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Time and venue: Unless otherwise stated, Geometry Seminars are on Wednesdays at 3pm, Geometry and Analysis Seminars are on Wednesdays at 4:30pm. These take place on Zoom or in person, usually in Roger Stevens LT 11. When on Zoom we will still meet (when possible) to view the zoom seminar. The joint geometry and analysis seminars are also listed here and are co-organised with Ben Sharp. Programme Semester 1: 21 September 2022 - 4:30pm - MALL 1 and 2 Ragini Singhal (King's College London) Deformations of G_2-instantons on nearly G_2 manifolds Abstract. In this talk we will study the deformation theory of G_2-instantons on nearly G_2 manifolds. There is a one-to-one correspondence between nearly parallel G_2 structures and real Killing spinors, thus the deformation theory can be formulated in terms of spinors and Dirac operators. We prove that the space of infinitesimal deformations of an instanton is isomorphic to the kernel of an elliptic operator. Using this formulation we prove that abelian instantons are rigid. Then we apply our results to describe the deformation space of the canonical connection on the four normal homogeneous nearly G_2 manifolds. We also see that the deformation spaces obtained are integrable in most of the cases. 12 October 2022 - 3pm - Roger Stevens LT 11 Calum Ross (University College London) A Gluing Construction for Calorons Abstract. Calorons are instantons on $\mathbb{R}^{3}\times S^{1}$, e.g. anti-self-dual connections which are periodic in one direction. Certain explicit caloron configurations have an interpretation as a ``superposition'' of monopoles. I will present a gluing construction which formalises this constituent monopole picture. This construction starts from a singular background and glues in ``fundamental calorons'' to desingularise the configuration. I will introduce what these constituents are, sketch the basics of the construction, and outline some applications to understanding caloron moduli spaces. This is joint work with Lorenzo Foscolo. 19 October 2022 - 4:30pm - Roger Stevens LT 11 Steven Lynch (Imperial College London) Uniqueness of ancient solutions to curvature flows Abstract. Ancient solutions of the mean curvature flow and Ricci flow arise naturally as blowup limits at singularities. In recent years there has been a lot of progress towards the classification of such solutions. We will discuss some uniqueness results for convex ancient solutions of mean curvature flow, and ancient solutions of Ricci flow which have nonnegative curvature operator. In both cases we show that, under an optimal curvature growth assumption (Type I growth), the solution evolves self-similarly by scaling. The arguments for the two flows are surprisingly similar, despite being completely extrinsic and intrinsic, respectively. Joint Geometry and Analysis Seminar 26 October 2022 - 3pm - Roger Stevens LT 11 Ravil Gabdurakhmanov (University of Leeds) Calderon's problem for maps between manifolds Abstract. The classical (geometric) Calderon problem concerns the reconstruction of a Riemannian manifold from the Dirichlet-to-Neumann operator on the boundary. The latter acts on functions on the boundary as follows, it sends a function to the normal derivative of its harmonic extension. We will first discuss some known results for the classical Calderon problem and then a generalisation of this problem to maps between manifolds. We will discuss the difference with the classical problem and difficulties that arise. We will continue with stating our main result and sketch of its proof. In addition, we will discuss Calderon's problem for the connection Laplacian and our result on this auxiliary problem. 2 November 2022 - 4:30pm - Roger Stevens LT 11 Lashi Bandara (Brunel University London) Boundary value problems for first-order elliptic operators with compact and noncompact boundary. Abstract. The index theorem for compact manifolds with boundary, established by Atiyah-Patodi-Singer in the mid-70s, is considered one of the most significant mathematical achievements of the 20th century. An important and curious fact is that local boundary conditions are topologically obstructed for index formulae and non-local boundary conditions lie at the heart of this theorem. Consequently, this has inspired the study of boundary value problems for first-order elliptic differential operators by many different schools, with a class of induced operators adapted to the boundary taking centre stage in formulating and understanding non-local boundary conditions. That being said, much of this analysis has been confined to the situation when adapted boundary operators can be chosen self-adjoint. Dirac-type operators are the quintessential example. Nevertheless, natural geometric operators such as the Rarita-Schwinger operator on 3/2-spiniors, arising from physics in the study of the so-called Delta baryon, falls outside of this class. Analytically, this requires analysis beyond self-adjoint operators. In recent work with Bär, the compact boundary case is handled for general first-order elliptic operators, using spectral theory to choose adapted boundary operators to be invertible bi-sectorial. The Fourier circle methods present in the self-adjoint analysis are replaced by the bounded holomorphic functional calculus, coupled with pseudo-differential operator theory and semi-group techniques. This allows for a full understanding of the maximal domain of the interior operator as a bounded surjection to a space on the boundary of mixed Sobolev regularity, constructed from spectral projectors associated to the adapted boundary operator. Regularity and Fredholm extensions are also studied. For the noncompact case, a preliminary trace theorem as well as regularity theory are handed by resorting to the case with compact boundary. This necessitates deforming the coefficients of the interior operator in a compact neighbourhood. Therefore, even for Dirac-type operators, allowing for fully general symbols in the compact boundary case is paramount. Under slightly stronger geometric assumptions near the noncompact boundary (automatic for the compact case) and when the interior operator admits a self-adjoint adapted boundary operator, an upgraded trace theorem mirroring the compact setting is obtained. Importantly, there is no spectral assumptions other than self-adjointness on the adapted boundary operator. This, in particular, means that the spectrum of this operator can be the entire real line. Again, the primarily tool that is used in the analysis is the bounded holomorphic functional calculus. Joint Geometry and Analysis Seminar 9 November 2022 - 4:30pm - Roger Stevens LT 11 Maxwell Stolarski (University of Warwick) Closed Ricci Flows with Singularities Modeled on Asymptotically Conical Shrinkers Abstract. Shrinking Ricci solitons are Ricci flow solutions that self-similarly shrink under the flow. Their significance comes from the fact that finite-time Ricci flow singularities are typically modeled on gradient shrinking Ricci solitons. Here, we shall address a certain converse question, namely, “Given a complete, noncompact gradient shrinking Ricci soliton, does there exist a Ricci flow on a closed manifold that forms a finite-time singularity modeled on the given soliton?” We’ll discuss work that shows the answer is yes when the soliton is asymptotically conical. No symmetry or Kahler assumption is required, and so the proof involves an analysis of the Ricci flow as a nonlinear degenerate parabolic PDE system in its full complexity. We’ll also discuss applications to the (non-)uniqueness of weak Ricci flows through singularities. Joint Geometry and Analysis Seminar 16 November 2022 - 3pm - Roger Stevens LT 11 Josh Cork (University of Leicester) Near BPS and BPS gauged skyrmions Abstract. Various physical field theories admit topological energy bounds, that is, the energy (a positive definite functional of the fields) is bounded below by a quantity which depends only on some topological class that the fields live in (for example, the degree of a map, a Chern class of a vector bundle, etc.). Typically one is interested in how close you can get to attaining the topological lower bound; fields which attain the topological bound are often called BPS, and analysis of such solutions often reveals rich underlying geometry. However sometimes there are obstructions which force fields to have strictly greater energy than the lower bound, and analysis of critical points of the energy is typically much harder. That said there can be surprising links with BPS solutions to other field theories, allowing for remarkably accurate approximations. We shall focus on topological energy bounds, and BPS and near BPS solutions, in a gauged Skyrme model: a gauge field theory with origins in Nuclear physics. We shall do this both indirectly, through an approximation via Yang-Mills instantons, and directly, through some explicit analysis of some restricted classes of solutions. This talk is based on ongoing joint work with Derek Harland (Leeds), and past work also with Thomas Winyard (Heriot-Watt). 23 November 2022 - 4:30pm - Roger Stevens LT 11 - (Zoom) Alessandra Pluda (University of Pisa) Network flow: resolution of singularities and stability Abstract. The curve shortening flow is an evolution equation in which a curve moves with normal velocity equal to its curvature, and can be interpreted as the gradient flow of the length. In this talk we consider the same flow for networks (finite unions of sufficiently smooth curves whose endpoints meet at junctions). We will explain how to define the flow in a classical PDE framework, and then we will list some examples of singularity formation, both at finite and infinite time, and explain the resolution of such singularities obtained by geometric microlocalanalysis techniques. Finally we will show how starting from a suitable Lojasiewicz-Simon inequality it is possible to prove the stability of the flow in the sense that a network sufficientlyclose in the $H^2$-norm to a minimal one exists for all times and converges smoothly. This seminar is mainly based on two recent papers in collaboration with Jorge Lira (Universidade Federal do Ceará), Rafe Mazzeo (Stanford University), Mariel Saez (P. Universidad Catolica de Chile) and Marco Pozzetta (Università di Napoli Federico II). Joint Geometry and Analysis Seminar 7 December 2022 - 15:00 Yorkshire Durham Geometry Day is taking place in Durham, see here for more details. 14 December - 4:30pm - Roger Stevens LT 11 - (Zoom) Isabel Fernández (Universidad de Sevilla) Free Boundary Minimal Annuli in the Ball Abstract. A compact minimal surface in R3 is called "free boundary in the unit ball" if the boundary of the surface meets orthogonally the boundary of the ball. These surfaces appear as critical points of the area functional among all surfaces in the ball whose boundaries lie in the boundary of the ball. In this talk, we will prove the existence of free boundary minimal annuli immersed in the ball, solving a problem posed by Nitsche in 1985. This is a joint work with Laurent Hauswirth and Pablo Mira. Joint Geometry and Analysis Seminar Previous Leeds Geometry Seminars |