## Information

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

 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.