Contact the organisers

The Imperial Junior Geometry seminar for 2020/21 is run from 5:30PM to 6:30PM every Friday. Until further notice the seminar will be run online through Zoom.

Zoom meeting ID: 942 7734 2429

## Upcoming Talks:

• ### Title: Symplectic Topology of Riemann Surfaces and Representation Theory

#### Abstract:

In this talk, I will give an overview of some of the fundamental tools used in symplectic topology, restricting to the case of Riemann surfaces. This is not only for simplicity — (symplectic) topology in this dimension has close links to the representation theory of an interesting class of algebras. The aim of this talk is to explain how symplectic topology can be used to answer difficult questions in representation theory. Time permitting, I will mention how one hopes to use representation theory to answer difficult questions in symplectic topology. There will be no background assumed for this talk.

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## Past Talks:

• ### Title: Introduction to K-Stability

#### Abstract:

In this talk, I will provide a brief introduction to the area of K-stability. Roughly, K-stability is an algebro-geometric stability condition on varieties, whose motivations come from both algebraic and complex differential geometry. On the algebraic geometry side, K-stability provides the correct notion of stability for forming moduli spaces of Fano varieties. On the differential geometry side, K-stability is conjecturally equivalent to the existence of a Kähler metric of constant scalar curvature. It was in this latter context that K-stability first arose, in the work of Tian and Donaldson. After giving some motivation for why K-stability is important, I will describe the definition in basic terms, then finish by giving a few examples of how K-stability can be computed.
• ### Title: Moduli spaces of vector bundles on curves

#### Abstract:

Moduli functors and moduli problems are ubiquitous in mathematics. They appear as soon as we want to understand what behaviour objects have when put in families. In this talk we will introduce the notion of semistable sheaf on a curve, and we will see that the moduli problem that parametrises families of semistable vector bundles of fixed rank and degree has a coarse moduli space, meaning that there is some scheme whose points parametrise equivalence classes of semistable vector bundles of rank r and degree n.
• ### Title: Introduction to the Moduli Space of Curves

#### Abstract:

This will be a slightly less technical introduction to the moduli space of curves. I will explain how to construct the space for genus 0 curves and how to compactify it. The boundary complex turns out to be another moduli space, of tropical curves. Time permitting, I will generalize the above constructions to stable maps. No prior knowledge is required except for some very basic algebraic geometry.
• ### Title: Elliptic Operators on Non-Compact Manifolds

#### Abstract:

A lot is known for elliptic operators on compact manifolds, such as the Laplacian. One useful fact among many is the Hodge Theorem: every de Rham cohomology class contains a unique harmonic representative. On non-compact manifolds this is no longer true, but the statement can be adapted to an interesting class of complete non-compact manifolds, called asymptotically cylindrical, which was done by Lockhart and McOwen. In the talk, I will review the Hodge theorem for compact manifolds, and then discuss in which sense it carries over to asymptotically cylindrical manifolds. As a toy example, I will carry out computations for the easiest case of an asymptotically cylindrical manifold, namely $$\mathbb{R}^n$$, which brings out surprising links to the representation theory of $$\mathrm{SO}(n)$$. Exactly the same calculations apply to asymptotically locally Euclidean manifolds.
• ### Title: Non-Commutative (Crepant) Resolutions

#### Abstract:

There is a classical geometric way to resolve singularities, recently the idea of finding an algebraic (or categorical) resolution instead appeared. I will explain what a non-commutative (crepant) resolution (NC(C)R) is, providing motivation and some of the benefits and uses as well as some examples. This talk will be more algebraic than geometric in nature and I aim to at least provide a sketch of the results and concepts required, whenever needed.
• ### Title: Spectral Curves from Monopoles

#### Abstract:

Monopoles, i.e. solutions to the Bogomolny equations in 3 dimensions, can be uniquely obtained from data defined by certain algebraic curves in the holomorphic tangent bundle of the projective line. We will discuss this result, and special properties of monopoles from a differential geometric point of view.
• ### Title: Holonomy and Deformations of Polyhedral Manifolds

#### Abstract:

In many classical settings, such as hyperbolic geometry, it is known that small deformations of a smooth geometric structure on a manifold are parametrised by small perturbations of the holonomy representation. This is a particular (affine) representation of the fundamental group. Hyperbolic Dehn surgery allows this fact to be extended even to hyperbolic 3-manifolds with certain singularities. In this talk, we will explore this idea in the setting of polyhedral manifolds—manifolds with a metric induced by a Euclidean triangulation. We will introduce the concepts of the developing map, monodromy and holonomy in this setting, via simple examples. We will see that desirable geometric properties can be deduced from the holonomy. And finally, we will think about how to understand the relationship between holonomy and deformations of polyhedral manifolds.
• ### Title: Foliations and Birational Geometry

#### Abstract:

We'll show what foliations are and why we can use birational geometry to study them. We'll then use recent work to establish a minimal model program for foliations on surfaces and threefolds.
• ### Title: Group Invariant Machine Learning via Geometric Techniques

#### Abstract:

Applications of machine learning have become ubiquitous in our everyday lives, and are steadily becoming more common even in basic research. In a nutshell, machine learning seeks to use computers to approximate highly complex, analytically intractable functions, with simple functions; such a function might map the intersection matrix of a complete intersection Calabi–Yau manifold to its Hodge number, for example.

In this talk I will discuss joint work with Ben Aslan and Daniel Platt, in which we apply methods from differential geometry and geometric group theory to design more efficient and accurate machine learning algorithms in the cases where the function to be approximated is invariant under the action of some group.

Notes accessible here.
• ### Title: Adiabatic Limits by Example

#### Abstract:

Understanding geometric objects via fibering them by lower dimensional structures is a widespread tool in geometry. Examples include Mirror Symmetry, Deligne's proof of the Weil conjectures, and many more. In differential geometry, adiabatic limits are an established tool to study fibred manifolds. The idea of an adiabatic limit is that one rescales the metric on a fibred manifold such that the size of the fibres shrinks to zero, while the base stays of constant size. In many cases, the geometry of the entire manifold splits into the geometries of the fibres and the base. This has applications to constant scalar curvature Kähler metrics, Calabi–Yau and $$\smash{G_2}$$ metrics, instantons on $$\smash{S^4}$$, harmonic forms on fibred manifolds (leading to an analytic interpretation of the Leray spectral sequence), and many more examples. In this talk we will have a look at the general technique and some examples.

Notes accessible here.
• ### Title: Obstructing Lagrangian Concordance for Braids

#### Abstract:

Two smooth knots are concordant when they mutually bound a cylinder. The symplectic version is called Lagrangian concordance and is a relation of Legendrian knots. We can ask: when is a Legendrian knot concordant to another? In this talk, I will define Lagrangian concordance and talk about what we know and don't know about it. Time permitting, I'll sketch one way to obstruct this relation for certain kinds of knots.
• ### Title: An Invitation to $$p$$-adic Geometry via Rigid Analytic Spaces

#### Abstract:

I will discuss one approach to doing analytic geometry over nonarchimedean fields which goes by the name of rigid analytic geometry. Since this is a geometry seminar, I will try to highlight the differences between complex analytic geometry and $$p$$-adic geometry when introducing the theory.
• ### Title: Introduction to elliptic surfaces

#### Abstract:

Elliptic surfaces play a key role in the theory of algebraic surfaces in the sense that they are very accessible to direct computations. I will introduce Kodaira’s classification of singular fibres and construct many explicit examples of both geometric and arithmetic nature. If time permits, I will also discuss a result by Beauville classifying elliptic surfaces with the minimal number of singular fibres.
• ### Title: 2 Proofs that there is a Unique Conic through $$5$$ Points

#### Abstract:

The problem of counting the number of smooth conics in $$\mathbb{P}^2$$ through $$5$$ general points can be solved very simply using linear algebra. By the end of this talk I will present a longer and unnecessary second proof, using some more complicated machinery. Along the way this will give me an excuse to provide a very narrow introduction to Gromov–Witten theory. The advantage of this overkill second method is that almost the exact same proof can be used to determine the number of rational plane curves of degree $$d$$ through $$3d-1$$ points for any $$d$$. Until Gromov–Witten theory came along, this is a problem which had only been solved up to $$d=5$$.
• ### Title: An introduction to Heegaard Floer homology

#### Abstract:

Heegaard Floer homology is a package of invariants defined for closed, oriented 3-manifolds. The simplest version arises as the homology of a certain chain complex associated to a Heegaard splitting. We will outline the construction of the various different flavours of Heegaard Floer homology and explore some of their key properties. Time permitting, we will also introduce the related theory of knot Floer homology, a powerful knot invariant which categorifies the Alexander polynomial and is able to detect features such as whether a knot is fibred.
• ### Title: Introduction to Gauge Theory

#### Abstract:

In physics, gauge theory is the study of gauge fields and their associated matter fields, such as the electromagnetic field and the associated electron field, which interact according to Maxwell's equations of electromagnetism (such an interaction being called a "coupling" by physicists). Such gauge theories were developed in generality by Yang and Mills in the language of principal bundles, associated vector bundles, and connections, and this Yang–Mills theory now underpins the standard model of particle physics.

Mathematical gauge theory arose in the 1970s and 1980s as various mathematicians, most notably Michael Atiyah, demonstrated that interesting geometry constructions and invariants could be derived from physically meaningful gauge-theoretic equations, such as the Yang–Mills equations. Since the 1980s, mathematical gauge theory — the study of connections and curvature on vector bundles and principal bundles — has produced many interesting new geometric problems, techniques, structures, and solutions.

In this talk I will give a mathematicians account of the history of gauge theory, introducing the key players such as the connections, curvature, and the Yang–Mills equations, and go on to discuss the great successes of the theory. These include the deep relationships between gauge-theoretic structures and algebraic geometry through the Hitchin–Kobayashi correspondence, the novelty of moduli spaces of solutions to these equations, such as Higgs bundle moduli spaces, some of the first examples of compact non-symmetric hyper-Kähler manifolds, and the many powerful topological invariants that have arisen out of gauge-theoretic equations such as Donaldson invariants and Seiberg–Witten invariants.

Notes accessible here.
• ### Title: Dirac and Elliptic Operators

#### Abstract:

The Laplacian is arguably the most important among differential operators. Its peculiar properties (e.g. regularity, maximum principles, Fredholmness) actually belong to a wide class of differential operators, which go by the name of “Elliptic operators”. In this talk, we will focus on instances of Dirac operators – elliptic operators that can be thought of as “square roots of a Laplacian” – in geometric contexts. In particular, we will see how, through the Atiyah–Singer Index Theorem, the study of some Dirac operators enables us to prove some main theorems of complex geometry.
• ### Title: Introduction to the Minimal Model Program

#### Abstract:

The Minimal Model Program (MMP) conjectures that every variety can be constructed by combining three basic types of varieties: Fano, Calabi–Yau and of general type. This talk will explore the main statements and techniques employed in the study of the MMP.
• ### Title: Symplectic Toric Manifolds

#### Abstract:

With its roots in algebraic geometry, the theory of toric varieties was first explored in the 70s and has since become valuable in a wide range of settings. The goal of this talk is to give an intuitive, example based introduction to toric manifolds from a symplectic viewpoint. We will discuss how to see the geometry of these spaces through their combinatorial properties and look to understand how they arise physically.
• ### Title: Divisor–Line Bundle Correspondence

#### Abstract:

The Divisor–Line Bundle correspondence is an invaluable tool in Complex and Algebraic Geometry but can be mysterious on one's first encounter. In this talk, we will see the correspondence three different ways with a particular focus on the underlying geometry. Time permitting, we will apply the correspondence to the study of Intersection Theory and Birational Geometry of surfaces.

Notes accessible here.

#### Abstract:

The du Val or A-D-E singularities, also known as rational double points, are isolated surface singularities which can be resolved by blowing up a finite number of times, the final resolution replacing the singular point with a tree of smooth rational curves. They crop up in many areas of geometry, and for example can be thought of as negligible singularities in the classification of surfaces. The intersection pattern of this tree of curves will be dual to a Dynkin diagram of type A, D or E.

I will introduce them via some examples and link them to two other basic mathematical objects: platonic solids and simple Lie groups. The du Val singularities can be characterised as orbifold singularities $$\smash{\mathbb{C}^2/G}$$, where $$G$$ is a binary polyhedral group (finite subgroup of $$\smash{\mathrm{SL}_2(\mathbb{C})}$$) which can be related to its Dynkin diagram via a McKay quiver. I will also describe the link between du Val singularities and simple Lie groups of type A, D and E.
• ### Title: The Kodaira Embedding Theorem

#### Abstract:

Kodaira’s Embedding Theorem characterises compact complex manifolds that can be embedded in a projective space as those admitting a positive line bundle on it.
Combining it with Chow’s Theorem, asserting that every complex projective variety is algebraic, it allows us to reduce problems of analysis to ones of algebra.
After reviewing a few needed tools, such as linear systems and maps to projective spaces and blow-ups, I will present the classical proof of the result, relying on Kodaira’s Vanishing Theorem.

Notes accessible here.
• ### Title: Homogeneous and Symmetric Spaces

#### Abstract:

Riemannian homogeneous spaces are objects that lie in the intersection of two huge domains in maths: Riemannian geometry and Lie theory. Due to their dual nature, one can study them using a wide variety of tools and methods from both domains. One especially nice subclass of Riemannian homogeneous spaces consists of symmetric spaces, whose high degree of symmetry really takes the interplay between Riemannian geometry and Lie theory one step further and ultimately allows a complete classification. We will discuss homogeneous and symmetric spaces and how their geometry can be studied by means of Lie theory, touch on Cartan's classification of symmetric spaces, and look at them through the lens of holonomy and Berger's theorem.

Notes accessible here.
• ### Title: Connections and Curvature

#### Abstract:

On Euclidean space, we have a very clear idea of what parallel means. On manifolds this notion is not immediately obvious, as there is no canonical way to compare tangent spaces at different points. To solve this, one might ask for a way to ‘connect’ them. This is precisely what a ‘connection’ does! We will introduce vector and principal $$G$$-bundles, and show how to define connections, covariant derivatives and curvature. We will show how connections can be used to define parallel transport and give exciting examples of how connections are used in the Gibbons–Hawking ansatz to construct interesting metrics.
• ### Title: Introduction to Schemes

#### Abstract:

Schemes are a fundamental part of algebraic geometry and are incredibly useful for studying number theory. Unfortunately, introductions to schemes can often be very technical, dry, and very hard to understand. In this talk, I won't go through all of the technical details, but I will try to give everyone some motivation behind why we need to work with schemes, and also give everyone an idea of how to think about schemes. I'll start by drawing some parallels with varieties, before defining "affine schemes", drawing a few examples of some schemes, and by the end, we'll work through an example to demonstrate how they can be very useful in a way that varieties can't be.

## Contact the Organisers:

 John McCarthy Qaasim Shafi Jaime Mendizabal