## Information

The KCL/UCL Junior Geometry Seminar is a joint seminar of King's College London and University College London. Speakers present topics from Differential Geometry, Algebraic Geometry, Topology, Geometric Analysis, Geometric Group Theory, and related topics.

The target audience is young researchers—in particular PhD students—from all London universities. The atmosphere is friendly and informal, most talks are accessible to a wide audience, and snacks are provided.

To receive updates about upcoming talks you can subscribe to our mailing list by emailing "subscribe" to maths-juniorgeom-subscribe@ucl.ac.uk. For inquiries please contact Matthew Habermann (m.habermann.17@ucl.ac.uk) and Daniel Platt (daniel.platt.17@ucl.ac.uk).

## Schedule for Online Seminars in Term 3

 Thu 30.04.20 17.00-18.15 Location: Zoom Title: Resolution of 4-dimensional symplectic orbifolds Speaker: Lucía Martín Merchán Abstract: An orbifold is a space which is locally modelled on balls of the euclidean space quotient by a finite group. These have been very useful in many geometrical contexts; in the setting of symplectic geometry, symplectic orbifolds have been introduced mainly as a way to construct symplectic manifolds by resolving their singularities. Some authors have provided methods for resolving particular types of symplectic orbifold singularities; but there are no techniques to resolve general symplectic orbifolds. In this talk we give a method to resolve arbitrary symplectic 4-orbifolds making use of techniques of resolution of quotient singularities and gluing symplectic forms. This is a joint work with J. Rojo. Thu 07.05.20 Title: No talk Thu 14.05.20 17.00-18.15 Location: Zoom Title: Spin(7) metrics admitting Kähler reduction Speaker: Udhav Fowdar Abstract: Spin(7) manifolds are Ricci flat 8-dimensional Riemannian manifolds. Together with G2, they form the class of exceptional holonomy manifolds. The first examples were constructed by Bryant and Salamon in 1989 and since then many more have appeared, though very few explicit. In this talk, I will describe a method of finding many explicit local examples of Spin(7) metrics admitting 2-torus actions such that the quotient 6-manifolds are Kähler. The key ingredients will be the symplectic reduction and the Gibbons-Hawking ansatz. The talk will be non-technical and I will give a comprehensive background. Thu 21.05.20 17.00-18.15 Location: Zoom Title: Deformation theory of nearly G_2 manifolds Speaker: Ragini Singhal Abstract: In this talk we will discuss the deformation theory of nearly G_2 manifolds. We will start by describing some identities for 2 and 3 forms on such manifolds. After introducing a Dirac type operator we will prove a result on the cohomology of nearly G_2 manifolds. Along the way we will reprove a result of Alexandrov—Semmelman on the space of infinitesimal deformation of nearly G_2 structures. Finally we will discuss the progress so far for the second order deformations. This is all a joint work with Shubham Dwivedi (University of Waterloo). Thu 28.05.20 17.00-18.15 Location: Zoom Title: Cohomogeneity one Spin(7)-manifolds Speaker: Fabian Lehmann Abstract: An 8-dimensional Riemannian manifold with holonomy group contained in Spin(7) is Ricci-flat, but not Kahler. The condition that the holonomy reduces to Spin(7) is equivalent to a complicated system of non-linear PDEs. In the non-compact setting, symmetries can be used to reduce this complexity. In the case of cohomogeneity one manifolds, i.e. where a generic orbit has codimension one, the non-linear PDE system reduces to a nonlinear ODE system. I will discuss examples where existence questions of Spin(7) holonomy metrics give rise to an interesting problem for ODE systems. Thu 04.06.20 17.00-18.15 Location: Zoom Title: $SU(2)^2\times U(1)$-invariant $G_2$-instantons Speaker: Matt Turner Abstract: $G_2$-instantons in 7-dim are analogous to ASD-instantons in 4-dim; this motivates the idea of using $G_2$-instantons to construct enumerative invariants of $G_2$-manifolds. Therefore it is natural to search for examples of $G_2$-instantons on all known constructions of $G_2$-manifolds. This becomes easier when the manifolds have a high level of symmetry. In this talk, we consider $G_2$-manifolds which have a cohomogeneity one action of $SU(2)^2\times U(1)$, while also being asymptotically conical. We will discuss the progress that has been made to find instantons on these manifolds. Thu 11.06.20 17.00-18.15 Location: Zoom Title: t.b.c. Speaker: t.b.c. Abstract:

## Schedule for 2019/20

 Thu 10.10.19 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Cutting up surfaces and commutators in free groups Speaker: David Sheard Abstract: When is an element $g\in G$ of a group a product of commutators $[a,b]$? What is the minimal number of commutators $n$ such that $g=[a_1,b_1]\cdots[a_n,b_n]$? What are all possible solutions $(a_1,b_1,\dots,a_m,b_m)$ to the equation $g=[a_1,b_1]\cdots[a_m,b_m]$? Very difficult, yet important, questions --- but ones whose answers seem to lie in the darkest recesses of combinatorial group theory. Not so! At least for $G$ a free group, these questions can all be answered elegantly and beautifully by cutting up and colouring surfaces. In this talk I shall present solutions to these problems with an emphasis on drawing nice pictures. Thu 17.10.19 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Character Varieties of Surface Groups Speaker: John McCarthy Abstract: The points of a character variety classify the representations of a group up to equivalence. Such representation-theoretic objects admit alternative descriptions coming from symplectic geometry, algebraic geometry, and gauge theory. In this we will investigate the character varieties of surface groups, and discuss their relationship to important moduli spaces appearing in algebraic and differential geometry. Thu 24.10.19 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Calibrated Submanifolds Speaker: Benjamin Aslan Abstract: Geodesics on a Riemannian manifold encode crucial information about its geometry such as the curvature. The same is true for the higher dimensional analogue of geodesics: minimal surfaces. On a manifold with special holonomy, there is distinguished class of minimal submanifolds, called calibrated submanifolds. The hope is that the space of calibrated submanifolds contains even finer information about the ambient manifold, potentially leading to new invariants. In this talk, we will review some basic results of calibrated geometry and then go on and see in which geometries the space of deformations of a calibrated submanifold is automatically smooth and compute its dimension. Thu 31.10.19 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: J-holomorphic curves and Lagrangians Speaker: Chris Evans Abstract: We investigate the properties and interactions of the two most important submanifolds of symplectic spaces. The discussion will focus on geometry and intuition. We aim to cover some of the core ideas in Gromov's original paper: Gromov compactness and bubbling, existence of discs on Lagrangians, and non-standard symplectic structures. Thu 07.11.19 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: An introduction to derived categories Speaker: Bradley Doyle Abstract: To any algebraic variety (or any ring, scheme, stack, etc) one can associate an associated category called the derived category. I will explain what this category is and provide motivation for why it is useful. After this I will cover some basic properties and results, finally I will explain one technique, semiorthogonal decompositions, that is used to study them. In the special case of projective space this will provide a connection with quiver algebras. Despite the derived category being used to study geometric objects this talk will be more algebraic than geometric. Thu 14.11.19 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Handle decomposition and Kirby calculus Speaker: Angela Wu Abstract: Here are two interesting questions: 1. What's the best way to visualize a four dimensional manifold? 2. How can you hold a glass of wine in the palm of one hand and without changing your grip or moving your feet, rotate the glass through 720 degrees about a fixed axis without spilling the wine? The answers to both of these questions will be presented in this talk! Thu 21.11.19 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: From triangles to algebraic stacks, a crash course in geometry Speaker: Luigi Lunardon Abstract: A stack is a category fibered in groupoids over a certain Grothendieck topology such that isomorphisms are a sheaf and every descent datum is effective. This definition is concise and correct, but definitely not enlightening. Moreover, it doesn't motivate the necessity of introducing stacks in the study of moduli problems, nor give any geometric intuition about them. The aim of this talk is to motivate algebraic stacks as a natural object in algebraic geometry and explain which properties we wish them to have. To do this, we start with a classifying problem for triangles, and then translate it in the language of category theory. Thu 28.11.19 Title: Seminar Cancelled Thu 05.12.19 Title: Seminar Cancelled Thu 12.12.19 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Existence and regularity of area minimizers , a brief intoduction to Geometric Measure Theory Speaker: Konstantinos Leskas Abstract: An interesting and extensively studied mathematical problem is , given a boundary in R^n find a manifold that minimizes area with that boundary. We will start by cosidering the case of graphs and discussing existence and regularity in this setting. Then we will pass to the more general problem, see what can go wrong even in the class of graphs and motivate ourselves to work with a more general class of surfaces, suitable for establishing existence. We finish this talk with Allard's regularity theorem for area minimizers , relating it with the PDE approach in the begining. If time permits we may discuss about optimal regularity results in codimension 1. Thu 16.01.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Perf(X) vs D^b(X) Speaker: Federico Barbacovi Abstract: Smoothness is a central concept in geometry. Whether you are working in algebraic geometry or differential geometry, if your scheme/manifold is not smooth you may have some troubles. When working with derived categories, one finds out that the singularities of a scheme are captured by the difference between perfect complexes and bounded complexes. However, understanding how distant Perf(X) and D^b(X) are is not always an easy task to accomplish. In this talk I will survey the topics mentioned above and introduce the category of matrix factorizations. Thu 23.01.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Polyhedral Manifolds Speaker: Tom Sharpe Abstract: Polyhedral manifolds are in some sense discrete versions of Riemannian manifolds, where metric singularities of conical type are allowed in codimension 2. In this setting, interesting invariants such as curvature and holonomy can be defined and extracted using only finite data. In my talk I will define the key notions in this field, introduce you to special class of polyhedral manifolds, and tell you some things we do and don't know about them. Thu 30.01.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Cohomogeneity-one manifolds Speaker: Jakob Stein Abstract: A natural generalisation of homogeneous spaces, say for a Lie group K, are spaces with a 1-dimensional quotient under a K-action. These are the so-called cohomogeneity-one manifolds, and there are many examples of them. The large degree of symmetry is useful when we consider cohomogeneity-one Riemannian manifolds, with isometric K-actions: cohomogeneity-one metrics have given rise to many examples of Riemannian metrics satisfying special properties e.g. Ricci flat, nearly Kahler. We will see examples, sketch some basic theory, and discuss applications. Thu 06.02.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: How can geometry help us predict the movement of planets? Speaker: Tim King Abstract: Suppose that you have a really big computer, and want to estimate where each of the planets will be in a week's time. I will describe how well you can do (i) if you ignore all the geometry and (ii) if you don't. Happily for geometers, strategy (ii) works out better. This will be an introductory talk which assumes knowledge of differential geometry, but not of Hamiltonian/Lagrangian mechanics. Keywords: Hamiltonian, Symplectic integrator Thu 13.02.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Aspherical Complex Surfaces, Proven using CAT(K) metrics Speaker: Jenny Swinson Abstract: We will have a look at a family of complex surfaces, obtained as ramified covers of the complex projective plane, ramified at a particularly special line arrangement. These complex surfaces (4-dimensional) have particular topological properties that can be shown by constructing particular singular metrics, zooming in on tangent cones around singular points, and expressing these tangent cones as bundles over...nice, simple 2-(actual)-dimensional surfaces, with spherical metrics and some conical singularities! I will explain all terminology as we go along. Thu 20.02.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Mean Curvature Flow with Surgeries Speaker: Albert Wood Abstract: It is well-known that the proof of the Poincare conjecture was completed using Ricci flow with surgeries, but how was this done, and what are surgeries? In this talk I will explain by analogy, using a theorem of submanifold geometry that is proved via my favourite geometric flow - the mean curvature flow. I will give a quick introduction to this flow, explain what surgery is and what it involves in this context, and deduce a couple of topological theorems. Thu 27.02.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: t.b.a. Speaker: t.b.a. Abstract: Thu 05.03.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Hyper-Kähler reduction Speaker: Jaime Mendizabal Abstract: A Kähler manifold carries Riemannian, complex and symplectic structures, which are compatible in a certain way. Hyper-Kähler manifolds carry a Riemannian structure, plus three complex structures (and symplectic forms) which satisfy the quaternionic relations. In particular, they are Kähler in three (and, in fact, infinitely many) ways, and their tangent spaces are quaternionic. They can also be viewed as having a holomorphic symplectic structure. This can be used to define a quotient of the manifold by a group that preserves the hyper-Kähler structure, in a way analogous to symplectic reduction for symplectic manifolds (or the closely related GIT quotient for Kähler manifolds). We will review symplectic reduction and define hyper-Kähler manifolds and reduction, and will end by show some examples. Thu 12.03.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: Homogeneous and inhomogeneous hypersurfaces in symmetric spaces Speaker: Alberto Vazquez Rodriguez Abstract: A hypersurface is (extrinsically) homogeneous if it is a regular orbit of a cohomogeneity one action. Also, a hypersurface is said to be isoparametric if it and all its nearby equidistant hypersurfaces have constant mean curvature. We will review some known classification results of these kinds of hypersurfaces in symmetric spaces. In particular, homogeneous hypersurfaces have been classified in symmetric spaces of rank one except in quaternionic hyperbolic spaces, where this problem is equivalent to an involved quaternionic linear algebra problem. ​ In this talk, I will present the classification of homogeneous hypersurfaces and the construction of inhomogeneous isoparametric examples in quaternionic hyperbolic spaces. Thu 19.03.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: t.b.a. Speaker: t.b.a. Abstract: Thu 26.03.20 17.00-18.15 Location: UCL, Department of Mathematics, 25 Gordon Street, Room 707 Title: t.b.a. Speaker: Lucía Martín Merchán (Universidad Complutense Madrid) Abstract: