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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 2: 1 February 2023 Moved due to strike action. 8 February 2023 - 15:00 - Roger Stevens LT11 Jason Cantarella (University of Georgia) "Topological" polymers and random graph embeddings or "Can you hear the structure of a polymer network?" Abstract. Biological structures such as collagen are often composed of complex networks of polymers. The chemistry and physics of these substances are a function of the random shapes that these networks assume in solution as they are perturbed by thermal motions. In this talk, we'll discuss the surprisingly deep and interesting geometry and topology involved in predicting the statistical properties of the shape ensemble from the graph structure and the type of bonds in the model. We will first give a generalization of the Gaussian theory of elastic networks, seeing how the spectrum of the combinatorial and Riemannian Laplace operators allow us to predict the average size of molecules in solution. We validate the model with experimental data from recently synthesized network polymers from the Tezuka lab at Tokyo Tech. We'll then switch to the freely-jointed network model, where the distances between monomers are fixed. In this model, the conformal geometry of the Poincare disk will be used to develop new algorithms for ring polymer ensemble sampling. The theme of the talk is that there are many more applications for differential geometry and algebraic topology in physics and biology than one might expect. We'll include some open questions. The talk will cover joint work with Clayton Shonkwiler (Colorado State), Tetsuo Deguchi and Erica Uehara (Ochanomizu University), and Henrik Schumacher (Chemnitz). Special joint Pure Maths/Theoretical Biophysics Seminar 15 February 2023 - 15:00 - Cancelled due to strike action, Jakob will speak in the YDGD instead. Jakob Stein (University College London) $G_2$-instantons on the spinor bundle of the 3-sphere. Abstract. Special Yang-Mills fields known as $G_2$-instantons appear naturally in when considering higher-dimensional analogues of anti-self-dual instantons in dimension four. However, examples are hard to construct in general. In this talk, I will discuss a recent joint work with Matt Turner (Bath), constructing of a new family of $G_2$-instantons using co-homogeneity one techniques, and describing the space of $G_2$-instantons on the spinor bundle of the 3-sphere, equipped the metric of Bryant-Salamon, using the deformation theory of Driscoll. 22 February 2023 - 16:30 - Roger Stevens LT11 - Moved due to strike action Theodora Bourni (University of Tennessee Knoxville) Joint Geometry and Analysis Seminar 1 March 2023 - Strike action expected 8 March 2023 - 16:30 - Roger Stevens LT11 Wei-Bo Su (University of Warwick) Glueing constructions in Lagrangian mean curvature flow. Abstract. Glueing construction has been used extensively to construct solutions to nonlinear geometric PDEs. In this talk, I will focus on the glueing construction of solutions to Lagrangian mean curvature flow. Specifically, I will explain the construction of Lagrangian translating solitons by glueing a small special Lagrangian 'Lawlor neck' into the intersection point of suitably rotated Lagrangian Grim Reaper cylinders. If time permits, I will also discuss the main idea of an ongoing joint project with Chung-Jun Tsai and Albert Wood, where we investigate the construction of solutions to Lagrangian mean curvature flow with infinite time singularities. Joint Geometry and Analysis Seminar 15 March 2023 - 15:00 - Roger Stevens LT11 - Cancelled due to strike action Julian Scheuer (Goethe Universität Frankfurt) Quermassintegral preserving curvature flows in the sphere Abstract. It is known since the seminal work by Huisken on the volume preserving mean curvature flow (VPMCF) from 1987, that the VPMCF for hypersurfaces of the sphere may lose convexity and hence a study of this flow has been avoided so far. The same problem arises for every nonlocal flow which preserves any of the other quermassintegrals. Without the presence of a global term, Guan and Li have constructed a flow within the sphere, which preserves the volume and converges smoothly to a geodesic sphere. However, the corresponding quermassintegral preserving versions of this flow are not understood until today, because it seems impossible to prove curvature estimates. In this talk, we introduce a quermassintegral preserving flow for hypersurfaces of the sphere, which is constrained by a global and a local term. This flow is shown to converge to a geodesic sphere and hence provides the first such example of a quermassintegral preserving curvature flow in the sphere, for which one can obtain smooth estimates. This is joint work with Esther Cabezas-Rivas. 29 March 2023 - 15:00 - Roger Stevens LT11 Yang Li (Massachusetts Institute of Technology) A variational program for the Thomas-Yau conjecture Abstract. The Thomas-Yau conjecture is concerned with relating the existence question of special Lagrangians to stability conditions and Fukaya categories. After a very brief recap on Floer theory, we will sketch a variational framework to tackle the conjecture, and the main bulk of the talk is on the analytical aspects. It turns out that a very weak minimizer exists under a suitable Thomas-Yau semistability condition, and the main open problem is to prove the Euler Lagrange equation, which is the special Lagrangian condition. 26 April 2023 - 16:30 - Chemistry West Block Lecture Theatre E (G.76) Note the unusual room! Sugata Mondal (University of Reading) Hot spots and other stories Abstract. I will recall the hot spots conjecture due to J. Rauch and review past and recent results on this conjecture. My talk will focus mainly on planar domains. If time permits, I will try to explain a few related problems. Joint Geometry and Analysis Seminar 3 May 2023 - 16:30 - Roger Stevens LT11 Carla Cedarbaum (Universität Tübingen) Explicit minimizing sequences related to the stability of the Riemannian Penrose Inequality Abstract. In 2015, Mantoulidis and Schoen suggested a rather explicit construction of asymptotically flat Riemannian 3-manifolds of non-negative scalar curvature with minimal, strictly outward minimizing inner boundary relying on a conformal flow of metrics. This construction was tailored to compute the so-called Bartnik mass functional for minimal surfaces. It turned out to be extremely useful also for understanding the stability (or almost rigidity) of the Riemannian Penrose inequality, a central geometric inequality in General Relativity proved by Bray and by Huisken and Ilmanen. After reviewing their construction, I will discuss how it can be refined to prove instability of the Riemannian Penrose inequality. This is joint work with Armando Cabrera Pacheco, building on ideas developed by Cabrera Pacheco, McCormick, Miao, Xie (in alphabetic order) and the speaker. Joint Geometry and Analysis Seminar Previous Leeds Geometry Seminars |