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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 2: 24 January - 15:00 - Worsley 9.59 - in person Samuel Borza (Scuola Internazionale Superiore di Studi Avanzati) Note the unusual room! The measure contraction property in the sub-Finsler Heisenberg group Abstract. The Heisenberg group is a source of inspiration for many fields in mathematics and physics, including quantum theory, metric geometry, and harmonic analysis. I will discuss the sub-Finsler geometry of the Heisenberg group and explain how it is related to the isoperimetric problem in the non-Euclidean (Finsler) plane. We will then explore approaches to studying the curvature of the sub-Finsler Heisenberg group, focusing particularly on the measure contraction property that appears in the analysis of metric measure spaces. This is a joint work with Kenshiro Tashiro, Mattia Magnabosco, and Tommaso Rossi. Geometry Seminar 7 February - 15:00 - Roger Stevens LT11 - in person Myles Workman (University College London) Minimal Hypersurfaces: Bubble Convergence and Index Abstract. The regularity theories of Schoen--Simon--Yau and Schoen--Simon for stable minimal hypersurfaces are foundational in geometric analysis. Using this regularity theory, in low dimensions, Chodosh--Ketover--Maximo and Buzano--Sharp studied singularity formation in sequences of minimal hypersurfaces through a bubble analysis. I will review this background, before talking about my recent work in this bubble analysis theory. In particular I will show how to obtain upper semicontinuity of index plus nullity along a bubble converging sequence of minimal hypersurfaces. Joint Geometry and Analysis Seminar 14 February - 15:00 - Roger Stevens LT11 - on zoom Amirmasoud Geevechi (University of Toronto) Slow Motion of Vortex filaments in the Abelian Higgs Model Abstract. Abelian Higgs model is a system of partial differential equations for describing the interaction of the Higgs field and the electromagnetic field, a toy model extracted from the Standard model. The equations enjoy some local symmetry or also called gauge symmetry. The solutions to the time-independent and 2D version of the equations have been constructed by Jaffe and Taubes in 1980. These are called vortex configurations. In this talk, I will mention the main result of my thesis with Prof. Robert Jerrard about how one can glue the 2D solutions and add perturbations to them in order to construct time-dependent solutions in (3 + 1)D. The final result is that one can construct solutions in (3 + 1)D arbitrary close to wave maps to the moduli space of vortex configurations, for long time. This is a generalization of a result by David Stuart in 1994 where the dynamics in (2 + 1)D has been approximated by a Hamiltonian dynamics on the moduli space. We will see that suitable gauge conditions are crucial in various steps of the construction in order to make the equations massive and stable. This is the so-called Higgs mechanism. Geometry Seminar 21 February - 16:30 - Roger Stevens LT11 - in person Luciano Luzzi Junior (IMPA) Note the unusual time! Balanced metrics, Zoll deformations and isosystolic inequalities in ℂPn Abstract. The k-systole of a Riemannian manifold is the infimum of the volume over all homologically non-trivial k-cycles. An astonishing result by M. Berger and M. Gromov states that in the complex projective space ℂP2, the Fubini-Study metric locally maximizes the codimension two systole over all metrics with unitary volume. The objective of this talk is to discuss the behavior of the codimension two systole in the context of higher-dimensional complex projective spaces. In particular, we intend to argue that in ℂPn for n ≥ 3, the Fubini-Study metric locally minimizes the systole over the set of volume-normalized balanced metrics. As an application, we will characterize the systolic behavior of almost-Hermitian 1-parameter Zoll deformations of the Fubini-Study metric. Joint Geometry and Analysis Seminar 28 February - 15:00 - Roger Stevens LT11 - on zoom Melanie Rothe (Technische Universitaet Darmstadt) Lawson’s Bipolar Minimal Surfaces in the 5-Sphere Abstract. Unlike Euclidean space ℝn, the n-sphere 𝕊n allows for closed minimal surfaces. Interestingly, the latter turned out to be highly relevant in the geometric optimization of Laplacian eigenvalues or the Willmore energy under fixed topology. However, regarding each topological class, the list of known examples is very limited. Concerning higher codimensions, H. B. Lawson has shown in 1970 that every minimal surface in 𝕊3 induces a minimal surface in 𝕊5, its so-called bipolar surface. For closed surfaces, the example of the bipolar Lawson surfaces ˜τm,k⊆𝕊5 showed that the topology can thereby undergo a change. From this perspective, we give a topological classification of the bipolar Lawson surfaces ˜ξm,k and ˜ηm,k. Additionally, we provide upper and lower area bounds, and find that these surfaces are not embedded for m≥2 or k≥2. Geometry Seminar 13 March - 15:00 - Roger Stevens LT11 - in person Josh Daniels-Holgate (Hebrew University of Jerusalem) Mean curvature flow from conical singularities Abstract. We discuss some regularity results for mean curvature flow from smooth hypersurfaces with conical singularities. We then discuss how to use these results to tackle two conjectures of Ilmanen. Joint Geometry and Analysis Seminar 20 March - 15:00 - Roger Stevens LT11 - in person Francesca Tripaldi (Scuola Normale Superiore di Pisa) Multicomplexes on Carnot groups Abstract. Carnot groups are the easiest examples of subRiemannian manifolds. We show that, on Carnot groups, the de Rham complex has the structure of a multicomplex. The extra algebraic structure given by the multicomplex has several applications, such as extracting sharper subcomplexes that are homotopic to the de Rham complex. Joint Geometry and Analysis Seminar 24 April - Yorkshire Durham Geometry Day in York 1 May - 13:30 - Roger Stevens LT23 - on zoom Aaron Naber (Northwestern University) Note the unusual time! Nonlinear Harmonic Maps and the Energy Identity. Abstract. We will begin this talk with a broad introduction to nonlinear harmonic maps with a discussion of some basic examples and background. We will then focus our attention on a type of singularity formulation, resulting in so called defect measures. Though not typically studied in such generality, these defect measures are a general construction for understanding the loss of energy in limits of H1 functions. In the case of nonlinear harmonic maps, there is a precise conjecture as to the form of these defect measures called the Energy Identity. This was recently proved together with Daniele Valorta and will be the focus of the talk. Joint Geometry and Analysis Seminar Geometry Seminar 2022-2023: Semester 1/Semester 2 Previous Leeds Geometry Seminars |