Forthcoming events in this series


Thu, 30 Nov 2023
15:00
L4

A gentle introduction to Ricci flow

John Hughes
(University of Oxford)
Abstract

Richard Hamilton introduced the Ricci flow as a way to study the Poincaré conjecture, which says that every simply connected, compact three-manifold is homeomorphic to the three-sphere. In this talk, we will introduce the Ricci flow in a way that is accessible to anyone with basic knowledge of Riemannian geometry. We will give some examples, discuss finite time singularities, and give an application to a theorem of Hamilton which says that every compact Riemannian 3-manifold with positive Ricci curvature admits a metric of constant positive sectional curvature.

Thu, 16 Nov 2023
15:00
L4

Compactness problems in new gauge theories

Alfred Holmes
(University of Oxford)
Abstract

Two areas of current research in Mathematical Gauge Theory are the study of higher dimensional instantons on manifolds with special holonomy (for example, Calabi-Yau three folds, Gand Spin(7) manifolds), and low dimensional gauge theories (for example the Kapustin-Witten, Haydys-Witten and ADHM Seiberg-Witten equations). A common feature of these two sets of theories is that the moduli spaces of solutions are in general not compact. In both cases, compactness issues arise because of solutions to a certain non-linear equation called the Fueter equation. In this talk, I'll explain how this non compactness gives a relationship between these high and low dimensional gauge theories.

Thu, 02 Nov 2023
15:00
L4

Generalising fat bundles and positive curvature

Alberto Rodriguez Vazquez
(KU Leuven)
Abstract

Alan Weinstein, introduced the concept of "fat bundle" as a tool to understand when the total space of a fiber bundle with totally geodesic fibers allows a metric with positive sectional curvature. 

In recent times, certain weaker notions than the condition of having a metric with positive sectional curvature have been studied due to the apparent scarcity of spaces that meet this condition. Positive kth-intermediate Ricci curvature (Rick > 0) on a Riemannian manifold Mn is a condition that bridges the gap between positive sectional curvature and positive Ricci curvature. Indeed, when k = 1, this condition corresponds to positive sectional curvature, and when k = n−1, it corresponds to positive Ricci curvature. 

In this talk, I will discuss an ongoing project with Miguel Domínguez Vázquez, David González-Álvaro, and Jason DeVito, which aims to create new examples of compact Riemannian manifolds with Ric2 > 0. We achieve this by employing a certain generalisation of the "fat bundle" concept.

Thu, 01 Jun 2023

15:00 - 16:00
L6

A Lagrangian Klein Bottle You Can't Squeeze

Matthew Buck
(University of Lancaster)
Abstract

Given a non-orientable Lagrangian surface L in a symplectic 4-manifold, how far
can the cohomology class of the symplectic form be deformed before there is no
longer a Lagrangian isotopic to L? I will properly introduce this and a
related question, both of which are less interesting for orientable
Lagrangians due to topological conditions. The majority of this talk will be
an exposition on Evans' 2020 work in which he solves this deformation
question for a particular Klein bottle. The proof employs the heavy machinery
of symplectic field theory and more classical pseudoholomorphic
curve theory to severely constrain the topology and intersection properties of
the limits of certain pseudoholomorphic curves under a process called
neck-stretching. The treatment of SFT-related material will be light and focus
mainly on how one can use the compactness theorem to prove interesting things.

Thu, 09 Feb 2023
15:00
L6

The HKKP filtration for algebraic stacks

Andres Ibanez Nunez
Abstract

In work of Haiden-Katzarkov-Konsevich-Pandit (HKKP), a canonical filtration, labeled by sequences of real numbers, of a semistable quiver representation or vector bundle on a curve is defined. The HKKP filtration is a purely algebraic object that depends only on a poset, yet it governs the asymptotic behaviour of a natural gradient flow in the space of metrics of the object. 

In this talk, we show that the HKKP filtration can be recovered from the stack of semistable objects, thus generalising the HKKP filtration to other moduli problems of non-linear origin. In particular, we will make sense of the notion of a filtration labelled by sequence of numbers for a point of an algebraic stack.

Thu, 02 Feb 2023
15:00
L6

Higher Geometry by Examples

Chenjing Bu
Abstract

We give an introduction to the subject of higher geometry, by giving many examples of higher geometric objects, and looking at their properties. These include examples of 2-rings, 2-vector spaces, and 2-vector bundles. We show how these concepts help solve problems in ordinary geometry, as one of the many motivations of the subject. We assume no prerequisites on the subject, and the talk should be applicable to both differential and algebraic geometry.

Thu, 08 Dec 2022
15:00
L3

On the stability of minimal submanifolds in conformal spheres

Federico Trinca
(Oxford University)
Abstract

Minimal submanifolds are the critical points of the volume functional. If the second derivative of the volume is nonnegative, we say that such a minimal submanifold is stable.

After reviewing some basics of minimal submanifolds in a generic Riemannian manifold, I will give some motivations behind the Lawson--Simons conjecture, which claims that there are no stable minimal submanifolds in 1/4-pinched spheres. Finally, I will discuss my recent work with Giada Franz on the nonexistence of stable minimal submanifolds in conformal pinched spheres.

Thu, 24 Nov 2022
15:00
L3

Desingularisation of conically singular Cayley submanifolds

Gilles Englebert
(Oxford)
Abstract

Cayley submanifolds in Spin(7) geometry are an analogue and generalisation of complex submanifolds in Kähler geometry. In this talk we provide a glimpse into calibrated geometry, which encompasses both of these, and how it ties into the study of manifolds of special holonomy. We then focus on the deformation theory of compact and conically singular Cayleys. Finally we explain how to remove conical singularities via a gluing construction.

Thu, 10 Nov 2022
15:00
L3

Compactified Universal Jacobians over Stacks of Stable Curves via GIT

George Cooper
(Oxford)
Abstract

Associated to any smooth projective curve C is its degree d Jacobian variety, parametrising isomorphism classes of degree d line bundles on C. Letting the curve vary as well, one is led to the universal Jacobian stack. This stack admits several compactifications over the stack of marked stable curves, depending on the choice of a stability condition. In this talk I will introduce these compactified universal Jacobians, and explain how their moduli spaces can be constructed using Geometric Invariant Theory (GIT). This talk is based on arXiv:2210.11457.

Thu, 24 Mar 2022

15:00 - 16:00
Virtual

Derived blow-ups using Rees algebras and virtual Cartier divisors

Jeroen Hekking
(KTH Stockholm)
Abstract

The blow-up B of a scheme X in a closed subscheme Z enjoys the universal property that for any scheme X' over X such that the pullback of Z to X' is an effective Cartier divisor, there is a unique morphism of X' into B over X. It is well-known that the blow-up commutes along flat base change.

In this talk, I will discuss a derived enhancement B' of B, namely the derived blow-up, which enjoys a universal property against all schemes over X, satisfies arbitrary (derived) base-change, and contains B as a closed subscheme. To this end, we will need some elements from derived algebraic geometry, which I will review along the way. This will allow us to construct the derived blow-up as the projective spectrum of the derived Rees algebra, and state its functor of points in terms of virtual Cartier divisors, using Weil restrictions.

This is based on ongoing joint work with Adeel Khan and David Rydh.

Thu, 10 Mar 2022

15:00 - 16:00
C2

Gauge theories in 4, 8 and 5 dimensions

Alfred Holmes
(University of Oxford)
Abstract

In the 1980s, gauge theory was used to provide new invariants (up to
diffeomorphism) of orientable four dimensional manifolds, by counting
solutions of certain equations up to to a choice of gauge. More
recently, similar techniques have been used to study manifolds of
different dimensions, most notably on Spin(7) and G_2 manifolds. Using
dimensional reduction, one can find candidates for gauge theoretic
equations on manifolds of lower dimension. The talk will give an
overview of gauge theory in the 4 and 8 dimensional cases, and how
gauge theory on Spin(7) manifolds could be used to develop a gauge
theory on 5 dimensional manifolds.

Thu, 24 Feb 2022

15:00 - 16:00
C2

TBC

TBC
Thu, 17 Feb 2022

15:00 - 16:00
C2

Torsion points on varieties and the Pila-Zannier method - TALK POSTPONED UNTIL WEEK 5

Francesco Ballini
(Oxford University)
Abstract

In 2008 Pila and Zannier used a Theorem coming from Logic, proven by Pila and Wilkie, to give a new proof of the Manin-Mumford Conjecture, creating a new, powerful way to prove Theorems in Diophantine Geometry. The Pila-Wilkie Theorem gives an upper bound on the number of rational points on analytic varieties which are not algebraic; this bound usually contradicts a Galois-theoretic bound obtained by arithmetic considerations. We show how this technique can be applied to the following problem of Lang: given an irreducible polynomial f(x,y) in C[x,y], if for infinitely many pairs of roots of unity (a,b) we have f(a,b)=0, then f(x,y) is either of the form x^my^n-c or x^m-cy^n for c a root of unity.

Thu, 27 Jan 2022

15:00 - 16:00
Virtual

Ricci curvature lower bounds for metric measure spaces.

Dimitri Navarro
(Oxford University)
Abstract

In the '80s, Gromov proved that sequences of Riemannian manifold with a lower bound on the Ricci curvature and an upper bound on the dimension are precompact in the measured Gromov--Hausdorff topology (mGH for short). Since then, much attention has been given to the limits of such sequences, called Ricci limit spaces. A way to study these limits is to introduce a synthetic definition of Ricci curvature lower bounds and dimension upper bounds. A synthetic definition should not rely on an underlying smooth structure and should be stable when passing to the limit in the mGH topology. In this talk, I will briefly introduce CD spaces, which are a generalization of Ricci limit spaces.