Wed, 01 Mar 2023
16:00
L6

Algorithms and 3-manifolds

Adele Jackson
(University of Oxford)
Abstract

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.

Wed, 15 Feb 2023
16:00
L6

[Cancelled]

Filippo Baroni
(University of Oxford)
Wed, 08 Feb 2023
16:00
L6

Minimal disks and the tower construction in 3-manifolds

Ognjen Tosic
(University of Oxford)
Abstract

A fundamental result in 3-manifold topology is the loop theorem: Given a null-homotopic simple closed curve in the boundary of a compact 3-manifold $M$, it bounds an embedded disk in $M$. The standard topological proof of this uses the tower construction due to Papakyriakopoulos. In this talk, I will introduce basic existence and regularity results on minimal surfaces, and show how to use the tower construction to prove a geometric version of the loop theorem due to Meeks--Yau: Given a null-homotopic simple closed curve in the boundary of a compact Riemannian 3-manifold $M$ with convex boundary, it bounds an embedded disk of least area. This also gives an independent proof of the (topological) loop theorem.

Wed, 25 Jan 2023
16:00
L6

Group cohomology, BNS invariants and subgroup separability

Monika Kudlinska
(University of Oxford)
Abstract

Group cohomology is a powerful tool which has found many applications in modern group theory. It can be calculated and interpreted through geometric, algebraic and topological means, and as such it encodes the relationships between these different aspects of infinite groups. The aim of this talk is to introduce a circle of ideas which link group cohomology with the theory of BNS invariants, and the property of being subgroup separable. No prior knowledge of any of these topics will be assumed.

Wed, 18 Jan 2023
16:00
L6

Condensed Mathematics

Sofía Marlasca Aparicio
(University of Oxford)
Abstract

Condensed Mathematics is a tool recently developed by Clausen and Scholze and it is proving fruitful in many areas of algebra and geometry. In this talk, we will cover the definition of condensed sets, the analogues of topological spaces in the condensed setting. We will also talk about condensed modules over a ring and some of their nice properties like forming an abelian category. Finally, we'll discuss some recent results that have been obtained through the application of Condensed Mathematics.

Thu, 27 Apr 2023

14:00 - 15:00
(This talk is hosted by Rutherford Appleton Laboratory)

All-at-once preconditioners for ocean data assimilation

Jemima Tabeart
(University of Oxford)
Abstract

Correlation operators are used in data assimilation algorithms
to weight the contribution of prior and observation information.
Efficient implementation of these operators is therefore crucial for
operational implementations. Diffusion-based correlation operators are popular in ocean data assimilation, but can require a large number of serial matrix-vector products. An all-at-once formulation removes this requirement, and offers the opportunity to exploit modern computer architectures. High quality preconditioners for the all-at-once approach are well-known, but impossible to apply in practice for the
high-dimensional problems that occur in oceanography. In this talk we
consider a nested preconditioning approach which retains many of the
beneficial properties of the ideal analytic preconditioner while
remaining affordable in terms of memory and computational resource.

Mon, 14 Nov 2022
16:00
L4

The Weil bound

Jared Duker Lichtman
(University of Oxford)
Abstract

The Riemann hypothesis (RH) is one of the great open problems in
mathematics. It arose from the study of prime numbers in an analytic
context, and—as often occurs in mathematics—developed analogies in an
algebraic setting, leading to the influential Weil conjectures. RH for
curves over finite fields was proven in the 1940’s by Weil using
algebraic-geometric methods, and later reproven by Stepanov and
Bombieri by elementary means. In this talk, we use RH for curves to
prove the Weil bound for certain (Kloosterman) exponential sums, which
in turn is a fundamental tool in the study of prime numbers.

Wed, 09 Nov 2022
16:00
L4

Persistent homology in theory and practice

Katherine Benjamin
(University of Oxford)
Abstract

Persistent homology is both a powerful framework for data science and a fruitful source of mathematical questions. Here, we will give an introduction to both single-parameter and multiparameter persistent homology. We will see some examples of how topology has been successfully applied to the real world, and also explore some of the pure-mathematical ideas that arise from this new perspective.

Wed, 26 Oct 2022
16:00
L4

$\ell^2$ and profinite invariants

Ismael Morales
(University of Oxford)
Abstract

We review a few instances in which the first $\ell^2$ Betti number of a group is a profinite invariant and we discuss some applications and open problems.

Wed, 19 Oct 2022
16:00
L4

$\ell^2$-invariants and generalisations in positive characteristic

Sam Fisher
(University of Oxford)
Abstract

We survey the theory of $\ell^2$-invariants, their applications in group theory and topology, and introduce a positive characteristic version of $\ell^2$-theory. We also discuss the Atiyah and Lück approximation conjectures, two of the central problems in this area.

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