KCL/UCL Junior Geometry Seminar Archive 2020-2022

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


Cayley fibrations in the Bryant–Salamon manifold

Federico Trinca (University of Oxford)
King's College, Strand Building, Room S-3.20 • 17:00 - 18:00

Abstract: In 1955, Marcel Berger classified all the possible holonomy groups for simply connected, irreducible and nonsymmetric Riemannian manifolds. The exceptional holonomy group ...

18 November 2021


Introduction to Higgs bundles and the Hitchin map

Riccardo Carini (Imperial College London - LSGNT)
King's College, Strand Building, Room S-3.20 • 17:00 - 18:00

Abstract: Higgs bundles were first introduced by Hitchin in the 80’s in the study of Yang-Mills self-duality equations. Since then, they have become central objects of study in Kähler and hyperkähler geometry, ...

4 November 2021


Surprising surfaces in positive characteristics

Marta Benozzo (University College London - LSGNT)
King's College, Strand Building, Room S-3.20 • 17:00 - 18:00

Abstract: The minimal model program aims to classify all varieties up to birational equivalence. It is completely understood for surfaces. Actually, we even have a classification of them. Enriques and ...

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


Quaternion-Kähler manifolds

Ivan Solonenko (King's College London - LSGNT)
King's College, Strand Building, Room S-3.20 • 17:00 - 18:00

Abstract: A quaternion-Kähler manifold is a 4n-dimensional Riemannian manifold whose holonomy group is contained in Sp(n)Sp(1), one of the 7 groups appearing in Berger’s list of ...

7 October 2021


Geometric applications of the Allen-Cahn equation and Min-Max techniques

Myles Workman (University College London - LSGNT)
King's College, Strand Building, Room S-3.20 • 17:00 - 18:00

Abstract: The Allen-Cahn equation is a reaction-diffusion equation that originally comes from mathematical-physics, but in recent decades has found a new home in the field of geometric analysis. The equation ...

Term 3

6 May 2021 - 27 June 2021


6 May 2021


Moduli spaces of stable sheaves on K3 surfaces via wall-crossing

Alessio Bottini (Tor Vergata University of Rome/Paris-Sud University)
Zoom • 17:00 - 18:00

Abstract: Moduli spaces of stable sheaves (and stable complexes) on K3 surfaces are a well-studied family of high-dimensional ...

13 May 2021


Convolutions on manifolds and applications to geometric deep learning

Daniel Platt (Imperial College London - LSGNT)
Zoom • 17:00 - 18:00

Abstract: Convolution is a natural operation combining two integrable functions on Euclidean space. This operation motivates ...

20 May 2021


Stability in differential and algebraic geometry

John McCarthy (Imperial College London - LSGNT)
Zoom • 17:00 - 18:00

Abstract: Research at the intersection of differential and algebraic geometry has been dominated by the principle that extremal objects ...

27 May 2021


From Feix-Kaledin metric on T*X to algebraic geometry

Anna Abasheva (Columbia University)
Zoom • 17:00 - 18:00

Abstract: Let X be a Kähler manifold and let’s consider the total space T*X of the cotangent bundle of X. In many nice cases there exists ...

03 June 2021


K3 Surfaces & a Modular Form

Qaasim Shafi (Imperial College London - LSGNT)
Zoom • 17:00 - 18:00

Abstract: A quadric surface in P^3 can be covered completely by lines. A cubic surface famously contains 27 lines. A quartic surface is an example of a K3 surface ...

10 June 2021


Maximum principle and geometric applications

Kostas Leskas (University College London - LSGNT)
Zoom • 17:00 - 18:00

Abstract: One of the main features that distinguishes linear, uniformly elliptic, 2nd order differential operators is the maximum principle ...

17 June 2021


Constant scalar curvature Kähler metrics

Alessio Di Lorenzo (University College London - LSGNT)
Zoom • 17:00 - 18:00

Abstract: Eugenio Calabi had a dream: that one could find on each Kähler manifold a metric that could be called "canonical" ...

24 June 2021


An L-infinity lifting of the first component of the semiregularity map

Emma Lepri (Sapienza Università di Roma)
Zoom • 17:00 - 18:00

Abstract: The subject of the talk is the construction of a lifting of the first component of the (modified) Buchweitz-Flenner semiregularity ...

Term 2

21 January 2021 - 1 April 2021


21 January 2021


Barrier methods for minimal submanifolds and
the Gibbons-Hawking ansatz

Federico Trinca (University of Oxford)
Zoom • 17:00 - 18:00

Abstract: A submanifold of a Riemannian manifold is minimal if it is critical points of the volume functional. Using the maximum principle ...

28 January 2021


A Physics approach to the Atiyah-Singer Index theorem

Andries Salm (University College London)
Zoom • 17:00 - 18:00

Abstract: Do you think your supervisor is too rigorous? Do you silently believe that the world is smooth and finite at the same time? Do ...

4 February 2021


Geometric approaches to Nielsen equivalence

David Sheard (University College London)
Zoom • 17:00 - 18:00

Abstract: In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer ...

11 February 2021


Mabuchi Geometry of the Space of Kähler/Sasaki Potentials

Thomas Franzinetti (Sorbonne Université)
Zoom • 17:30 - 18:30

Abstract: Given a compact Kähler (resp. Sasakian) manifold, one can endow the space of Kähler (resp. Sasakian) potentials with a so called Mabuchi metric. We show that its metric completion is negatively curved in the sense of Alexandrov.

18 February 2021


From Riemann to Russia: Alexandrov Geometry

Tom Sharpe (LSGNT - King's College London)
Zoom • 17:00 - 18:00

Abstract: Do you long for a coordinate-free approach to Riemannian geometry? Are you filled with dread whenever your metrics develop singularities? Or can you not imagine why anyone would want to leave the world of smoothly varying tensors? Either way, ...

25 February 2021


Geodesic orbit spaces

Jordan Hofmann (LSGNT - King's College London)
Zoom • 17:00 - 18:00

Abstract: Homogeneous spaces are at the heart of Riemannian geometry, providing a vast "playground" of examples, however this class of spaces is vast and a complete classification is probably out of reach. For this reason many subclasses have been studied ...

4 March 2021


Symplectic embeddings: the Hofer conjecture

Roman Krutowski (University of California, Los Angeles)
Zoom • 17:00 - 18:00

Abstract: The study of symplectic embeddings has a long history starting with the celebrated Gromov's non-squeezing theorem. One of the long-standing conundrums in the field was the ellipsoid embedding problem. The answer to it is now known under ...

11 March 2021


Weinstein handlebodies for toric manifolds

Angela Wu (Imperial College London)
Zoom • 17:00 - 18:00

Abstract: This talk is about two important classes of symplectic manifolds: toric manifolds, which are equipped with an effective Hamiltonian action of the torus, and Weinstein manifolds, which ...

18 March 2021


The Witten genus as the "volume" of the double loop space

Eugenio Landi (Università degli Studi Roma Tre)
Zoom • 17:00 - 18:00

Abstract: Witten introduced his genus in 1986 as a suitable index of a Dirac operator on the loop space of a manifold. In this talk I will recover the Witten genus using the localization theorem in ...

25 March 2021


Hamiltonian Floer Cohomology

Kai Hugtenburg (University of Edinburgh)
Zoom • 17:00 - 18:00

Abstract: Every day the Earth completes a rotation around its axis. In this case there are infinitely many 24-hour periodic orbits, there are even 2 fixed points! This is a slight approximation though ...

1 April 2021


K3-fibred G2 manifolds and their adiabatic limits

Corvin Paul (University College London - LSGNT)
Zoom • 17:00 - 18:00

Abstract: One of the most interesting open conjectures in geometry is the SYZ conjecture. It roughly states that (complex) 3D Calabi Yau manifold should admit a fibration over a (real) 3D base such that ...

Term 1

28 September 2020 - 18 December 2020


1 October 2020


Gauge Theory and Invariants

Jakob Stein (University College London)
MS Teams • 17:00 - 18:00

Abstract: The uses of "gauge theory", that is — connections on principal/vector bundles — to obtain invariants of smooth manifolds, has been a topic of mathematical research since ...

8 October 2020


Intrinsically Harmonic Forms

Ivan Solonenko (King's College London)
MS Teams • 17:00 - 18:00

Abstract: Given a closed differential form on an oriented smooth manifold M, does there exist a Riemannian metric on M with respect to which the form becomes harmonic? Do forms allowing such metrics ...

15 October 2020


The Conway Knot is not Slice

Laura Wakelin (Imperial College London)
MS Teams • 17:00 - 18:00

Abstract: A knot is said to be slice if it bounds a smoothly embedded disc in B^4. For a long time, the question of sliceness had been answered for all knots with up to 12 crossings – except ...

22 October 2020


Introduction to Geometric Flows

Joshua Daniels-Holgate (University of Warwick)
Zoom • 17:00 - 18:00

Abstract: When first introduced to topology, you were probably shown an animation depicting the family of smooth deformations that turn a coffee cup into a donut. One might hope for a way of doing this ...

29 October 2020


Hyper-Kähler Geometry

Jaime Mendizabal Roche (University College London)
Zoom • 17:00 - 18:00

Abstract: Hyper-Kähler manifolds are, in a sense, the quaternionic analogues of Kähler manifolds. As such, many points of view can be taken when studying them. They have a Riemannian metric which ...

5 November 2020


Lefschetz Hyperplane Theorem and the Topology of Algebraic Varieties

Riccardo Carini (Imperial College London)
Zoom • 17:00 - 18:00

Abstract: When dealing with nonsingular varieties over the complex numbers, an extremely powerful method is to regard them as ...

12 November 2020


Singularities in Algebraic Geometry

Federico Bongiorno (Imperial College London)
Zoom • 17:00 - 18:00

Abstract: During the course of a PhD, it is generally very difficult to avoid encountering singular varieties. They arise naturally as limits of smooth varieties but dealing with them requires ...

19 November 2020


Hochschild (co)homology

Federico Barbacovi (University College London)
Zoom • 17:00 - 18:00

Abstract: When studying geometric problems, it is often useful to produce some algebraic invariant that controls the geometry what we are interested in. Sometimes, these algebraic tools are ...

26 November 2020


The Grothendieck Ring of Varieties, and some new Euler Characteristics

Jesse Pajwani (Imperial College London)
Zoom • 17:00 - 18:00

Abstract: The Grothendieck ring of varieties is, depending on your point of view, either a very natural and obvious object to ...

3 December 2020


Equivariant Cohomology of GKM Manifolds

Teresa Lüdenbach (King's College London)
Zoom • 17:00 - 18:00

Abstract: Torus actions and the manifolds which admit them have been vastly studied, in particular by Goresky, Kottwitz and MacPherson ...

10 December 2020


A brief introduction to Mirror Symmetry

Soham Karwa (Imperial College London)
Zoom • 17:00 - 18:00

Abstract: Mirror Symmetry is a striking conjecture/philosophy concerning a certain equivalence of spaces that has revolutionised geometry and ...