KCL/UCL Junior Geometry Seminar 2022-2023

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, Geometric Analysis, Geometric Group Theory, Topology, and related areas.

The target audience is young researchers — in particular PhD students. This seminar will run entirely online from January to April 2023, on Wednesday from 14:00 - 15:00 GMT. Anyone can join via the zoom link KCL/UCL Junior Geometry

The format of this seminar will be somewhat unusual. While talks will have the usual 45 minutes of content, there will be an added focus on facilitating questions and free discussion with the audience. Listeners are invited to ask about unfamiliar terminology, while speakers are encouraged to define important basics. All are rewarded with free food.

The goal of this seminar, above all, is to be useful to its audience - mathematically, socially, possibly even nutritionally (when in person). If you would like to hear about a particular topic, give joint talks or help facilitate a discussion during a talk that interests you, feel free to let us know.

For any reason, please contact us at Michela Barbieri (michela.barbieri.21@ucl.ac.uk) or Ananya Satoskar (ananya.satoskar.21@ucl.ac.uk).

Subscription

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Other Junior Seminars in London

KCL/UCL Junior Number Theory: https://sites.google.com/view/yyuan304/london-junior-number-theory-seminar

Imperial Junior Geometry: https://www.imperial.ac.uk/geometry/seminars/junior-geometry-seminar/



Term 2

January 2023 - March 2023


1 February 2023


Spaces of Polyhedra

Ananya Satoskar (King's College London)
Online:
Zoom Link 14:00 - 15:00 GMT

Abstract: This talk is an exploration of W. P. Thurston’s beautiful paper ‘Shapes of Polyhedra and Triangulations of the Sphere’, which examines moduli spaces of polyhedra arising from triangulations of the sphere. We will seek to understand the first result of the paper (namely, that these spaces are locally isometric to a complex hyperbolic space) using Alexandrov’s work on convex polyhedra. I will define all required basic definitions in a friendly and intuitive manner and use these to build the proof in real-time using a slightly different, more combinatorial approach to the material based on lecture notes by R. Schwartz. The talk is largely self-contained; there will be many enjoyable pictures and, if time permits, some pleasant geometric digressions.

Term 1

October 2022 - December 2022


5 December 2022


Geometry and holonomy of cones

Enric Sole Farre (UCL,LSGNT)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: Given a Riemannian manifold (M,g), a natural object to consider is its associated metric cone M x R^+, dr^2+r^2 g_M). The geometric properties of the cone can then be rephrased in terms of properties of M itself, which can be then used to define new (or rediscover) geometries on M. In my talk, I will explore this question from the perspective of special holonomy and describe the associated special geometries.


28 November 2022


Geometric Invariant Theory

Thamarai Venkatachalam (Imperial, LSGNT)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: GIT is a theory for obtaining quotients in algebraic geometry sense. If we have a (reductive) group acting on an algebraic variety, GIT gives a recipe for the construction of the quotient variety with respect to this action. In this talk, we will see the general construction of GIT quotients. We will also see a few examples to see how it works exactly in very special cases.


21 November 2022


A Hitchhikers guide to gluing constructions

Andries Salm (UCL)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: In the quest of finding manifolds with special metrics, people often want to combine known solutions and glue them together. For example, the first compact G2 manifolds were constructed this way. Although this method looks simple, and might be reduced to a single sentence in certain physics papers, how hard is it actually? In this talk we work out a simple example of this gluing construction and talk about the technicalities one must be wary of.


14 November 2022


Points are Galois representations

Diego Chicharro Gordo (King's, LSGNT)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: Algebraic varieties have many cohomology groups attached to them: Betti cohomology, de Rham cohomology, étale cohomology, and crystalline cohomology. But they are not too different. In fact, under reasonable assumptions, there are comparison isomorphisms relating them. This led Grothendieck to the idea of having a "universal cohomology theory" which any other (good) cohomology theory is a particular case of. This is (conjecturally) realised with his construction of the category of pure motives, whose objects can be thought of as the "essence" of algebraic varieties. In this talk, I will try to explain and motivate this construction and show that when we restrict our attention to the simplest varieties one can think of, namely points, one recovers the familiar notion of Galois representation that is well-known to number theorists.


7 November 2022


An introduction to Topological Data Analysis – or how we can actually have applications of Pure Maths!

Ines Garcia Redondo (Imperial, LSGNT)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: In this talk I will give an introduction of Topological Data Analysis, or more precisely, to one of its main tools: Persistent Homology. The idea behind Persistent Homology is to study the changes on the homology groups of a sequence of nested simplicial complexes. The theory of persistence started around 2005 and has since developed very solid foundations. In the talk I plan to cover some of these, while discussing with you why those where necessary or relevant to the theory. After that, we will relate these pretty much theoretical tools to data problems, and discuss why persistence might be useful in other contexts such as biology of medicine.


31 October 2022


An Introduction to Knot Floer Homology

Sara Veneziale (Imperial, LSGNT)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: Apart from being very popular with fishermen and climbers, knots are rich mathematical objects, whose classification is still of great interest. Classifying different knots comes down to defining invariants, which can be as easy as the crossing numbers or as complicated as homology theories. In this talk, we quickly go through the basics of knot theory and define a very powerful recent invariant, Knot Floer Homology, which was introduced by Ozsváth and Szabó and independently by Rasmussen. Time-permitting we will delve into the details of the Ozsváth-Szabó construction of this invariant, whose approach is more combinatorial and allows it to be calculated by a computer program.


24 October 2022


Arithmetic Hyperbolic Manifolds

Mads Christensen (UCL)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: A hyperbolic manifold is a quotient of hyperbolic space H^n by a discrete group of isometries. In particular, an arithmetic hyperbolic manifold is a quotient of H^n$ by a group which is in some sense arithmetic. For example, taking the quotient of H^2 by an action of SL_2(Z) yields more or less the (level 1) modular curve, which is a beloved object of number theorists around the world. Starting from the beginning, I will try to explain some of the rich structure of these arithmetic hyperbolic manifolds which make them particularly nice to study.

17 October 2022


On Banach's isometric subspaces problem

Danya Mamaev (PDMI RAS, LSGNT)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: Let V be a (real) normed vector space and suppose that for a fixed 2 < = k < dim V all k-dimensional vector subspaces of V are isometric. Is it true that the norm on V is induced by an inner product?

This question was asked by Stephan Banach in 1932, and partial (affirmative) answers were given by Stanislaw Ulam et al. (1935), Arie Dvoretzky (1959), Misha Gromov (1967), and Luis Montejano et al. (2019). Recently in the preprint https://arxiv.org/abs/2204.00936

Sergei Ivanov, Anya Nordskova and myself gave a positive answer to this question in the case k = 3, but the problem remains open for k + 1 = dim V = 4l with l >= 2 and k + 1 = dim V = 134.

In the talk I will explain why the problem is natural to ask, give several reformulations of the problem, sketch the proofs in some cases, and, if time permits, give a vague idea of what goes on in our preprint. No prerequisites other than basic linear algebra will be needed to understand most of the talk.

10 October 2022


Formal minimal surfaces, informally

Nick Manrique (Imperial, LSGNT)
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: Minimal surfaces are critical points of the area functional, and have been of great interest to differential geometers since the time of Euler and Bernoulli. Their study usually involves situations where we fix some target manifold and ask whether it contains minimal surfaces of certain kinds, and if so how many, and so on. In this talk, we will approach things from the opposite direction: given some surface, can it be realised as a minimal surface in some ambient space? When we insist that the ambient space is hyperbolic, it's possible to obtain an elegant answer to this question via the concept of a formal minimal surface. After introducing these, we will attempt to say something about global aspects of their geometry, some recent results and open problems.

03 October 2022


K3 Surfaces

Sebastian Monnet (KCL, LSGNT) https://www.smonnet.com
UCL, Room 337 David Sacks, Rockefeller Building 17:30 - 19:30

Abstract: K3 surfaces are currently very trendy in algebraic geometry and arithmetic geometry. According to an anonymous source, they are "the simplest type of surface we don't understand", which makes them good to study, since we might actually make some progress. The goal of the talk is to explain the terminology required to define K3 surfaces, give a few examples, and discuss some key properties.