Junior Geometry Seminar

2023/2024

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. **In term 3, we will be hosting a whole day of talks - see below for details!**

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 or give a talk, please contact one of the organisers:

To receive updates about upcoming talks you can subscribe to our mailing list by sending an email with subject "Subscribe" to maths-juniorgeom-subscribe@ucl.ac.uk . Alternatively, you can register here.

If you want to stop receiving updates, send an email with subject "Unsubscribe" to maths-juniorgeom-subscribe@ucl.ac.uk.

June

3rd

10am - 6.00pm

Location: 500, Department of Mathematics, 25 Gordon Street, UCL

Join us on June 3rd, in for an exciting day of talks on various aspects of geometry.
The schedule is:

- 10:00 - 11:00: Abigail Hollingsworth, University of Warwick
- 11:00 - 11:15: Coffee break
- 11:15 - 12:15: Arne Wolf, LSGNT
- 12:15 - 13:30: Lunch
- 13:30 - 14:30: Austin Hubbard, University of Bath
- 14:30 - 15:00: Coffee break
- 15:00 - 16:00: Mahdi Haghshenas, LSGNT
- 16:00 - 16:15: Coffee break
- 16:15 - 17:15: Aleksandra Bozovic, LSGNT

June

3rd

10am - 11am

Abigail Hollingsworth, University of Warwick

Location: 500, Department of Mathematics, 25 Gordon Street, UCL

We will define an ideal triangulation of a three-manifold and build the shape variety of this three-manifold
with this ideal triangulation.
Throughout this talk we will use the example of the figure-eight knot complement, drawing an ideal triangulation
of this three-manifold and defining its shape variety.
From the real locus of the shape variety, we will explore two conjectures.
First, we will look at taut triangulations and find that some lie on the real locus, and others do not. By
looking at the special properties of the taut triangulations that do not lie on the real locus, we explain
the conjecture: triangulations on the real locus are flexible.
We then look at representations of the knot group that coincide with points on the real locus of the shape
variety. We will see some loops have images in PSL(2,\RR), others in PSL(2,\CC), and explore a conjecture
relating when the image of the representation itself is a subgroup of PSL(2,\RR) or PSL(2,\CC).

June

3rd

11.15am - 12.15pm

Arne Wolf, LSGNT

Location: 500, Department of Mathematics, 25 Gordon Street, UCL

It is well-known that the kernel of the graph Laplacian captures the topological properties
(number of cycles and connected components) of a graph. In a similar fashion, the kernel of
a persistent Laplacian captures the information contained in the persistent homology of a
given simplicial complex. Our main goal is to understand what we can deduce from the remaining
eigenvalues and -vectors in the more general cellular sheaf setting, which theoretically
incorporate further information of the faces of a simplicial complex. In this talk, I will
discuss work in progress towards this aim and present a recently-established theoretical
foundation for this goal, where we show that the eigenvalues are stable under small perturbation
of the sheaf and simplicial complex. The upshot of this result is that we can reasonably assume
that the additional information encoded by the other eigenvalues and -vectors are a faithful
representation of other geometric or topological properties of the underlying simplicial complex,
although precisely what this information represents remains to be investigated (current work in
progress proceeds with a machine learning approach). Joint work with Shiv Bhatia, Daniel Ruiz
Cifuentes and Anthea Monod.

June

3rd

1.30pm - 2.30pm

Austin Hubbard, University of Bath

Location: 500, Department of Mathematics, 25 Gordon Street, UCL

Conical symplectic singularities and their crepant resolutions play a starring role in the
theory of symplectic duality, pertaining to 3-D mirror symmetry. Examples of conical symplectic
singularities include: Kleinian singularities. $\CC^{2n}/\Gamma$, for $\Gamma\subseteq \mathrm{Sp}(2n)$
(affine) Nakajima quiver varieties. We discuss an application of GIT techniques developed by
Arzantev-Derental-Hausen-Laface to the construction of crepant resolutions of conical symplectic
singularities. We will present our recent result classifying and enumerating all crepant resolutions
of hyperpolygon spaces: $\MM_n(0)$, a particular family of conical symplectic singularities. The
novelty of our result lies in the construction of all the non-projective crepant resolutions of $\MM_n(0)$.

June

3rd

3.00pm - 4.00pm

Mahdi Haghshenas, LSGNT

Location: 500, Department of Mathematics, 25 Gordon Street, UCL

The expansion of the universe and singular spacetimes were both remarkable predictions of general relativity. Many cosmological spacetimes model the expanding universe, such as FLRW, Kasner and de Sitter spacetimes. The study of Big Bang-type singularities and the future and past stability of these metrics have been a highly active area of mathematical research.
After a brief introduction, we explain some of the classical results regarding the Big Bang singularities. In particular, we explain and compare Hawking's Incompleteness Theorem with its Riemannian analogue, the Bonnet–Myers theorem. Additionally, we will delve into more recent works on the singularity's stability and future stability, and outline some of the tools required to prove these results.

June

3rd

4.15pm - 5.15pm

Aleksandra Bozovic, LSGNT

Location: 500, Department of Mathematics, 25 Gordon Street, UCL

The Mandelbrot set is possibly one of the most famous images in mathematics,
and is even popular outside of mathematics due to its striking visual qualities,
in particular its self-similarity. It originates from complex or holomorphic
dynamics, a field which studies iterated holomorphic maps, and in particular from
the study of the quadratic family $\{z\mapsto z^{2}+c|c\in \mathbb{C}\}$, which can
be considered a step up in complexity after the Möbius maps, whose dynamics are
easily classified), yet still, many open questions remain surrounding its
properties. In this talk, I shall give an introduction to the dynamics
of rational maps in one dimension in general, with a particular focus on the
quadratic family. Next, I shall explain one aspect of the Mandelbrot set's
self-similarity, the existence of little Mandelbrot copies, homeomorphic to the
Mandelbrot set itself, through the theory of quadratic-like renormalization. Broadly
speaking, renormalization in dynamics is the phenomenon where restricting the
iterates of elements of a family of maps to a subset of their domains again gives
us a map in the same family (up to an appropriate conjugation), called the
renormalization, which can be used to study the original map. In the end, I shall
present some conjectures and open questions related to the Mandelbrot set and discuss
how they are related to quadratic-like renormalization.

January

8

5.30 - 6.30pm

Myles Workman, University College London

Location: B20 Jevons LT, Drayton House

The regularity theories of Schoen--Simon--Yau and Schoen--Simon for stable minimal hypersurfaces are
foundational in geometric analysis. Using this regularity theory, in low dimensions,
Chodosh--Ketover--Maximo and Buzano--Sharp studied singularity formation in sequences of minimal
hypersurfaces through a bubble analysis.
I will review this background, before talking about my recent work in this bubble analysis theory.
In particular I will show how to obtain upper semicontinuity of index plus nullity along a bubble
converging sequence of minimal hypersurfaces. The strategy we employ is inspired by recent
developments in the bubbling theory of harmonic maps by DaLio--Gianocca-Rivière.

January

15

5.45 - 7.00pm

Diego Artacho de Obeso, Imperial College London

Location: B03 Ricardo LT, Drayton House

Spin geometry is an area of Mathematics which exhibits a very nice interplay between
algebra, topology, analysis and geometry. After this talk, you will know what the
following things are and why they are interesting: Clifford algebras (algebra), spin
structures (topology), spinors (geometry) and Dirac operators (analysis).

January

22

5.45 - 7.00pm

Nick Nho, University of Cambridge

Location: B03 Ricardo LT, Drayton House

In this talk, we will see how the A_{2} quiver arises in various geometric contexts
(ideal triangulations of the pentagon, CY 3-folds, spectral curves). We will also talk about how they are all related.

January

29

5.45 - 7.00pm

Adele Jackson, University of Oxford

Location: B03 Ricardo LT, Drayton House

Given a mathematical object, what can you compute about it?
In some settings, you cannot say very much. Given an arbitrary
group presentation, for example, there is no procedure to
decide whether the group is trivial. In 3-manifolds, however,
algorithms are a fruitful and active area of study (and
some of them are even implementable!). One of the main tools
in this area is normal surface theory, which allows us to
describe interesting surfaces in a 3-manifold with respect to
an arbitrary triangulation. I will discuss some results in
this area, particularly around Seifert fibered spaces.

February

5

5.45 - 7.00pm

Robert Maher, University of Birmingham

Location: B20 Jevons LT, Drayton House

FJRW theory is an enumerative curve count for Landau-Ginzburg models. Analogous to Gromov-Witten theory for Calabi-Yau or Fano varieties,
there is a mirror to FJRW theory that has become known as Saito-Givental theory. In this talk, I will motivate Saito-Givental theory by
describing closed r-spin intersection numbers as the simplest example of FJRW theory. I will then explain closed Saito-Givental theory
as a cohomological field theory by providing explicit formulas for the flat coordinates of the Frobenius manifold associated to any
simple or elliptic singularity. This is an extension of work done by Noumi-Yamada. Using this, as well as recent work of Gross-Kelly-Tessler,
I will construct Saito-Givental theory for open string invariants via the existence of a Lie group of wall-crossing transformations in
rank 2, in addition to describing modularity properties in the case of elliptic orbifolds.

February

12

5.45 - 7.00pm

Alexander Fruh, University of Birmingham

Location: B20 Jevons LT, Drayton House

The study of Higgs bundles on an algebraic curve originated from Hitchin's work on the self-dual Yang Mills equations but have
connections to a vast array of topics in differential and algebraic geometry (and even number theory). If vector
bundles can be thought of as global analogues of vector spaces, Higgs bundles can be thought of as global analogues
of linear endomorphisms. The Hitchin fibration on the moduli space of Higgs bundles is defined using the characteristic
polynomial, and is a key tool in the study of this space. The Hitchin fibres can be described via the spectral correspondence
which relate Higgs bundles on the curve to rank 1 sheaves on its spectral cover; this is in effect a diagonalisation process.

There is also a notion of Higgs bundles associated to a real group G. In the case that G is quasi-split, there is a spectral correspondence such that the Hitchin fibres can be described as abelian (or semi-abelian) varieties. Spectral correspondences have also been calculated for a number of non-quasi-split cases, all of which exhibit behaviour which is in a certain sense 'non-abelian'. I will give a spectral correspondence for the group G=U(p,q), building on the descriptions for the quasi-split cases by Schaposnik (for p=q) and Peón-Nieto (for p=q+1)

There is also a notion of Higgs bundles associated to a real group G. In the case that G is quasi-split, there is a spectral correspondence such that the Hitchin fibres can be described as abelian (or semi-abelian) varieties. Spectral correspondences have also been calculated for a number of non-quasi-split cases, all of which exhibit behaviour which is in a certain sense 'non-abelian'. I will give a spectral correspondence for the group G=U(p,q), building on the descriptions for the quasi-split cases by Schaposnik (for p=q) and Peón-Nieto (for p=q+1)

February

19

5.45 - 7.00pm

Megan Masters, LSGNT - UCL

Location: B20 Jevons LT, Drayton House

In this talk, we will introduce Coxeter groups, and build simplicial complexes on which the groups act.
Then, we will take unions of these complexes to create buildings, which we view as representations of
algebraic groups. We will see how the concepts of galleries and shadows in Coxeter complexes help
answer questions about these algebraic groups. I will finish by briefly talking about Hecke algebras in
the context of Coxeter groups. In particular, I will focus on Kazhdan-Lusztig theory and the work I am
trying to do. No prior knowledge of geometric group theory will be assumed or required.

February

26

5.45 - 7.00pm

Álvaro González Hernández, Warwick University

Location: B20 Jevons LT, Drayton House

In introductory algebraic geometry courses, mathematics students are typically shown how to desingularise a cone by blowing up its singular point. If you have ever done this by hand and you did not find it too hard, consider now the challenge of doing this not just once, but 16 times!

In this talk, I will showcase how we can blow-up singularities in a "smarter" way, by explaining the theory that allow us to desingularise Kummer surfaces. These are K3 surfaces with 16 nodes that arise as quotients of Abelian surfaces and, besides having a very interesting birational geometry, they also have applications in number theory, as they allow us to find rational points on genus 2 curves.

This talk will provide a nice introduction to the theory of algebraic surfaces and their singularities through an example, but I also intend to mention my contributions to this topic, by describing how Kummer surfaces behave when we work over a field of characteristic 2 and explaining what we can say about their desingularisation in that case.

In this talk, I will showcase how we can blow-up singularities in a "smarter" way, by explaining the theory that allow us to desingularise Kummer surfaces. These are K3 surfaces with 16 nodes that arise as quotients of Abelian surfaces and, besides having a very interesting birational geometry, they also have applications in number theory, as they allow us to find rational points on genus 2 curves.

This talk will provide a nice introduction to the theory of algebraic surfaces and their singularities through an example, but I also intend to mention my contributions to this topic, by describing how Kummer surfaces behave when we work over a field of characteristic 2 and explaining what we can say about their desingularisation in that case.

March

4

5.45 - 7.00pm

Gilles Englebert, University of Oxford

Location: B20 Jevons LT, Drayton House

The SYZ conjecture is a geometric way of understanding mirror symmetry via the existence
of dual special Lagrangian fibrations on mirror Calabi-Yau manifolds. Motivated by
this conjecture, it is expected that G_{2} and Spin(7)-manifolds
admit calibrated fibrations as well. After giving a quick introduction to the world of calibrated
geometry, I will explain how to construct examples of such fibrations on compact
Spin(7)-manifolds obtained via gluing of complex fibrations. The key ingredient is the stability of the
fibration property under deformation of the ambient Spin(7)-structure, with the main difficulty being the analysis
of singular fibres. On the way we talk about the deformation theory of compact and conical Cayley submanifolds
and ways to desingularise singular ones.

March

11

5.45 - 7.00pm

Siao Chi Mok, University of Cambridge

Location: B20 Jevons LT, Drayton House

Despite its complicated sounding name, this talk is merely to understand the following problem: Fix a smooth projective variety X and a normal crossings divisor D, and consider the configuration space of n distinct points on X\D. Two undesirable issues can happen in the limit: the points might approach D, or collide into each other. If we forget the divisor D and consider the space of n distinct points on X, then the issue of colliding points has been resolved by Fulton and MacPherson (1994) with their Fulton—MacPherson configuration spaces.

In this talk, I am going to explain how one can compactify the configuration space in two steps – first using logarithmic geometry to resolve the first issue (resulting in an intermediate space (X|D)^n), then lifting the Fulton—MacPherson compactification to this setting to separate the points, giving the space FM_n(X|D). I will describe a combinatorial stratification of both spaces, as well as some potential applications of this project.

In this talk, I am going to explain how one can compactify the configuration space in two steps – first using logarithmic geometry to resolve the first issue (resulting in an intermediate space (X|D)^n), then lifting the Fulton—MacPherson compactification to this setting to separate the points, giving the space FM_n(X|D). I will describe a combinatorial stratification of both spaces, as well as some potential applications of this project.

March

18

5.45 - 7.00pm

Yifan Zhao, LSGNT - Imperial

Location: B20 Jevons LT, Drayton House

Often, the set that classifies certain objects (e.g., all Riemann surfaces) can itself be given a geometric structure. The geometry of this set (which we call the moduli space) gives further insight into the original classification problem.

In this talk, I will first introduce moduli spaces in the algebro-geometric setting, and then focus on two nice examples: Hilbert schemes of points on surfaces (parameterizing subspaces of surfaces) and moduli spaces of sheaves on surfaces (parameterizing coherent sheaves on surfaces). I will also discuss how these two examples play a big role in hyperkähler geometry and, if time permits, recent work relating them to the "deformation to the normal cone" construction.

In this talk, I will first introduce moduli spaces in the algebro-geometric setting, and then focus on two nice examples: Hilbert schemes of points on surfaces (parameterizing subspaces of surfaces) and moduli spaces of sheaves on surfaces (parameterizing coherent sheaves on surfaces). I will also discuss how these two examples play a big role in hyperkähler geometry and, if time permits, recent work relating them to the "deformation to the normal cone" construction.

September

25

5.30 - 6.30pm

Martin Ortiz, LSGNT - Imperial

The concept of a sheaf is central in modern geometry: one can see spaces and sheaves as two sides of the same coin.
In this talk I will give some examples that illustrate what a sheaf is, what one can do with them, and why
they are particularly useful in algebraic geometry.

October

2

5.30 - 6.30pm

Michela Barbieri, LSGNT - UCL

In this talk I'm going to try to give an overview of some (co)homology stuff that will
come up in the topics courses. I'll start with a classical algebraic topology perspective
and discuss singular/simplicial/cell (co)homologies. This sounds like a lot of
information but in fact these are all the same, with simplicial/cell just being
computational tools to compute singular (co)homology. Loosely speaking, the ith singular
homology of a topological space X (denoted H^i(X) ) counts the 'i-dimensional holes'
in your topological space X. This (co)homology theory already lets us prove some pretty
cool topology theorems, such as the Brower fixed point theorem, the Hairy ball theorem,
and the Borsuk Ulam Theorem. Maybe I'll discuss the proof of one of these.

There are other homology theories you're going to come across during the topics courses, such as de Rham and sheaf, so time permitting I'll mention these and how they're related to the others.

There are other homology theories you're going to come across during the topics courses, such as de Rham and sheaf, so time permitting I'll mention these and how they're related to the others.

October

9

5.30 - 6.30pm

Nick Manrique, LSGNT - Imperial

One of the most foundational and beautiful results in the theory of Riemann surfaces is the
uniformisation theorem, which classifies the possibilities for their geometry completely.

In this talk we will explore some of the ideas which go into its proof, discovering some cool stuff about Dolbeault cohomology and PDEs along the way. We will also touch on some corollaries of uniformisation on other geometric structures in real dimension 2, and ask about generalisations if time permits.

A working knowledge of complex analysis and basic manifolds will be helpful.

In this talk we will explore some of the ideas which go into its proof, discovering some cool stuff about Dolbeault cohomology and PDEs along the way. We will also touch on some corollaries of uniformisation on other geometric structures in real dimension 2, and ask about generalisations if time permits.

A working knowledge of complex analysis and basic manifolds will be helpful.

October

16

5.30 - 6.30pm

Jaime Mendizabal, LSGNT - UCL

Characteristic classes are cohomology classes associated naturally to bundles, and condense important information about them into
algebraic invariants. In this talk we will review some properties of some of the main types of these characteristic classes and
discuss a few of their consequences.

October

23

5.30 - 6.30pm

Enric Solé-Farré, LSGNT - UCL/Imperial

I will review deRham cohomology and explain how the choice of a metric can be used to find
representatives for the cohomology classes, that can be suitably interpreted as minimizers
of a Dirichlet-type energy functional.

I will then explain how these results fit more generally in the context of elliptic operators and discuss Dolbeaut cohomology as a second example.

I will then explain how these results fit more generally in the context of elliptic operators and discuss Dolbeaut cohomology as a second example.

October

30

5.30 - 6.30pm

Aporva Varshney, University College London

In this talk, I will give a gentle introduction to intersection theory on varieties by
thinking about curves on surfaces. We'll look at the curious notion of self-intersection
and construct the Hirzebruch surfaces, touching upon aspects of birational geometry such as
the minimal model program and singularity theory along the way. There will be lots of drawings!

November

6

5.30 - 6.30pm

Lucas Day, University College London

In this talk, we will explore the classification of manifolds from dimension zero to four: i.e. the journey
from trivial to very non-trivial! The classification of closed surfaces will be explained through geometric
topology (with diagrams), employing the invariants of Euler characteristic and orientability.

We will then discuss the work of Thurston and Perelman in the classification of 3-manifolds, before finishing the talk with a survey of some of the main results (and open questions!) in the classification of 4-manifolds.

We will then discuss the work of Thurston and Perelman in the classification of 3-manifolds, before finishing the talk with a survey of some of the main results (and open questions!) in the classification of 4-manifolds.

November

13

5.30 - 6.30pm

Daniil Mamaev, LSGNT - Imperial

An important fact in elementary symplectic topology is that symplectic structures have no local invariants.
This manifests in Darboux's theorem (all symplectic manifolds locally look the same), Moser's stability theorem
(perturbing a symplectic form within its cohomology class does not change its diffeomorphic type), Weinstein's Lagrangian neighbourhood theorem
(a neighbourhood of Lagrangian submanifold L looks like a neighbourhood of the zero section in the cotangent
bundle of L), and more.

All this theorems can be proved using a simple argument, called Moser's trick, stating that families of symplectic forms with exact derivative come from families of diffeomorphisms. We will prove Moser's trick and then deduce (some of) the theorems above from it.

All this theorems can be proved using a simple argument, called Moser's trick, stating that families of symplectic forms with exact derivative come from families of diffeomorphisms. We will prove Moser's trick and then deduce (some of) the theorems above from it.

November

20

5.30 - 6.30pm

Thamarai Valli Venkatachalam, LSGNT - Imperial

Toric varieties are combinatorially rich objects in algebraic geometry.
All their data can be read from the combinatorial objects("fans") associated with them.
In this talk, I will give a brief introduction to toric varieties, and we will see how
the combinatorial data helps us in computing their class group and Picard group explicitly.

November

27

5.30 - 6.30pm

Edward Hirst, Queen Mary University of London

A selection of techniques are introduced from various subfields in machine learning, with discussion on how they
are appropriately designed for use in mathematics and mathematical physics. Some results will be presented
where machine learning has shown success in these areas, with focus on problems in algebraic geometry and
the use of these geometries in string and M-theory.

December

4

5.30 - 6.30pm

Ananya Satoskar, LSGNT - King's

The mapping class group of a manifold is the group of isotopy classes of its automorphisms; this has been widely studied in many contexts and has connections to almost every field in pure mathematics. In the 1970's, now-classic work by Birman and Hilden established a way to study mapping class groups using branched covers of the manifold, leading to beautiful and unexpected results in the theory of representations of braid groups.

In this talk, I will define the mapping class group for surfaces, along with its symplectic representation and some interesting subgroups (the Torelli group, congruence subgroups). I will then use the Birman-Hilden isomorphism to explore a classical result of Artin and Arnol'd about the generators of the level-2 braid congruence subgroup. All necessary background will be defined in the talk and there will be many pictures.

In this talk, I will define the mapping class group for surfaces, along with its symplectic representation and some interesting subgroups (the Torelli group, congruence subgroups). I will then use the Birman-Hilden isomorphism to explore a classical result of Artin and Arnol'd about the generators of the level-2 braid congruence subgroup. All necessary background will be defined in the talk and there will be many pictures.

December

11

5.30 - 6.30pm

Inés García Redondo, LSGNT - Imperial

In this talk, I will introduce one of the most well-known methods in Topological Data Analysis: Persistent Homology.
I will begin by motivating the definition of Persistent Homology as a powerful tool to 'study the shape' of data.
We will delve into persistence barcodes and diagrams, discussing their significance in data analysis. If time permits,
I will introduce a generalization of persistent homology to multidimensional indexing posets, aptly named Multiparameter Persistent Homology
(very original, I know). I'll touch on why this generalization drastically changes the theory in this setting, allowing me to have a topic for my PhD.