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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 co-organised with Ben Sharp. Programme Semester 1: 9 October - 15:00 - Roger Stevens LT11- in person Prachi Sahjwani (University of Cardiff) Stability of geometric inequalities in various spaces Abstract. In this talk, I will discuss the stability of two inequalities: the "Alexandrov-Fenchel inequalities in hyperbolic space" and "Minkowski's inequality in a general warped product space." I will give a brief overview of both inequalities and their respective stability problems. To explain what I mean by stability, I will first discuss it in the context of the isoperimetric inequality, which is a special case of the Alexandrov-Fenchel inequalities. I will also briefly discuss the proofs in both cases. This is joint work with Prof. Dr. Julian Scheuer and is based on the work of Wang/Xia on Alexandrov-Fenchel inequalities and the work of Brendle/Hung/Wang and Scheuer on Minkowski's inequality. Joint Geometry and Analysis Seminar 16 October - 15:00 - Roger Stevens LT11 - in person Arjun Sobnack (University of Warwick) Delayed parabolic regularity for the Curve Shortening Flow Abstract. The Curve Shortening Flow (CSF) is a parabolic partial differential equation (PDE) which arises as the L2–gradient flow of the length functional. It is a general phenomenon that parabolic PDEs do not observe an instantaneous L1–smoothing property (as they do for L1 < p< ∞). Nonetheless, in a recent preprint P. M. Topping & S. propose the framework that, two planer curves evolving under the CSF to trap a fixed area start observing a smoothing estimate after waiting for a time depending only on the area, which is a type delayed L1-smoothing estimate. The aim of this talk is to introduce the principle of delayed parabolic regularity for the CSF and to demonstrate that it is sharp, and also to present some applications of localised versions of the estimates. Joint Geometry and Analysis Seminar 30 October - 15:00 - Chemistry West Lecture Theatre F - in person Stuart Hall (University of Newcastle) Note the unusual room! Rigidity results for Grassmannians and some other symmetric spaces Abstract. I will report on recent joint work with Paul Schwahn and Uwe Semmelmann where we show that the canonical Einstein metric on 'odd' Grassmannians is rigid. Time permitting, I will discuss an approach to demonstrating this for the spaces SU(n)/SO(n) and SU(2n)/Sp(n). Geometry Seminar 6 November - 15:00 - Chemistry Lecture Theatre B (2.17) - in person Kasia Wyczesany (University of Leeds) Note the unusual room! Zoo of dualities and related inequalities Abstract. In this talk, we will discuss order reversing quasi involutions, which are dualities on their image, and their properties. We will relate to the classical example called ``polarity'', which corresponds to duality between finite-dimensional normed spaces, and generates many interesting questions. We prove that any order reversing quasi involution is of a special form and discuss how this unified point of view on order reversing quasi involutions helps to deepen the understanding of the underlying structures and principles. This talk is based on joint work with Shiri Artstein-Avidan and Shay Sadovsky. Joint Geometry and Analysis Seminar 13 November - 15:00 - Chemistry Lecture Theatre B (2.17) - in person Philipp Reiser (Université de Fribourg) Note the unusual room! Positive Ricci curvature and connected sums Abstract. The connected sum operation is a simple but useful tool in geometric topology to connect two given manifolds. However, if both manifolds are equipped with Riemannian metrics of positive Ricci curvature, it is surprisingly difficult to determine whether this condition can be preserved under the connected sum. In this talk, I will review previous work by Perelman and Burdick on this problem, and then discuss a new construction for Riemannian metrics of positive Ricci curvature on connected sums of certain fibre bundles. Joint Geometry and Analysis Seminar 20 November- 15:00 - Roger Stevens LT11 - in person Gautam Chaudhuri (University of Leeds) 2-vortex dynamics on complex toric curves Abstract. Vortices are topological solitons on Riemann surfaces arising in the Abelian Higgs model. Low energy dynamics of vortices are well approximated by geodesic motion on an associated moduli space, but in many cases the metric on this moduli space has no explicit global description. In this talk, we will look at how the metric itself can be approximated in the dissolving limit, i.e. when the volume of the base Riemann surface tends to a critical minimum value. We will then see some new results on how this approximation can be used to model the dynamics of 2-vortices on complex toric curves, yielding explicit approximations of vortex dynamics and characterising the relative motion of the vortices. Joint Geometry and Analysis Seminar 27 November - 15:00 - Roger Stevens LT11 - on zoom Maxime Fortier Bourque (Université de Montréal) Probing the spectra of hyperbolic surfaces with the Selberg trace formula Abstract. I will explain how the Selberg trace formula can be used to prove good upper bounds on the eigenvalues of the Laplacian and their multiplicities or on the lengths of closed geodesics and their multiplicities, for arbitrary closed hyperbolic surfaces. This approach is heavily inspired by the theory of sphere packings. For very symmetric surfaces, we can use a version of the trace formula twisted by the irreducible representations of the automorphism group to prove rigorous lower and upper bounds on the Laplace spectrum. This method yields counterexamples to an upper bound on the multiplicity of the first nonzero Laplacian eigenvalue conjectured by Colin de Verdière in 1986. This is joint work with Émile Gruda-Mediavilla, Bram Petri, and Mathieu Pineault. Geometry Seminar 4 December - 15:00 - Roger Stevens LT11 - in person Marco Guaraco (Imperial College London) On the Plateau problem and the Allen-Cahn equation Abstract. I will discuss recent progress on the approach to the Plateau problem via the Allen-Cahn equation. After presenting the setting, which was introduced by Frohlich and Struwe, I will briefly talk about the boundary regularity result, in joint work with Stephen Lynch, as well as more recent results and related open problems. Joint Geometry and Analysis Seminar 11 December - Yorkshire Durham Geometry Day in Durham Geometry Seminar 2023-2024: Semester 1/Semester 2 Geometry Seminar 2022-2023: Semester 1/Semester 2 Previous Leeds Geometry Seminars |