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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
Password: geometry


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Upcoming Talks:

  • Title: Dirac and Elliptic Operators

    Speaker: Alessio Di Lorenzo

    Date: Friday, 27 November 2020

    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 Gauge Theory

    Speaker: Jaime Mendizabal

    Date: Friday, 04 December 2020

    Abstract:

    TBA
  • Title: TBA

    Speaker: Laura Wakelin

    Date: Friday, 11 December 2020

    Abstract:

    TBA

Past Talks:

  • Title: Introduction to the Minimal Model Program

    Speaker: Federico Bongiorno

    Date: Friday, 20 November 2020

    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

    Speaker: Teresa Ludenbach

    Date: Friday, 13 November 2020

    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

    Speaker: Soham Karwa

    Date: Friday, 06 November 2020

    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.
  • Title: ADE Singularities

    Speaker: Hannah Tillmann-Morris

    Date: Friday, 30 October 2020

    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

    Speaker: Riccardo Carini

    Date: Friday, 23 October 2020

    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

    Speaker: Ivan Solonenko

    Date: Friday, 16 October 2020

    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

    Speaker: Joshua Daniels-Holgate

    Date: Friday, 09 October 2020

    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

    Speaker: Jesse Pajwani

    Date: Friday, 02 October 2020

    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