivortex pairs arising in the low energy approximation to the O(3)-sigma model. The model is described by a Lagrangian admitting a family of solutions called BPS static. These are a set of equations from which a metric for the moduli space of solutions modulo gauge equivalence can be obtained with the meromorphic Strachan-Samols procedure. A solution to the field equations can be classified according to the number of vortices and antivortices on it. These are lumps of energy whose positions determine the fields. In the asymmetric model, vortices and antivortices have different effective mass. The main result I will talk about is that if the ambient manifold is a sphere, Samol's metric in the subspace of 1-vortex and 1-antivortex pairs is incomplete while the volume is finite.

2 October 2019
Chris Halcrow (Leeds)
Nucleon interactions from soliton models
Abstract. Topological solitons are spatially localised solutions to some nonlinear PDEs whose existence and stability are owed to the topology of the system. To study their interactions, dynamics, quantum mechanics and physical consequence, you must first understand their moduli spaces. Physically, the moduli space is parameterised by the degrees of freedom in the system, loosely interpreted as the `positions' and `orientations' of the solitons. In this talk I will review the structure of a 2-instanton moduli space, which has a nice geometric description due to Hartshorne. This is used to model the 2-Skyrmion interaction which in turn is used to model the 2-nucleon interaction. I will argue that the `spin-orbit interaction' - traditionally a phenomenological nuclear interaction - arises as a simple consequence of the geometry of the relevant moduli space.

9 October 2019
Jonny Evans (Lancaster)
Constructing local models of Lagrangian torus fibrations
Abstract. Lagrangian torus fibrations are central to our current understanding of mirror symmetry. For example, the SYZ conjecture asserts that mirror pairs of Calabi-Yau manifolds admit dual Lagrangian torus fibrations. Early expectations were that a sufficiently nice Calabi-Yau 3-fold (like the quintic) should admit a Lagrangian torus fibration over a homology 3-sphere M such that the discriminant locus (the points in the base living below the singular fibres upstairs) forms a trivalent graph in M. In practice, the Lagrangian torus fibrations that people (Joyce, Castano-Bernard, Matessi) found have codimension 1 discriminant loci. In recent work (with Mirko Mauri), we figured out a general way to construct Lagrangian torus fibrations (in local models) where the discriminant locus is a Y (in particular on the local model for the problematic "negative vertices" of the trivalent graph).

16 October 2019
Yuuji Tanaka (Oxford)
On two generalisations of Hitchin's equations in four dimensions
Abstract. This talk is about two kinds of generalisations of Hitchin's prominent equations on Riemann surfaces to ones in dimension four, which have the origin in N=4 super Yang-Mills theory in four dimensions. We call them Vafa-Witten and Kapustin-Witten equations. First we will recall Hitchin's equations on Riemann surfaces, introduce the two above-mentioned equations on four-manifolds, and then survey recent progress on the studies of the moduli spaces of solutions to these equations from both analytic and algebraic points of view.

30 October 2019
Alexei Kovalev (Cambridge)
A compact G_2-calibrated 7-manifold with b_1=1
Abstract. We construct a compact formal 7-manifold with a closed G_2-structure and with first Betti number b_1=1, which does not admit any torsion-free G_2-structure, that is, it does not admit any G_2-structure such that the holonomy group of the associated metric is a subgroup of G_2. We also construct associative calibrated (hence volume-minimizing) 3-tori with respect to this closed G_2-structure and, for each of those 3-tori, we show a 3-dimensional family of non-trivial associative deformations. We also construct a fibration of our 7-manifold over S^2 x S^1 with generic fibre a (non-calibrated) coassociative 4-torus and some singular fibres. Joint work with Marisa Fernandez, Anna Fino and Vicente Munoz.

13 November 2019
Dmitry Jakobson (McGill University)
Zero and negative eigenvalues of conformally covariant operators, and nodal sets in conformal geometry
Abstract. We study conformal invariants that arise from nodal sets and negative eigenvalues of conformally covariant operators, which include the Yamabe and Paneitz operators. We give several applications to curvature prescription problems. We establish a version in conformal geometry of Courant's Nodal Domain Theorem. We prove that the Yamabe operator can have an arbitrarily large number of negative eigenvalues on any manifold of dimension greater than 2. We show that 0 is generically not an eigenvalue of the conformal Laplacian. If time permits, we shall discuss related results for manifolds with boundary, and for weighted graphs. This is joint work with Y. Canzani, R. Gover, R. Ponge, A. Hassannezhad, M. Levitin, M. Karpukhin, G. Cox and Y. Sire.

20 November 2019 Joint Analysis/Geometry Seminar
Neshan Wickramasekera (Cambridge) 4.30pm Roger Stevens LT11
On the behaviour of stationary integral varifolds near multiplicity 2 planes
Abstract. Consider a singular n-dimensional minimal submanifold (i.e. a stationary integral n-varifold) V in an open ball in R^{n+k} lying close to a plane of some integer multiplicity q passing through the centre of the ball. In 1972 Allard proved a fundamental regularity theorem, generalising earlier pioneering work of De Giorgi, that implies that if q=1 then near the centre of the ball the varifold is smoothly embedded. This celebrated De Giorgi--Allard theory in fact says that near the centre of the ball the varifold is the graph of a smooth function over the plane with small gradient and satisfying estimates on all derivatives. This result implies that for any stationary integral varifold, the (relatively open) set of points of mass density <2 is fairly regular; if non-empty, it is an embedded submanifold away from a closed set whose Hausdorff dimension is at most (n-1), and in the absence of triple-junction singularities (e.g. when the varifold is the limit of embedded minimal submanifolds) it is embedded everywhere if n=2 and is embedded away from a closed set of Hausdorff dimension at most (n-3) if n>2.

It is a long standing open question what one can say about V when q is greater or equal to 2. We will discuss some work (joint with Spencer Becker-Kahn) that considers this question when q=2. The work gives a necessary and sufficient topological condition on the region under which, near the centre of the ball: (a) V is a Lipschitz 2-valued graph with small Lipschitz constant and (b) each tangent cone to V is unique, and is equal to either a single plane of multiplicity 1 or 2, or a pair of distinct multiplicity 1 planes or a union of four multiplicity 1 half-planes meeting along an (n-1)-dimensional axis. This condition is automatically satisfied if V is a Lipschitz 2-valued graph (of arbitrary Lipschitz constant) or if the codimension is 1, V corresponds to a current without boundary in the ball and the regular part of V is stable. The analysis involves, among other things, a new energy non-concentration estimate for a class of q-valued harmonic functions that approximate stationary integral varifolds close to multiplicity q planes, and a novel non-variational argument based on this estimate to establish mononotonicity of the Almgren frequency function for this class when q=2.


27 November 2019
Nuno Romão (University of Augsburg)
Pochhammer states on Riemann surfaces
Postponed to a later date

4 December 2019
Ovidiu Munteanu (University of Connecticut)
Spectral estimates for noncompact Kahler manifolds
Abstract. I will describe a sharp upper bound for the bottom spectrum of the Laplace operator on a complete noncompact Kahler manifold with Ricci curvature bounded from below. The comparison space is the complex hyperbolic space, but the bound is also achieved by many other Kahler manifolds as well. However, we do have rigidity results in certain situations. A generalization to the p-Laplace operator will also be described.

11 December 2019
Yorkshire and Durham Geometry Day (to be held at the University of Durham)

Previous Geometry Seminars:

Academic Year 2018/19: Autumn 2018, Spring 2019
Academic Year 2017/18: Autumn 2017, Spring 2018
Academic Year 2016/17: Autumn 2016, Spring 2017
http://www1.maths.leeds.ac.uk/~pmtgk/seminar/current.html
Last modified: 3 Sep 2019