Contact the organisers

The Imperial Junior Geometry seminar for 2021/22 is run from 5:30PM to 6:30PM every Friday. The seminar is in room 402 in the Imperial CDT space, and will be run hybrid online through Zoom or recorded for those who cannot attend in person.

## Past Talks:

• ### Title: Gromov-Hausdorff limits of Kahler manifolds

#### Abstract:

Gromov-Hausdorff convergence is a very general notion of convergence in the space of metric spaces modulo isometries. If we restrict to Riemannian manifolds with certain non-collapsing properties, the limit of converging subsequences is itself a C^{1,\alpha} Riemannian manifold with singularities.

When we consider the case of sequences of polarised Kahler manifolds (which can be thought to be algebraic varieties), we expect the limit to be a polarised Kahler manifold itself, and hence to have an algebraic equivalent.

By results of Donaldson and Sun, it is indeed the case, and we can roughly say that the geometric convergence (Gromov-Hausdorff) corresponds to the algebraic convergence (in the Chow variety, which is a parameter space of subvarieties with fixed degree and dimension).

This exhibits the Gromov-Hausdorff limit of Kahler manifolds as an algebraic variety, and thus enables us to use algebraic techniques to gain information about the geometry of this limit.

In this talk, we’ll survey the notions and arguments around Donaldson-Sun’s Theorem, and some applications.

TBA

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• ### Title: Why $$1 \neq 32$$: Excess Intersection

#### Abstract:

Given five general lines in the plane there is a unique smooth conic which is tangent to all of them. On the other hand, the space of all conics is a copy of $$\mathbb{P}^5$$ and the condition of being tangent to a line defines a quadric hypersurface in $$\mathbb{P}^5$$. By Bezout’s theorem, intersecting five of these hypersurfaces should give you $$2^5=32$$ points corresponding to $$32$$ conics tangent to all five lines. In this talk I’ll explain what went mysteriously wrong…

TBA
• ### Title: Determinant bundles in algebraic geometry

#### Abstract:

When studying moduli spaces in algebraic geometry, one method to check projectivity is to identify an ample line bundle. In some situations, it is possible to construct this line bundle using the theory of determinants, and this analytical description of the ample line bundle allows an alternative analytic description of the moduli space. This was used by Donaldson in 1987 to provide an alternative proof that every stable vector bundle on a projective manifold admits a Hermite--Einstein metric. I will describe the determinant bundle construction, and discuss some applications of it in algebraic geometry.

TBA
• ### Title: Knot Floer homology obstructs ribbon concordance

#### Abstract:

For a pair of knots K_0, K_1 in S^3, a concordance C from K_0 to K_1 is a smoothly embedded annulus in S^3 x [0,1] with boundary -K_0 x {0} \cup K_1 x {1}. C is called a ribbon concordance if it has only index 0 and 1 critical points. It has long been conjectured that ribbon concordance is a partial order on knots; this was proved by Agol in January 2022. We will review an earlier paper by Zemke which supports the result by showing that knot Floer homology obstructs ribbon concordance. This also has topological applications involving the Seifert genus of knots.
• ### Title: Compactifying with Logarithmic Geometry

#### Abstract:

The first character in this talk is an irreducible closed subvariety Z in the n dimensional torus (C^*)^n. We describe a class of toric varieties {X_\sigma} associated to Z. We encounter tools for understanding the chow class of Z in such toric varieties. Our approach will allow us to “discover” tropical geometry.
The second character in this talk is a closed subvariety Z’ in a logarithmic scheme. The simplest example of a logarithmic scheme is a toric variety. We meet analogues of phenomena in the toric situation. Our story touches on the technology of Artin fans, piecewise polynomials and the logarithmic Chow group.
• ### Title: Intersection theory

#### Abstract:

There exist two approaches to intersection theory: a homological one through Chow groups and a cohomological one through K-theory. These are higher codimension generalisations of Weil divisors and line bundles respectively. This talk will deal with both approaches and explain how they are related.

• ### Title: Weakness

#### Abstract:

Analysts spend a lot of time talking about 'weak' things. Weak limits, weak derivatives, weak solutions, etcetera etcetera. As we will see weakening notions like derivatives and solutions turns out to be a very useful tool in PDE theory. Can we apply similar ideas in geometry? Is it possible to weaken the notion of a surface or of curvature, and why might this be useful?
• ### Title: Moment Maps and Moment Polytopes

#### Abstract:

In this talk, we will look at toric manifolds from the symplectic point of view. Specifically, we will define the moment map, whose image, the moment polytope, determines the symplectic toric manifold.
• ### Title: Kodaira embedding theorem

#### Abstract:

The Kodaira embedding theorem characterises smooth projective varieties among general Kaehler manifolds. More precisely, it establishes an equivalence between the differential geometric notion of positivity of a holomorphic line bundle and the algebraic concept of ampleness. In this way it provides us with a useful criterion on holomorphic line bundles to have enough sections to define an embedding into projective space. In the talk we will recall the relevant differential geometric as well as algebro geometric notions before delving into a light sketch of the proof of this far reaching theorem.
• ### Title: The Total Coordinate ring of a Toric Variety

#### Abstract:

Projective space can be written as a quotient $\mathbb{P}^n = \mathbb{C}^{n+1}-0/\mathbb{C}^\ast$ and as a consequence we have global homogeneous coordinates on $\P^n$. E.g., we can talk about the zero set of a homogeneous ideal in $\mathbb{C}[x_0,\dots,x_n]$. More generally, there is a correspondence between graded modules over $\mathbb{C}[x_0,\dots,x_n]$ and sheaves on $\P^n$. It turns out that this construction generalizes to arbitrary toric varieties. We will see how a toric variety can be identified with a quotient of $\mathbb{C}^k$ and that there is a similar correspondence between sheaves and graded modules over its "total coordinate ring".
• ### Title: Du Val Singularities and how to make them go away

#### Abstract:

Du Val singularities are a particularly nice class of surface singularity which have been classified quite explicitly. We will talk about a few different characterisations of them with examples, and also how to attempt to remove singularities with two different techniques: Resolution of singularities, which in the case of Du Val will be done with successive blowups, and at the end I will touch on smoothing singularities.

Recording: https://ucl.zoom.us/rec/share/xtdATIGc_wm2ETVhyJhCMlzYJxaeUmzN3rsxMEWmZuKlfmt76clMu2FwpcPdJoxw.wP57qCmh5utXMrCZ

• ### Title: Splitting principle, characteristic classes and Hirzebruch-Riemann-Roch theorem

#### Abstract:

We introduce the splitting principle, which is a convenient tool for defining and studying characteristic classes, like Chern classes, Chern characters and Todd classes. We then present the Hirzebruch-Riemann-Roch theorem and discuss how it generalizes various classical Riemann-Roch-type statements.
• ### Title: Riemannian Holonomy Groups

#### Abstract:

An elementary introduction to the holonomy principle, Riemannian holonomy groups, and Berger’s list. Overview of examples, including sketch constructions of metrics with special holonomy. We will also see a specific physical application in a joint work with Barton-Singer (Herriot-Watt).

Recording: https://ucl.zoom.us/rec/share/uR09Y4TGjMkLZXBtanV6qQUX284Ghhv_SmHGOuub53LZPUe-sW56HqVaz9U3HSGf.W_xVhml0P7FU1nCp

Access Passcode: Geometry21-22
• ### Title: Holomorphic line bundles, divisors and maps to projective spaces

#### Abstract:

It is well-known that compact complex manifolds admit no global holomorphic functions: a natural generalisation is to look at holomorphic sections of non-trivial line bundles.
I’ll show how line bundles can be associated to codimension-one analytic subvarieties (divisors), and how their sections can be used to define maps to projective spaces.

https://ucl.zoom.us/rec/play/vTz7eWPIYpUqsYze3zaxZph8aZEK40mQIWzVFD0CBp8Q8P875OQCuLhp453i7wiPL-fTbLo6tNv9Xv5W.ZrUaJ3mGEtJwxTiD?continueMode=true

Access Passcode: Geometry21-22
• ### Title: Connections, curvature and Chern classes

#### Abstract:

Did you ever get scared when hearing the word “connections”?
The aim of this talk is trying to get rid of this feeling.
On general manifolds there is no trivial way to define parallelism. To define it, we need to introduce a new notion of derivative, here is where connections come into play.
After getting an idea of the role of connections in differential geometry, we will explore a related notion: curvature. As the name suggests, it is a way to measure how far from being flat a manifold is. Curvature is also used to define Chern classes, a tool used in any kind of geometry.
• ### Title: Connections, curvature and Chern classes

#### Abstract:

Did you ever get scared when hearing the word “connections”?
The aim of this talk is trying to get rid of this feeling.
On general manifolds, there is no trivial way to define parallelism. To define it, we need to introduce a new notion of derivative, here is where connections come into play.
After getting an idea of the role of connections in differential geometry, we will explore a related notion: curvature. As the name suggests, it is a way to measure how far from being flat a manifold is. Curvature is also used to define Chern classes, a tool used in any kind of geometry.
• ### 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.
• ### Title: Trees, Quasi-Trees, and Hyperbolic Spaces

#### Abstract:

Quasi-trees are spaces that can be thought of as lying somewhere between trees and hyperbolic spaces. They retain many of the strong geometric properties of trees, while at the same time admitting many interesting group actions, most notably by acylindrically hyperbolic groups. The aim of this talk will be to illustrate quite how "tree-like" quasi-trees are, as well as to give some idea of their importance in geometric group theory.
• ### Title: Balanced Metrics on 6-Dimensional Cohomogeneity One Manifolds

#### Abstract:

In the context of Hermitian geometry, the Hull–Strominger system is a system of non-linear PDEs on heterotic string theory.
We will begin the talk by introducing and giving a description of the geometry of cohomogeneity one manifolds. Then, we will look for solutions to the Hull–Strominger system in three dimensions in the cohomogeneity one setting.
This leads us to consider simply connected $$3$$-dimensional balanced manifolds endowed with an invariant nowhere-vanishing holomorphic $$(3,0)$$-form which are of cohomogeneity one under the almost effective action of a connected Lie group $$G$$.
We show that one of such $$M$$ would have to be compact and have a certain principal orbit type, up to $$G$$-equivariant diffeomorphism.
• ### Title: Donaldson-Thomas Theory

#### Abstract:

We will start this talk with an introduction to Donaldson-Thomas (DT) theory. DT theory counts ideal sheaves to answer questions starting “how many curves on the 3 dimensional variety X...”. We will look at a linear system as a simple example of sheaf counting. We will then discuss the general DT construction, its relationship to Gromov-Witten theory and a handful of interesting results.
Subject to time, we explain why a logarithmic Donaldson-Thomas theory should prove useful. We discuss the logarithmic linear system as a simple example of logarithmic Donaldson-Thomas spaces. The example will show how solving a tropical moduli problem solves an algebraic moduli problem for free.
• ### Title: Positivity and Campana Orbifolds

#### Abstract:

Campana orbifolds are pairs $$(X, \Delta)$$ that interpolate between the world of compact geometry and of logarithmic geometry. They come equipped with a sheaf of orbifold differential forms that generalises the ordinary cotangent sheaf and the logarithmic cotangent sheaf. After introducing these objects, I will turn my attention to the quest for positivity. In particular the quest for finding projective varieties with ample cotangent bundles; I will highlight some recent work by Brotbeck–Deng and Brotbeck–Darondeau in providing examples of varieties with ample cotangent bundles and positive logarithmic cotangent bundles. Finally I would like to say something about positivity of Campana orbifolds.
• ### Title: The Kummer Construction for Manifolds with Special Holonomy

#### Abstract:

In 1955 Berger gave a complete classification of Riemannian manifolds in terms of their holonomy. Finding examples for each class turned out to be much harder. Especially for the exceptional cases, finding examples took at least 29 years. In this talk we will explain a glueing construction for these types of manifolds. We will work out an example and explain the subtleties needed in the analysis to make things work.
• ### 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.
• ### 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:

 Robert Crumplin Marta Benozzo