22/23 20/22 19/20

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. 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:

Also check out these other junior seminars in London:


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Next seminar

June
3rd
10am - 6.00pm

Junior Geometry: A Day of Talks

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
Titles and abstracts can be found below. To register, please fill in the form. Talks will take place in room 500 in the UCL maths department. Note that the university currently has increased security and may check for a UCL ID; in case of any issues, please contact one of the organisers.
June
3rd
10am - 11am

Taut triangulations and representations of the knot group.

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

Spectral Stability of Persistent Sheaf Laplacians

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

Crepant resolutions of symplectic singularities

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

Singularities and Stability of Cosmological Spacetimes

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

The Mandelbrot set and quadratic-like renormalization

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.

Schedule


Spring

January
8
5.30 - 6.30pm

Minimal Hypersurfaces: Bubble Convergence and Index

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

What is spin geometry?

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

Fun with the A2 quiver

Nick Nho, University of Cambridge
Location: B03 Ricardo LT, Drayton House
In this talk, we will see how the A2 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

Algorithms and 3-manifolds

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

Predictions in Open FJRW Theory Via Mirror Symmetry, Modularity and Wall-Crossing

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

Spectral data for U(p,q)-Higgs bundles

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)
February
19
5.45 - 7.00pm

Coxeter Complexes and Bruhat-Tits buildings

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

How to desingularise Kummer surfaces

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.
March
4
5.45 - 7.00pm

Stability of Cayley fibrations

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 G2 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

Logarithmic Fulton-MacPherson configuration spaces

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.
March
18
5.45 - 7.00pm

Some examples of moduli spaces in algebraic geometry

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.

Autumn

September
25
5.30 - 6.30pm

What is a Sheaf?

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

What is (co)homology?

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.
October
9
5.30 - 6.30pm

What is the uniformisation theorem?

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.
October
16
5.30 - 6.30pm

What are characteristic classes?

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

What is a harmonic form?

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.
October
30
5.30 - 6.30pm

What is intersection theory?

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

What are the geometries of low-dimensional manifolds?

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.
November
13
5.30 - 6.30pm

What is Moser's trick?

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.
November
20
5.30 - 6.30pm

What is a toric variety?

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

What is machine learning?

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

What is Birman-Hilden theory? T for 2 : Groups, Braids and Twists

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.
December
11
5.30 - 6.30pm

What is Persistent Homology?

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.