KCL/UCL Junior Geometry Seminar

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 — from all London universities. The atmosphere is friendly and informal, most talks are accessible to a wide audience.

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Term 2

10 January 2022 - 25 March 2022


24 March 2022


Higher rank K-theoretic Donaldson-Thomas theory of points

Sergej Monavari (Utrecht University)
Strand S-3.20 17:00 - 18:00

Abstract: A classical way to produce invariants is through intersection theory, usually on a smooth projective variety. We give a gentle introduction on how to use torus actions to refine invariants in several directions, for example K-theoretic and virtual invariants. As a concrete example, we explain how to extract meaningful invariants from the Quot schemes of quasi-projective smooth toric threefolds and how to refine them. We present and prove various closed formulas for different flavours of higher rank Donaldson-Thomas invariants of points, solving a series of conjectures proposed in String Theory. This is based on joint work with N. Fasola and A. Ricolfi.

17 March 2022


Categorical Torelli theorems for prime Fano threefolds

Augustinas Jacovskis (University of Edinburgh)
Strand S-3.20 17:00 - 18:00

Abstract: A lot of geometric information about a variety X can be recovered from its derived category D(X). If the variety is Fano, then X can in fact be reconstructed up to isomorphism from D(X). This begs the question of whether less information than D(X) can determine X up to isomorphism. In this talk I’ll discuss what happens when “less information” means a certain subcategory of D(X) called the Kuznetsov component. In particular, I will discuss joint work with Zhiyu Liu and Shizhuo Zhang which describes the situation for index 1 Fano threefolds.

10 March 2022


Introduction to G-structures

Enric Solé Farré (LSGNT)
Strand S-3.20 17:00 - 18:00

Abstract: The ideas of principal bundles and G-structure provide us with a new perspective on differential geometry. We will start by revising the basic definitions of principal bundle theory and move on to discussing how different geometries & their fundamental theorems fit in the context of G-structures. Time permitting, we will discuss how the G-structure perspective can be useful in the context of gauge theory.

3 March 2022


The Langlands program for function fields and the geometric Langlands conjecture

Tom Gannon (University of Texas at Austin)
Zoom & Strand S-3.20 17:00 - 18:00

Abstract: The Langlands program predicts that representations of the absolute Galois group of a field can be understood by relating them to certain other representations known as automorphic representations. In this talk, we will discuss the Langlands program when the ground field is the function field k of an algebraic curve X. We will see that many of the players of this conjecture can be understood through the geometry of X itself. After discussing the Langlands program for k and some more motivation, we will discuss how it gives rise to the geometric Langlands conjecture, a collection of conjectures which provides a sheaf version of automorphic functions.

24 February 2022


D-modules in algebraic geometry

Haiping Yang (Imperial College London)
King's College, Strand Building, Room S-3.20 17:00 - 18:00

Abstract: D-mods were first introduced in the 1970s, motivated as an algebraic way of encoding (linear) PDEs. Methods of tackling D-modules include powerful tools such as homological algebra and sheaf theory. I am going to introduce D-mods on algebraic varieties and compare some of their basic properties to O-mods, in particular, we will see they are more topological rather than algebro-geometric. Some further properties will provide a bridge between geometry and representation theory, including representations of semisimple Lie algebras, Poisson homology of Poisson manifolds.

17 February 2022


Introduction to higher codimension mean curvature flow

Artemis Vogiatzi (Queen Mary University of London)
King's College, Strand Building, Room S-3.20 17:00 - 18:00

Abstract: Geometric Analysis is about a study of geometric problems that can be formulated as problems of systems of partial differential equations. This talk will be an introduction to Mean Curvature Flow of higher codimension submanifolds. We’ll see the differences with the codimension 1 case and how we can deal with the new problems that arise.

10 February 2022


Mirrors to toric degenerations via intrinsic mirror symmetry

Evgeny Goncharov (University of Cambridge)
Zoom 17:00 - 18:00

Abstract: I will explain two different approaches to Gross-Siebert mirror symmetry. In the old approach, mirrors are seen as toric degenerations with dual intersection complexes related by a discrete Legendre transform. This setup has been shown to generalize the classical mirror symmetry constructions (e.g Batyrev-Borisov). The new approach constructs a mirror to an arbitrary log smooth degeneration of Calabi-Yaus. I'll sketch how the new approach generalizes the old one in dimension 2 using resolution of singularities, scattering diagrams, and logarithmic Gromov-Witten invariants.

3 February 2022


Kähler geometry of holomorphic submersions

Annamaria Ortu (Scuola Internazionale Superiore di Studi Avanzati)
King's College, Strand Building, Room S-3.20 17:00 - 18:00

Abstract: Kähler manifolds have a rich geometric structure: they are smooth manifolds endowed with a complex structure, a symplectic form and a Riemannian metric together with a compatibility condition. They can also be seen as smooth projective varieties and have a deep link with algebro-geometric stability. After reviewing the definitions and basic properties, we will introduce the problem of finding a canonical Kähler metric in terms of prescribing its curvature, and we will specialize this problem in the case of Kähler fibrations.

27 January 2022


Bubble tree convergence of gradient shrinking Ricci solitons

Louis Yudowitz (Queen Mary University of London)
UCL Lecture Room 706 17:00 - 18:00

Abstract: Introduced by Richard Hamilton in 1982, Ricci flow has been used to solve a variety of problems in geometry and topology. Arguably the most notable of these is Perelman’s proof of Thurston’s Geometrization Conjecture and the Poincare conjecture in the early 2000s. A vital part of these proofs and other applications understanding what types of finite time singularities can form under the flow. This is largely finished in 2 and 3 dimensions, but higher dimensions still pose issues. I will present a joint work with R. Buzano which examines possible degenerations of singularity models in dimensions 4 and above. This is done through a "bubble tree construction". I will also present a diffeomorphism finiteness theorem and gap result which follow as an application of the construction.

20 January 2022


Quiver varieties and moduli spaces attached to Kleinian singularities

Søren Gammelgaard (University of Oxford)
Zoom 17:00 - 18:00

Abstract: We discuss how Nakajima's quiver varieties can be used to understand the Hilbert schemes of points on the singularities C^2/G, for G a finite subgroup of SL_2(C). Time permitting, we will touch upon a type of "equivariant Quot scheme" associated to the same singularity, and possibly a generalisation to sheaves on a stacky compactification of the same singular surface. This is joint work with A. Craw, Á. Gyenge, and B. Szendrői.

Term 1

27 September 2021 - 17 December 2021


16 December 2021


Regularity theory for minimal submanifolds

Paul Minter (University of Cambridge)
Zoom 17:00 - 18:00

Abstract: Imagine you have two identical rings and a bubble (of minimal area) bounding them. If the rings are sufficiently close, the bubble forms a catenoid. If we then move the rings apart, eventually the bubble "pops", with the minimal area solution now becoming two (separate) disks. This simple example illustrates that submanifolds minimising an area can exhibit bad behaviour, especially in their limits. To find minimisers or critical points of area one must therefore work in a larger class of potentially singular "submanifolds". For this purpose, F. Almgren (1965) introduced the concept of a varifold as the candidate for a "weak submanifold" and proved an existence theory for critical points of area in this class. However, since the celebrated regularity theory of W. Allard (1972)---which shows that these weak solutions are smoothly embedded on an open dense subset---very little is known about the regularity of these stationary integral varifolds. The primary reason for this is the possibility of a degenerate type of singularity known as a branch point. In this talk I will discuss various aspects of the regularity theory for stationary integral varifolds, and in particular mention a recent new result concerning branch points for a large class of stationary integral varifolds (which is joint work with N. Wickramasekera).

2 December 2021


EGG stability

George Smith (Imperial College London)
King's College, Strand Building, Room S-3.20 17:00 - 18:00

Abstract: I live in a world of slightly horrible moduli spaces and, perhaps more philosophically, so do you. The framework of “Exceptional Generalised Geometry” (EGG) reshuffles the existence of Killing spinors into an equivalent condition involving generalised G-structures where G is a sub-group of the E_{7(7)} exceptional group. The moduli space of such G-structures forms an infinite dimensional pseudo-Kähler manifold which is very similar to the pseudo-Kähler moduli space of SL(3, C) structures on a 6-manifold. On both these moduli spaces we can define orbits of a diffeomorphism group and ask when these orbits intersect the zeros of a moment map. In the SL(3,C) case these zeros are integrable SL(3, C) structures, while in the exceptional geometry case the zeros are solutions to the supergravity equations of motion which contain G2 manifolds as a special case. The story is then appears similar Donaldsons story for constant scalar curvature Kähler manifolds, with a few key differences: The moduli space has split signature and the complexified group does not have fully complexified orbits. In this talk I will review the EGG construction and results, and highlight some open questions. Perhaps some of the audience will even have answers to these questions.

25 November 2021


18 November 2021


4 November 2021


28 October 2021


Tropical and algebraic curve counting

Patrick Kennedy-Hunt (University of Cambridge)
King's College, Strand Building, Room S-3.20 17:00 - 18:00

Abstract: In enumerative geometry we often try to count curves, but computations can be hard. Logarithmic geometry, and its cousin tropical geometry, provide insights and new approaches to computing these curve counts. Gromov--Witten invariants are closely related to these curve counts. I will start by explaining a result of Mikhalkin reducing the Gromov--Witten theory of toric surfaces to a combinatorial (perhaps you might say "tropical") problem. I will go on to discuss a new construction ("The Logarithmic Linear System") which seems a promising route to a new proof of Mikhalkin theorem.

21 October 2021


7 October 2021


Term 3

6 May 2021 - 27 June 2021


6 May 2021


13 May 2021


20 May 2021


27 May 2021


03 June 2021


10 June 2021


17 June 2021


24 June 2021


Term 2

21 January 2021 - 1 April 2021


21 January 2021


28 January 2021


4 February 2021


11 February 2021


18 February 2021


25 February 2021


4 March 2021


11 March 2021


18 March 2021


25 March 2021


1 April 2021


Term 1

28 September 2020 - 18 December 2020


1 October 2020


8 October 2020


15 October 2020


22 October 2020


29 October 2020


5 November 2020


12 November 2020


19 November 2020


26 November 2020


3 December 2020


10 December 2020