|
Time and venue: Unless otherwise stated, seminars are on Wednesdays at 3pm. 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: 20 September - 16:30 - Roger Stevens LT11 - in person Benoit Charbonneau (University of Waterloo) Symmetric instantons Abstract. With Spencer Whitehead, we developed a systematic framework to study instantons on R4 that are invariant under groups of isometries. In this presentation, I will describe this framework and some results obtained using it. Geometry Seminar 11 October - 15:00 - Roger Stevens LT11 - in person Thibault Langlais (University of Oxford) Spectral properties of twisted connected sums Abstract. In this talk, we consider families of Riemannian manifolds (MT,gT) constructed by gluing together two asymptotically cylindrical manifolds along a neck of length 2T. We present a method for constructing a right inverse for the associated Laplacian operator ΔT in the limit where T goes to ∞. As an application, we study the asymptotic properties of the lower spectrum of ΔT, and relate our results to conjectures arising from Physics. Geometry Seminar 18 October - 15:00 - Roger Stevens LT11 - in person Artemis Vogiatzi (Queen Mary University of London) Submanifolds of high codimension evolving by their mean curvature. Abstract. Assuming a quadratic curvature pinching condition, in regions where the curvature is large, the submanifold evolving by mean curvature flow becomes approximately codimension one in a quantifiable way, regardless of the original flow’s codimension. This fundamental codimension estimate along with a cylindrical type estimate, prove that at a singularity, there exists a rescaling that converges to a smooth codimension-one limiting flow in Euclidean space, which is weakly convex and either moves by translation or is a self-shrinker. Joint Geometry and Analysis Seminar 1 November - 15:00 - Roger Stevens LT11 - in person Albert Wood (King's College London) Cohomogeneity-one Lagrangian Mean Curvature Flow Abstract. Mean Curvature Flow, the negative gradient flow for the volume functional of submanifolds of Riemannian manifolds, is a well-studied field of modern geometric analysis. Of particular interest are classifications of self-similar solutions (shrinkers, expanders, and translators) and finite-time singularities; projects which when completed will hopefully allow one to apply the flow to prove results in Riemannian geometry and differential topology. Moreover, in a Calabi-Yau manifold the class of Lagrangian submanifolds is preserved by mean curvature flow, a fact which inspired Thomas and Yau to make influential conjectures about existence of special Lagrangians in Calabi-Yau manifolds. In this talk, we aim to make progress towards an understanding of self-similar solutions and singularities of Lagrangian mean curvature flow, by focusing on Lagrangians in C^n that are cohomogeneity-one under the action of a compact Lie group. Interestingly, each such Lagrangian lies in a level set \mu^{-1}(c) of the moment map \mu, and mean curvature flow preserves this containment. Using this, we classify all shrinking, expanding, and translating solitons, and in the zero level set \mu^{-1}(0), we classify the Type I and Type II blowup models of LMCF singularities. Finally, given any special Lagrangian in \mu^{-1}(0), we'll show that it arises as a Type II blowup, thereby yielding infinitely many new singularity models of Lagrangian mean curvature flow. The results presented in this talk are contained in the preprint ‘Cohomogeneity-One Lagrangian Mean Curvature Flow’, which is jointly written with Jesse Madnick, University of Oregon. Geometry Seminar 8 November - 15:00 - Roger Stevens LT11 - on zoom Isabella Ianni (Sapienza Università di Roma) Uniqueness and nondegeneracy for fractional Dirichlet problems Abstract. We discuss some recent uniqueness and nondegeneracy results obtained for non-negative solutions of some fractional semilinear problems in bounded domains with Dirichlet exterior condition. Symmetry properties of the solutions of the associated linearized equation are also investigated. The talk is mainly based on the following joint works: [1] A. Dieb, I. Ianni, A. Saldaña, Uniqueness and nondegeneracy for Dirichlet fractional problems in bounded domains via asymptotic methods, Nonlinear Analysis, 236, 2023, https://doi.org/10.1016/j.na.2023.113354 [2] A. Dieb, I. Ianni, A. Saldaña, Uniqueness and nondegeneracy of least-energy solutions to fractional Dirichlet problems, preprint arXiv:2310.01214 Joint Geometry and Analysis Seminar 15 November - 16:30 - Roger Stevens LT11 - on zoom Gavin Ball (University of Wisconsin - Madison) Note the unusual time! The Morse index of quartic minimal hypersurfaces Abstract. Given a minimal hypersurface S in a compact Riemannian manifold, its Morse index is the number of variations of S that are area-decreasing to second order. In practice, computing the Morse index of a given minimal hypersurface is difficult. Indeed, even for the simplest case in which the ambient space is the round sphere and S is homogeneous, the Morse index of S is not known in general. In this talk, I will describe recent work (joint with Jesse Madnick and Uwe Semmelmann) where we compute the Morse index of two such minimal hypersurfaces. In this setting the Morse index is determined by the Laplace spectrum, and for these examples we are able to give an algorithm to determine the spectrum. Moreover, we find that in both of our examples, the spectra contain eigenvalues not expressible in radicals, a phenomenon not present in other examples. Geometry Seminar 22 November - 15:00 - Roger Stevens LT11 - on zoom Theodora Bourni (University of Tennessee) Ancient pancakes for mean curvature and Ricci flow Abstract. In this talk we will describe how to construct certain "collapsed" ancient solutions. In the recent years we have developed a method to construct convex ancient solutions of mean curvature flow that are confined in a slab. In this talk we will revisit this construction and show how parallel techniques can be used to construct ancient solutions of Ricci flow that have "bounded girth". This is joint work with T. Buttsworth, R. Lafuente and M. Langford. Joint Geometry and Analysis Seminar 29 November - 15:00 - Roger Stevens LT11 - in person Daniel Platt (Imperial College London) New Spin(7)-instantons on compact manifolds Abstract. Spin(7)-instantons are certain interesting principal bundle connections on 8-dimensional manifolds. Conjecturally, they can be used to define numerical invariants of 8-dimensional manifolds. However, not many examples of such instantons are known, which holds back the development of these invariants. In the talk I will explain a new construction method for Spin(7)-instantons generating more than 20,000 examples. The construction takes place on Joyce's first examples of compact Spin(7)-manifolds. No prior knowledge of Spin(7) or special holonomy is needed, this will be introduced in the talk. This is joint work with Mateo Galdeano, Yuuji Tanaka, and Luya Wang. (arXiv:2310.03451) Geometry Seminar 7 December - 10:00 - Roger Stevens LT11 - in person Melanie Rupflin (University of Oxford) Note the unusual time and day! Maps between spheres with nearly minimal energy Abstract. Many interesting geometric objects are characterised as minimisers or critical points of natural geometric quantities such as the length of a curve, the area of a surface or the energy of a map. For the corresponding variational problems it is often important to not only analyse the existence and properties of potential minimisers, but to obtain a more general understanding of the energy landscape. It is in particular natural to ask whether an object with nearly minimal energy must essentially "look like" a minimiser, and if so whether this holds in a quantitative sense, i.e. whether one can bound the distance to the nearest minimiser in terms of the energy defect. In this talk we will discuss this and related questions for the simple model problem of the Dirichlet energy of maps from the sphere S2 to itself, where in stereographic coordinates the minimisers (to given degree) are given by meromorphic functions. Joint Geometry and Analysis Seminar 13 December - 15:00 - Roger Stevens LT11 - on zoom Udhav Fowder (Unicamp, Brazil) Flows of SU(2)-structures Abstract. An SU(2)-structure on a 4-manifold M is given a triple of 2-forms satisfying certain algebraic conditions: this is equivalent to saying that M is almost hyper-Hermitian manifold. An SU(2)-structure is said to be torsion free if and only if M is a hyper-Kähler manifold. In this talk I want to report on an ongoing joint work with Henrique Sá Earp on the classification of flows of SU(2)-structures (in particular, this will include the Ricci flow!). Time permitting, I will also explain how these ideas can be applied to other H-structures (in particular, this was recently done in the G2 case by Dwivedi-Gianniotis-Karigiannis). Joint Geometry and Analysis Seminar Geometry Seminar 2022-2023: Semester 1/Semester 2 Previous Leeds Geometry Seminars |