The KCL/UCL Junior Geometry Seminar is a joint seminar of King's College London and University College London, presenting 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 — with the goal of being useful to those that attend. Talks can be introductory, or more advanced. If you would like to see a particular topic, or give a talk, please email one of the organizers:
Other seminars in London which may be of interest: Imperial Junior Geometry Seminar, London Junior Number Theory Seminar, DOGS (Derived Obsessed Graduate Students), UCL Geometry Seminar, KCL Geometry Seminar, London Geometry and Topology Seminar, MAGIC Seminar.
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Schedule
We are now finished for the academic year! Stay tuned for further updates regarding next year.
Summer Sessions
We held two afternoon sessions on the 2nd and 9th of June, each with three talks, snacks, and pizza at the end. Schedules and abstracts are below. Thanks to all our speakers, and to those who attended!
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Session Two - Schedule
Monday 9th June, 2pm-5pm, A1/3 Physics Building UCL
2pm-2:45pm Ruth Wye (University of Bath)
Break + snacks
3pm-3:45pm Thais Gomes Ribeiro (University of Birmingham)
Break + snacks
4pm-4:45pm Thibault Langlais (University of Oxford)
Final questions + pizza
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Session Two - Abstracts
(Ruth Wye) A family of quiver varieties
Nakajima quiver varieties (NQVs) are a useful set of varieties to study, as they are irreducible symplectic singularities, so in particular are normal. Therefore, we can learn a lot about other varieties by showing they are isomorphic to an NQV. In this talk, we will look at one particular quiver, and see how varying the stability condition means the NQV is isomorphic to other varieties, including the symmetric product and Hilbert scheme of points on surfaces related to Kleinian singularities. No prior knowledge of quiver varieties will be assumed!
(Thais Gomes Ribeiro) Hypergeometric structures of invertible pencils of prime chains
An invertible polynomial is a polynomial with integer coefficients, whose monomials are determined by an invertible matrix. They were classified by Kreuzer and Skarke in terms of three atomic types: chains, loops and Fermats. In this talk, we will deal with a one-parameter deformation of a specific atomic type from two different perspectives: over the complex field and over finite fields. Geometrically, the atomic type chain is the most rigid one. This is because the pencil associated has a trivial diagonal symmetry group. For a prime chain, the triviality of this group is expected to imply that over the complex numbers, all the non-trivial periods of the pencil associated satisfy the same hypergeometric Picard-Fuchs equation. On the other hand, considering the same invertible pencil, but now over a finite field, the triviality of the symmetry group is expected to imply that there is a unique finite field hypergeometric sum determining the point counts, which shares the hypergeometric parameters with the Picard-Fuchs differential operator. The talk will start with one motivating example, and after that, we will explore a bit of the general case that we are working on now. This is a work in progress and part of it is joint with Adriana Salerno, Eli Orvis, Jessamin Dukes, Leah Sturman, Rachel Davis and Ursula Whitcher.
(Thibault Langlais) Some observations on the geometry of G2 and Calabi-Yau moduli spaces
The goal of this talk is to describe certain geometric properties of the moduli spaces of compact manifolds endowed with special holonomy metrics (mostly G2 and SU(3)). By means of motivation, we recall how the Weil-Petersson metric on Calabi—Yau moduli spaces can be studied by Hodge-theoretic methods. We then develop a formally analogous framework to describe the geometry of G2-moduli spaces and mention a few applications.
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Session One - Schedule
Monday 2nd June, 2pm-5pm, Room 500, 25 Gordon Street UCL
2pm-2:45pm Aporva Varshney (University College London)
Break + snacks
3pm-3:45pm Michael Kohn (University of Durham)
Break + snacks
4pm-4:45pm Anisa Hassan (Brunel University of London)
Final questions + pizza -
Session One - Abstracts
(Aporva Varshney) A geometer's poor attempt at understanding physics
The appearance of geometric concepts in theoretical physics has been a great boon for geometers, giving us ideas in enumerative geometry, theories of stability conditions, and mirror symmetry. In this butchery of physics, we will focus on derived categories and how they show up in string theory. Starting with an overview of the (perhaps dry) construction of a derived category, we'll slowly spice things up by introducing physical meanings to the mathematical terms. There will be very few pre-requisites: we'll only need vector bundles, some homological algebra and an appreciation for a lack of mathematical rigor.
(Michael Kohn) Towards a constructive proof of the four colour theorem
The famous computer-aided proof of the four-colour theorem is somewhat unenlightening. In recent years, research in low-dimensional topology has produced four approaches that may lead to a constructive proof. We will outline one approach via the study of foams, taking inspiration from their use in sln homology, a generalisation of Khovanov homology.
(Anisa Hassan) On the topology of the space of rough metrics
Motivated by numerous applications in PDE and analysis, we define and study the topology of the space of Rough Riemannian Metrics (RRMs) on a fixed smooth manifold M. These metrics generalise Riemannian metrics but are only assumed to be measurable in regularity and may lack smoothness or even continuity. The central object of our considerations is the space of rough metrics, denoted Met(M), equipped with an extended metric dM. We obtain several topological properties of Met(M), starting with completeness. Moreover, we investigate whether every rough Riemannian metric can be approximated by smooth metrics within the rough metric space, and we answer this in the negative. Specifically, we analyze the closure of the subspace of smooth metrics, which turns out to be the space of continuous metrics. Although this space is generally disconnected, we show equivalence classes corresponding to metrics of finite distance via dM are path-connected. These components arise naturally in applications, such as extending results from smooth to rough settings, notably in solving a low-regular Kato square root problem through the invariance of the quadratic estimates. Furthermore, we investigate conditions under which Met(M) becomes connected. In particular, we show that compactness of the manifold M implies the connectedness of the rough metrics space. Remarkably, the converse also holds, providing an alternative criteria to determine compactness of a manifold.
Past Talks
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Title: Crepant resolutions of quotient singularities
Speaker: Austin Hubbard (University of Bath)
Monday 24th March, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
We consider quotient singularities V/G, where V is a finite-dimensional vector space and G is a finite subgroup of SL(V). V/G has a canonical bundle K, and a resolution of singularities f:Y->V/G is crepant if f*K is isomorphic to the canonical bundle on Y. Crepant resolutions need not be unique, and much interesting work has been generated exploring the deep relationships between different crepant resolutions of a given singularity. A particular heading is provided by Bondal and Orlov's conjecture that any two crepant resolutions of a fixed V/G should have equivalent derived categories.
We will give an introduction to this topic, describing the moduli space of G clusters on V: G-Hilb(V) which often serves as a crepant resolutions in this setting, and which is used to construct various derived equivalences between crepant resolutions in the work of Bridgeland, King, and Reid. We will then describe some new work answering a question of Abramovich in a large number of cases, expanding on recent work of Abdelgadir and Segal, which asks whether there exists an affine variety Z with an action of a reductive group H such that a crepant resolution Y and the Deligne-Mumford stack [V/G] are constructed as GIT quotients of Z.
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Title: From Morse theory to Floer homology
Speaker: Elvar Atlason (University College London)
Monday 17th March, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
Morse theory gives a way to compute singular homology for manifolds. Unlike singular homology, Morse homology has infinite dimensional generalisations, known as Floer homology. The existence of such a homology theory gives a lower bound for the number of critcal points of certain functionals, proving the Arnold conjecture. We will introduce the flowline interpretation of Morse Theory, and discuss how this interpretation generalises to the infinite dimensional case.
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Title: Almgren’s isomorphism and geometric variational problems
Speaker: Nick Manrique (Imperial College London)
Monday 10th March, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
Does every compact manifold contain a closed geodesic? This is a classic problem which was first considered by Hadamard and Poincare at the end of the 19th century, but it wasn’t until Birkhoff introduced the idea of a min-max approach that it became tractable in generality. In this talk we’ll look at the geometric measure-theoretic perspective taken by Almgren, Federer, Fleming and many others on this problem, and more importantly on its higher-dimensional generalisations. The bulk of the talk should be accessible to anyone with a little geometric intuition and will be purely expository in nature.
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Title: Cluster varieties and mirror symmetry
Speaker: Ines Chung-Halpern (LSGNT)
Monday 3rd March, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
In this talk, I will introduce the notion of a cluster variety, and how we can associate combinatorial data to one in a way that generalises frameworks of toric geometry. I will discuss how Fano compactifications of cluster varieties can be obtained from this machinery, and how to find mirrors to certain log Calabi-Yau surfaces in the style of Batryev-Borisov duality for toric varieties.
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Title: Studying flows on three-manifolds using combinatorial topology
Speaker: Layne Hall (University of Warwick)
Monday 24th February, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
Pseudo-Anosov flows are a rich class of dynamical systems on three-manifolds. Their behaviour is strongly connected to the topology (think surfaces, foliations) and geometry of the underlying manifold. A fruitful modern development is that these flows are equivalent to a neat combinatorial structure on the manifold: a veering triangulation. We will give an introduction to this correspondence and then discuss some of its applications.
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Title: The Lickorish-Wallace theorem
Speaker: Lucy Phillips (LSGNT)
Monday 17th February, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
The Lickorish-Wallace theorem states that any closed, connected, orientable 3-manifold can be obtained by ±1-surgery on a link in S³. In this talk I will explain what this means and then go through two sketch proofs of this theorem. It will involve talk of Heegaard splittings, Dehn twists and handle decompositions, and plenty of pictures. The aim is for a gentle talk that is accessible to people who haven’t met these concepts before.
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Title: Exploring Dehn Surgery Space and Tetrahedra Shapes
Speaker: Abigail Hollingsworth (University of Warwick)
Monday 10th February, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
Let M be a compact three-manifold with torus boundary. We will define the tetrahedra shapes of the triangulation of M and Dehn Surgery on the torus boundary of M. We then explore the Dehn filling equation that links these shapes to the Dehn Surgery space. Throughout we will look at the figure-eight knot as an example.
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Title: Introduction to Variety of Minimal Rational Tangents
Speaker: Roktim Mascharak (LSGNT)
Monday 3rd February, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
In this talk I will try to give an introduction to the beautiful machinery of VMRT and if time permits I will show an application of this theory which proves it’s strength. The basic philosophy is to understand global geometry Fano manifolds via extracting data from family of curves which are “free” to move in some sense. This elegant theory developed by J M Hwang uses techniques from classical projective geometry and differential geometry to study structures of Fano Manifold.
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Title: Ultrasolid geometry and deformation theory
Speaker: Sofía Marlasca Aparicio (University of Oxford)
Monday 27th January, 5:30pm–6:30pm, Foyer Seminar Room 3 Cruciform Building UCL
We will introduce ultrasolid modules, a variant of the solid modules of Clausen and Scholze, which generalise complete modules over a field k. In this setting, we show some commutative algebra results like an ultrasolid variant of Nakayama's lemma. We also explore higher algebra in the form of animated and E∞ ultrasolid k-algebras. Finally, we will apply this to generalise the Lurie-Schlessinger criterion in deformation theory (no previous knowledge of deformation theory, condensed mathematics or derived geometry is assumed).
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Title: Matrix Factorizations and Knörrer Periodicity
Speaker: Calum Crossley (LSGNT)
Monday 20th January, 5:30pm–6:30pm, B.05 Chadwick Building UCL
Matrix factorizations crop up in the algebraic geometry of hypersurface singularities, when we try to find free resolutions of modules. Putting these chain-complex-like objects into a suitable homotopy category gives a deformation of derived categories of coherent sheaves, relating to both singularities and, unexpectedly, smooth global geometry. A crucial result in this context is Knörrer periodicity: there are no "non-obvious" matrix factorizations of xy. I will introduce these ideas, and talk about the various incarnations of Knörrer periodicity.
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Title: Surfaces and the Thurston norm in 3-manifolds
Speaker: Alessandro Cigna (King's College London)
Monday 13th January, 5:30pm–6:30pm, B.05 Chadwick Building UCL
A classical strategy for studying the topology of a manifold is to analyze its submanifolds. The world of 3-manifolds is rich and diverse, and we aim to explore the complexity of surfaces contained within a given 3-manifold. After reviewing the fundamental definitions, we will introduce the Thurston norm, a seminorm on the second real homology of a compact orientable 3-manifold. Expect engaging visuals and detailed examples!
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Title: Arithmetic Surfaces
Speaker: Jakab Schrettner (LSGNT)
Monday 2nd December, 5:30pm–6:30pm, S5.20 Strand Building KCL
An arithmetic surface is a scheme of relative dimension 1 over a Dedekind scheme, such as the spectrum of a number field or local field. They are an important tool in studying the reduction of algebraic curves over these fields, which in turn encodes lots of interesting arithmetic information. In this talk I will try to give an overview of the theory of arithmetic surfaces and reduction mod p, while drawing parallels with the theory of algebraic surfaces.
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Title: What is a G2 manifold?
Speaker: Parsa Mashayekhi (LSGNT)
Monday 25th November, 5:30pm–6:30pm, S5.20 Strand Building KCL
Marcel Berger classified the list of possible Riemannian holonomy groups in 1955. These correspond to "interesting" geometric structures on Riemannian manifolds. It took almost 40 years to prove that manifolds with these holonomy groups actually exist. Two exceptional groups on this list, G2 and Spin(7), appear only in dimensions 7 and 8, respectively. In this talk, after providing an introduction to Riemannian holonomy groups and G2 manifolds, I'll discuss the construction of compact G2 manifolds.
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Title: Bridging Characteristic Classes, Cobordism, and Formal Group Laws in Algebraic Topology
Speaker: Marco Milanesi (University of Warwick)
Monday 18th November, 5:30pm–6:30pm, S5.20 Strand Building KCL
This talk aims to connect three fundamental concepts in algebraic topology: characteristic classes of vector bundles, cobordism theory, and formal group laws. In particular, after giving definitions and examples for each notion, we will see how characteristic classes of a manifold's tangent bundle can be assembled into cobordism invariants, and how the complex cobordism ring encode information about all formal group laws. This perspective reveals deep interconnections between geometry, algebra, and topology.
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Title: What is exotic R⁴?
Speaker: Brad Wilson (LSGNT)
Monday 11th November, 5:30pm–6:30pm, S5.20 Strand Building KCL
During the early eighties, Freedman classified 4-manifolds up to homeomorphism. This was soon followed by a theorem of Donaldson, which placed strong restrictions on which 4-manifolds can have smooth structures. I will explain these theorems and some subsequent constructions of exotic 4-manifolds, including exotic R⁴.
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Title: What are spectral sequences?
Speaker: Bogdan Simeonov (LSGNT)
Monday 4th November, 5:30pm–6:30pm, S5.20 Strand Building KCL
I will introduce spectral sequences, an extremely useful tool for calculating (co)homology in a variety of settings. I will try not to be technical, but rather illustrate the theory via different examples, including the Serre spectral sequence in topology, the Hodge-to-de-Rham spectral sequence in algebraic geometry and the Morse-Bott spectral sequence in symplectic geometry.
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Title: What is a hyperbolic group?
Speaker: Will Cohen (University of Cambridge)
Monday 28th October, 5:30pm–6:30pm, S5.20 Strand Building KCL
The field of geometric group theory focuses on gleaning information about a group G from the metric spaces on which it acts by isometry. In particular, when a group acts on a hyperbolic metric space in a rigid way we call it hyperbolic, and hyperbolic groups have been at the centre of infinite group theory research since the 1990s. In this talk, I will introduce the field of geometric group theory, including demonstrating how to construct a "canonical" metric space corresponding to a group, on which it acts particularly nicely. I will then introduce the definition of hyperbolic groups and discuss some properties of hyperbolic groups that we can see by studying their actions on such canonical spaces.
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Title: What is the moduli space of relative stable maps to P^1?
Speaker: Cat Rust (Queen Mary University London)
Monday 21st October, 5:30pm–6:30pm, S5.20 Strand Building KCL
We will begin by introducing the moduli space of stable, n-marked, genus zero curves, M_{0,n}, and its compactification. We will then turn our focus to the moduli space, M(x), of relative stable maps to nodal curves, which gives a family of birational models for M_{0,n}. We will do many examples and calculations throughout, with the aim to give you a good feel for these moduli spaces!
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Title: What are canonical metrics?
Speaker: Enric Solé-Farré (LSGNT)
Monday 14th October, 5:30pm–6:30pm, S5.20 Strand Building KCL
What is the “best” metric a manifold can admit? While this well depends on your chosen definition of best, this question has driven Riemannian geometers over the last century. During this talk, we will explore different incarnations of this problem. We will focus on the Yamabe problem, the Einstein problem and special holonomy metrics.
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Title: What is knot theory?
Speaker: Maartje Wisse (LSGNT)
Monday 7th October, 5:30pm–6:30pm, S5.20 Strand Building KCL
We introduce the concept of a knot and explain its significance in low-dimensional topology as well as cover some of the common techniques and approaches people bring to this subject. Expect lots of diagrams!