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Time and venue: Wednesday 3pm, Roger Stevens LT07. Coffee/Tea served in the Common Room after 4pm. Programme: 3 October 2018 Yorkshire and Durham Geometry Day (to be held at the University of York) 10 October 2018 Luc Nguyen (Oxford) Existence and uniqueness of Green's functions to a nonlinear Yamabe problem Abstract. The Yamabe problem asks to find in a conformal class of metrics on a compact manifold $M$ a metric of constant scalar curvature. In this context, the Green's function with a given pole $p$ in $M$ corresponds to a complete asymptotically flat metric of zero scalar curvature on $M \setminus \{p\}$. We discuss existence and uniqueness of similar objects when the scalar curvature is replaced by other fully nonlinear conformal curvature quantities. Joint work with Yanyan Li. 17 October 2018 Alexey Bolsinov (Loughbourough) Argument shift method and Manakov-type operators: applications in differential geometry Abstract. The talk will be devoted to an interesting and rather unexpected relationship between some ideas and notions well known in the theory of integrable systems on Lie algebras and a rather different area of mathematics studying projectively and c-projectively equivalent metrics in Riemannian and Kaehler geometry. 7 November 2018 Huy Nguyen (QMUL) Convergence results in high codimension curve shortening flow 4pm (unusual time) Abstract. Mean curvature flow is the gradient descent flow of the area functional of a submanifold. While the codimension one case has been extensively studied, relatively little is known about the higher codimension case. For example, it is known that embedded plane curves shrink to round points but for space curves embeddedness is not even preserved. In this talk, I will give a (geometric) criterion to show that space curves shrink to a round point. 14 November 2018 Lauri Oksanen (UCL) Some questions in integral geometry related to hyperbolic inverse problems Cancelled due to speaker's illness. 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. 21 November 2018 Michael Magee (Durham) Uniform spectral gap for Riemann surfaces (and beyond) Abstract. I'll begin by discussing Selberg's eigenvalue conjecture. This is an analog of the Riemann hypothesis for a special family of Riemann surfaces that feature heavily in number theory, for example in Wiles' proof of the Taniyama-Shimura conjecture. I'll explain how in the last 10-15 years, number theorists have had to turn to Anosov dynamics to obtain the approximations to Selberg's conjecture that became relevant to emerging 'thin groups' questions about Apollonian circle packings and continued fractions. Then if I have time, I'll explain how I have proved an extension of Selberg's theorem to higher genus moduli spaces, and point out some interesting ingredients that were involved. 28 November 2018 Mark Haskins (Bath) Complete noncompact metrics of special and exceptional holonomy: the first 40 years Abstract. I will attempt to give a (biased) overview of recent progress on the construction of complete noncompact metrics of exceptional holonomy. Along the way I will describe some of the most important historical developments since the field began (in the late 1970s). Throughout its history, the field has seen a fruitful back-and-forth between physicists and mathematicians, some of which I will describe. I will try to explain some of the similarities and differences between the more familiar special holonomy metrics - hyperkaehler and Calabi-Yau metrics and the exceptional cases G_2 and Spin_7 holonomy, and why the latter are much more difficult to construct. In the early 2000s M theorists predicted the existence of various new complete noncompact Riemannian metrics with holonomy group the compact exceptional Lie group G_2. Very recently mathematicians have constructed many, but by no means all, of these physically predicted G_2 metrics and also other G_2 metrics not necessarily anticipated by physics. It will turn out the construction of these complete noncompact metrics of exceptional holonomy relies on some of the most recent developments on constructing complete noncompact hyperkaehler and Calabi-Yau metrics with controlled asymptotic geometry. If (by some miracle) time permits, I will describe some of the future prospects for the field. 5 December 2018 Yang Li (Imperial) Dirichlet problem for maximal graphs of higher codimension Abstract. Maximal submanifolds in Lorentzian type ambient spaces are the formal analogues of minimal submanifolds in Euclidean spaces, which arise naturally in adiabatic problems for G2 manifolds. We obtain general existence and uniqueness results for the Dirichlet problem of graphical maximal submanifolds in any codimension, which stand in sharp contrast to the analogous problem for graphical minimal submanifolds. |