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Spring 2019

Leeds Geometry Seminar

Organiser: Gerasim Kokarev

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

30 January 2019 AGIS Colloquium
Sabir Gusein-Zade (Lomonosov Moscow State University, Russia)
On a version of the Berglund-Hübsch-Henningson duality with non-abelian symmetry groups.
Abstract. P. Berglund, T. Hübsch and M. Henningson found a method to construct mirror symmetric Calabi-Yau manifolds using so-called invertible polynomials. They considered pairs (f,G) consisting of an invertible polynomial f and a (finite abelian) group G of diagonal symmetries of f. To a pair (f,G) one associates the Berglund-Hübsch-Henningson (BHH) dual pair (f^,G^). There were found some symmetries between dual invertible polynomials and dual pairs not related directly with the mirror symmetry (e.g., some of them hold when the corresponding manifolds are not Calabi-Yau). One of them was the so-called equivariant Saito duality as a duality between Burnside rings. A. Takahashi suggested a conjectural method to find symmetric pairs consisting of invertible polynomials and symmetry groups generated by some diagonal symmetries and some permutations of variables. The equivariant Saito duality was generalized to the case of non-abelian groups. It turns out that the corresponding symmetry holds only under a special condition on the action of the subgroup of the permutation group called here PC (parity condition). An inspection of data on Calabi-Yau threefolds obtained from quotients by non-abelian groups (taken from tables computed by X. Yu) shows that the pairs found on the basis of the method of Takahashi have symmetric pairs of Hodge numbers (and thus are hopefully mirror symmetric) if and only if they satisfy PC.

13 February 2019
Mariel Sáez Trumper (Pontificia Universidad Católica de Chile)
Fractional Laplacians and extension problems in conformal geometry
Abstract. In this talk I will discuss the construction of conformal fractional operators that arise from extension problems and some associated questions. In the first part of the talk I will give definitions and interpretations of the fractional Laplacian and the conformal fractional Laplacian in the general framework of representation theory on symmetric spaces and also from the point of view of scattering operators in conformal geometry. In the second part of the talk I will show constructions of boundary operators with good conformal properties that arise from co-dimension 2 extensions. I will also discuss some associated eigenvalue questions and Yamabe type problems that are natural for fractional operators. Joint work with Maria del Mar Gonzalez.

20 February 2019
Lauri Oksanen (UCL) Some questions in integral geometry related to hyperbolic inverse problems
Abstract. The problem to recover subleading terms in a wave equation given boundary traces of solutions to the equation can be reduced to the following problem in integral geometry: find a function (or a one form modulo a certain gauge invariance) given its light ray transfrom, that is, its integrals over all lightlike geodesics. It is an open question if the light ray transform is invertible even when the Lorentzian metric associated to the wave equation is close to the Minkowski metric. We describe some recent results in product geometries, and discuss also a broken version of the light ray transform that arises when recovering the first order terms in a non-linear wave equation.

27 February 2019
James Roberts (Bath) Fractional Harmonic Mappings of Riemannian Manifolds
Abstract. Fractional harmonic maps are analogues of both harmonic mappings of Riemannian manifolds and solutions of fractional Laplace equations. They were first introduced for the fractional power 1/2 on domains of dimension one, with a view to analysing their regularity, in connection with conformal geometry; on one dimensional domains they are critical points of energies which satisfy a type of conformal invariance. Subsequently, their regularity has been analysed on domains of arbitrary dimension for a range of fractional powers. Many properties of fractional harmonic maps, including their regularity, may be analysed by considering their extensions to a half-space equipped with a Riemannian metric which may degenerate/become singular along the boundary depending on the fractional power. I will introduce the notion of fractional harmonic maps from domains in Euclidean space into smooth compact Riemannian manifolds. I will also illustrate the connection between their extensions to the aforementioned half-spaces and harmonic mappings of Riemannian manifolds with (partially) free boundary data and present some regularity results for energy minimising fractional harmonic maps and free boundary harmonic maps in this context.

6 March 2019
Jean Lagacé (UCL) Explicit geometric control for Weyl's law on tori and applications
Abstract. For a fixed manifold M, Weyl's law describes the distribution of the eigenvalues of the Laplacian as the spectral parameter tends to infinity. While the distribution is eventually similar for all manifolds, it is certainly not so uniformly. In this talk, I will describe how, by including explicitly the injectivity radius in the remainder term in Weyl's law, one can make the estimates uniform in the class of flat tori. I will then show how these results can be used to study asymptotic eigenvalue optimisation, where the divergence result we obtain are in sharp contrast to the convergence results obtained for asymptotic eigenvalue optimisation on cuboids obtained by K. Gittins and S. Larson in 2017.

8 March 2019
Yorkshire and Durham Geometry Day (to be held at the University of Durham)

13 March 2019
Joe Oliver (Leeds) Harmonic maps into the Symplectic group
Abstract. In 1989 K. Uhlenbeck showed that all harmonic maps into the unitary group U(n) can be obtained from certain maps (extended solutions) into the based loop group of U(n). In 1997 another approach was taken by F.E. Burstall and M.A. Guest who used certain elements of the algebraic loop group of U(n) called canonical elements to provide a finer classification of these extended solutions. In 2018 M.J. Ferreira, B.A. Simoes and J.C. Wood gave an algorithm which is inductive on dimension to find formulae for extended solutions for the group O(n) from those for O(n-2) and thus finding all harmonic maps of finite uniton number and their extended solutions from a surface to O(n) in terms of free holomorphic data. Influenced by this, I will give parameterizations of S^1-invariant harmonic maps of finite uniton number from a surface to Sp(n) and their extended solutions up to complex dimension 6. I will also give an algorithm which is inductive on dimension to find formulae for S^1-invariant standard type extended solutions for the group Sp(n) from those for Sp(n-1) and thus finding all S^1-invariant standard type harmonic maps of finite uniton number and their extended solutions from a surface to Sp(n) in terms of free holomorphic data.

20 March 2019
Bailin Deng (Cardiff) Geometry Processing for Digital Fabrication
Abstract. In recent years, the emergence of digital fabrication tools such as 3D printers and laser cutters has allowed us to turn a digital design into a physical object. But effective use of these tools requires the design shape to satisfy specific requirements that are not considered by traditional 3D design tools. We argue that these fabrication requirements can be incorporated into the design process as geometric constraints related to the fabrication technologies and materials. This talk will present a few examples of formulating such constraints for different applications, as well as efficient numerical solution techniques for the resulting geometric optimisation problems.

15 May 2019
Yorkshire and Durham Geometry Day at Leeds

Previous Geometry Seminars:

Academic Year 2018/19: Autumn 2018
Academic Year 2017/18: Autumn 2017, Spring 2018
Academic Year 2016/17: Autumn 2016, Spring 2017
Last modified: 7 May 2019