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 in person at UCL in Term 3 (May to June 2023), on Thursday from 14:00 - 15:00 BST in Ricardo Lecture Theatre B03, Drayton House (address: 30 Gordon Street, London, WC1H 0AX). After the seminar we will head to the common room in the UCL maths depart for tea and biscuits. All are welcome! The goal of this seminar, above all, is to be useful to its audience. 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 Michela Barbieri (michela.barbieri.21@ucl.ac.uk).

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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 3

May 2023 - June 2023


8 June 2023


Reflection equivalence in Coxeter groups

Dr. David Sheard (King's College London)
Ricardo Lecture Theatre B03, Drayton House • 14:00 - 15:00 BST

Abstract: I will introduce Coxeter groups, a rich class of finitely generated groups with connections to geometry, combinatorics, number theory, Lie theory, and much more. One definition (if made suitable precise), is that they are groups generated by reflections which act discretely. Studying the geometry of how they act on various spaces is often the key to understanding them. I will focus on one question: how can we classify the sets of reflections which generate a Coxeter group (up to a suitable notion of equivalence)? The suitable notion of equivalence is a version of Nielsen equivalence adapted to the reflection setting, and it has an elegant interpretation in terms of hyperplane arrangements. I will prove that any reflect generating set of a Coxeter group is equivalent to a “geometrically simple” generating set, and survey how this relates to other results in surface and Fuchsian groups.

1 June 2023


What is a Higgs bundle?

Enya Hsiao (University of Heidelberg)
Ricardo Lecture Theatre B03, Drayton House • 14:00 - 15:00 BST

Abstract: Higgs bundles arose historically in 1987 as solutions to Hitchin's equations on compact Riemann surfaces. In this talk, we will explain what a Higgs bundle is, and introduce the main tools to study them, namely, stability conditions and the Hitchin fibration. If time permits, we will briefly discuss how Higgs bundles enter the study of character varieties and thereby higher Teichmüller theory.

25 May 2023


The topology of moduli spaces

Robert Crumplin
Ricardo Lecture Theatre B03, Drayton House• 14:00 - 15:00 BST

Abstract: In the recent few years, Chan-Galatius-Payne used tropical techniques to calculate the growth of the rational cohomology of M_{g,n}. I will explain the ideas behind their work, and if time permits, present a few new applications of this technique to deduce vanishing top weight cohomology in certain moduli spaces of stable log maps to pairs (X | D) for D a smooth divisor.

18 May 2023


Constructing Moduli Spaces in Gauge Theory

Jaime Mendizabal Roche (UCL)
Ricardo Lecture Theatre B03, Drayton House• 14:00 - 15:00 BST

Abstract: Many gauge theory equations give rise to interesting moduli spaces. One such case is that of monopoles, which are solutions to the Bogomolny equation over a 3-manifold. In this talk, we will introduce some important concepts in gauge theory and the study of monopoles, and illustrate some of the tools necessary to construct their moduli spaces as quotients of infinite-dimensional objects.

11 May 2023


Global Kuranishi charts for moduli spaces of stable maps and a product formula

Amanda Hirschi (University of Cambridge)
G25 John Adams Hall, Institute of Education• 14:00 - 15:00 BST

Abstract: The moduli space of stable pseudoholomorphic maps to a symplectic manifold has been an important object in symplectic geometry since its definition. Unfortunately, it is in general not cut out transversely, which makes it difficult to obtain geometrically useful information. I will explain a presentation of this moduli space as the zero locus of an orbisection. This allows for a straightforward definition of symplectic invariants. To show how this construction works in practice, I will sketch the proof of a formula for the Gromov-Witten invariants of a product of symplectic manifold and give some examples.

Term 2

January 2023 - March 2023


22 March 2023


Algebraic properties of differential operators on almost complex manifolds

Jiahao Hu (Stony Brook)
Online: Zoom Link • 14:00 - 15:00 GMT

Abstract: The exterior differential d on complex-valued differential forms of complex manifolds decomposes into the Cauchy-Riemann operator and its complex conjugate. Meanwhile on almost complex manifolds, the exterior d in general has two extra components, thus decomposes into four operators. In this talk, I will introduce these operators and discuss the structure of the (graded) associative algebra generated by these four components of d, subject to relations deduced from d squaring to zero. Then I will compare this algebra to the corresponding one in the complex (i.e. integrable) case, we shall see they are very different strictly speaking but similar in a weak sense (quasi-isomorphic). This is based on joint work with Shamuel Aueyung and Jin-Cheng Guu.

15 March 2023


On the Picard lattice of K3 surfaces and its isometries

Wim Nijgh (Leiden University)
Online: Zoom Link • 14:00 - 15:00 GMT

Abstract: Automorphisms of K3 surfaces are widely studied. One well-known result is a finiteness theorem relating the automorphisms of a K3 surface with the isometries of its Picard lattice. The goal of this talk is to explain this relation by giving an introduction to lattices and the Picard group of a K3 surface. If time permits, we will also show some original research for the case of K3 surfaces over non-algebraically closed fields.

8 March 2023


Berkovich Spaces and Non-Archimedean Geometry

Aporva Varshney (UCL)
Online: Zoom Link • 14:00 - 15:00 GMT

Abstract: Over the complex numbers, a variety can be considered as a complex analytic space, which reflects the geometry of the variety through "GAGA"-type theorems. Attempting to copy this process naïvely over the p-adics fails horribly: for example, the p-adics are totally disconnected as topological spaces, while the algebraic affine line is connected. This necessitates a more subtle definition of a non-Archimedean analytic space. In this talk, I will introduce Berkovich spaces, which are one way to overcome this problem. These spaces find many uses across both number theory and geometry, have nice topological properties and admit similar GAGA theorems. After constructing the analytic affine line, we will consider the key notion of a "skeleton", draw an explicit picture of the analytification of an elliptic curve and briefly discuss the links of the theory with mirror symmetry.

1 March 2023


Flops and Deformations

Charlotte Llewellyn (University of Glasgow)
Online: Zoom Link • 14:00 - 15:00 GMT

Abstract: Spherical objects were first introduced in 2000 by Seidel and Thomas as mirror symmetric analogues of Lagrangian spheres on a symplectic manifold. A well-known example of a spherical object is the structure sheaf of a (-1, -1) 3-fold flopping curve. In this talk I will explain how, using deformation theory, Toda was able to relate more general flopping curves to this construction. Time permitting, I will also explain how this result was generalised by Donovan and Wemyss using noncommutative deformations.

22 February 2023


An Introduction to Khovanov homology and Link Cobordisms

Michael Kohn (Durham University)
Online: Zoom Link • 14:00 - 15:00 GMT

Abstract: As an area of study, knot theory has been propelled over the last twenty years by two main viewpoints; Floer theories and Khovanov homology. The aim of the talk will be to introduce Khovanov homology, a field of study which combines algebraic topology with the combinatorics of knot diagrams and has ties to other perspectives on knot theory. Assuming very little knot theory, the talk will have three parts; we will first cover how to build a chain complex from a link diagram in such a way that taking its homology yields a powerful, bigraded invariant that extends a simple, known knot invariant. Next, we will look at how Khovanov homology can be upgraded to a functor from a certain category of links, and ways in which this gives us information about smooth manifolds. Time permitting, we will also explore some surprising applications of this invariant and relations to other knot invariants.

15 February 2023


Discriminants and semi-orthogonal decompositions

Michela Barbieri (University College London)
Online: Zoom Link • 14:00 - 15:00 GMT

Abstract: Geometric Invariant Theory (GIT) gives us a theory of how to take quotients in a way that is 'nice' in an algebraic geometry sense. In 2020, inspired by mirror symmetry, Kite-Segal conjectured the following: Given a wall in a toric Calabi-Yau GIT problem and what we call 'a minimal face of the primary polytope', the multiplicity on the A-side is equal to the multiplicity on the B-side. The theorem was proved in 2022 by Huang and Zhou. The goal of this talk is to explain all the words in this theorem and perhaps to motivate it. Fortunately for us, this doesn't require Hartshorne as a prerequisite. The statement and its proof turn out to be mostly a combinatorics problem thanks to toric geometry. Throughout the talk we will explain all the ingredients needed, including some GIT theory, toric geometry, and a refreshment on derived categories.

8 February 2023


Basic Examples of Homological Mirror Symmetry

Edwin Hollands (University College London)
Online: Zoom Link • 14:00 - 15:00 GMT

Abstract: Mirror symmetry arose from a connection between A-model and B-model string theory and uncovered a deep link between symplectic and algebraic geometry. Kontsevich suggested the study of this phenomenon as equivalence between certain derived categories: Fukaya categories and categories of coherent sheaves. In this talk we will cover just enough background and definitions to be able to calculate and understand some simple examples of this homological perspective on mirror symmetry.

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.