logo Luke Naylor

Presentations

Talk Event
Fourier-Mukai Transforms (slides)

Abstract: Shigeru Mukai first introduced a "Fourier functor" in the 80s, now known as the Fourier-Mukai transform, which has since been used extensively for derived equivalences in algebraic geometry. In this talk, I will cover some material around the following topics: Where does the "Fourier" come from? Some homological algebra around the definition and early propositions, and some roles it plays in algebraic geometry.

DOGS seminar,
UCL (online)
2024-01-16
Narrowing Down Possibilities for Tilt Walls (slides)

Abstract: Bridgeland stability conditions were introduced on arbitrary triangulated categories (such as D^b(X)) as an analogue to classical notions of stability for vector bundles. In certain settings, for a fixed Chern character, there are "chambers" of stability conditions, separated by "walls", which yield the same moduli of stable objects. One of these chambers corresponding to Gieseker stability of sheaves. Here I present a tool I have been developing to narrow down the possibilities for such walls on Picard rank 1 surfaces in a case when we know there are finitely many. As well as potential extensions to give a generating sequence when there are infinitely many walls on a principally polarized abelian surface.

Structure and
Symmetry Day
,
Edinburgh University
2023-12-06
Conway's approach to symmetries (slides)

Abstract: When I first saw the classification of wallpaper groups, as I'm sure it was the same with many others, it was a very algebraic approach. For example considering the translation subgroup, the point group, and ways they can fit together. This past semester I had the opportunity to tutor for Toby Bailey's course "Symmetry and Geometry", which is based on material from a book co-authored by one of his past lecturers John Conway titled "The Symmetries of things". The approach in this text is, instead of considering the groups themselves, to consider the space (which has the symmetries) quotiented by the group of symmetries giving so called orbifolds. This method generalises well to spherical and hyperbolic symmetry and gives clean picturesque ways of classifying certain types of tilings. This talk is a 50 minute picturesque summary and advert for the topics in this area.

Hodge Club,
Edinburgh University
2023-11-10
Hodge Club,
Edinburgh University
2022-04-22
From stability of vector bundles to stability in triangulated categories (slides)

Abstract: Classically, stability of vector bundles was studied. But more recently, this area has taken a homological algebra twist with Bridgeland stabilities becoming very popular. Here I will talk about this transition to triangulated categories, with an attempt to make the similarities with Mumford stability as clear as possible.

GEARS/ARTIN,
Glasgow University
2023-09-12
Bridgeland Stabilities and Finding Walls (slides)

Abstract: As many are aware, quite a few of us in the school are concerned with the business of stability conditions on sheaves. If you’ve talked to these people before, you very quickly realise that this is actually short for “stability conditions on the derived category of coherent sheaves”. In this talk I introduce this topic, particularly, the explicit stability conditions constructed by Bridgeland for K3 surfaces. But also how they relate to the more classical Mumford and Gieseker stabilities (which do not involve the derived). Later on, some stabilities on threefolds too. Along the way we will have a look at the notion of walls and why we want to find them. This will also expose the benefit of a couple of different viewpoints to find restrictions on the set of possible walls, some numeric, but also the geometry of some characteristic curves on the space of stability conditions.

Hodge Club,
Edinburgh University
2023-03-24
From Fourier to Fourier-Mukai

Abstract: When you look at the definition for the Fourier-Mukai Transform, it's hard to see what this has to do with integral transforms, let alone the one in its name. Yet, this analogy was obvious to Shigeru Mukai, who first introduced it in the 80s as "a Fourier functor". During this talk I will present the Fourier Transform, and the Fourier-Mukai transform, concentrating on the similarities and how they are used for the same purpose. Then, finally, a peek at what role this plays in finding walls for Bridgeland stability conditions.

GEARS,
Glasgow University
2022-10-25
Reading Group Talk Reading Group
Closed embeddings Algebraic Geometry
(Rising Sea)
,
2023-05-05
Constructive proof of syzygy theorem (application of Groebner bases) Commutative Algebra
(Eisenbud)
,
2023-03-03
Flat families Commutative Algebra
(Eisenbud)
,
2022-12-10
Example workthrough: primary decomposition Commutative Algebra
(Eisenbud)
,
2022-11-04
Poster Event
Tighter bounds on ranks of tilt semistabilizers – narrowing down on possibilities quicker (poster)

Abstract: When trying to narrow down the possibilities for the locations of tilt walls, the typical strategy is to find potential semistabilizers who’s Chern characters satisfy a set of inequalities which would be satisfied by genuine ones (including Bogomolov-Gieseker). In 2020, Benjamin Schmidt published a program to compute the possibilities, however the running time suffered from a conservative estimate for a bound on the ranks of tilt semistabilizers. The work I want to present is a set of refinements on this bound, one in the form of a pragmatic formula to calculate by hand which can be as small as a quarter of the original bound. As well as more complicated formulae with extra parameters which are better suited in the context of a newer computer program. This newer program gives instantaneous results in certain examples where the original takes over an hour, and can easily be tried from your browser at https://pseudowalls.gitlab.io/webapp/tilt.sycamore/.

Conference: DCMSCI,
Imperial College London
2023-07-03 to 2023-07-07

Teaching

Teaching Material Creation

I was involved with the logistics for moving to using GitHub Codespaces in the Computational Numerics course, following the release by GitHub in early 2023. In particular, the containerization. I've also added to the teaching material in that course to encourage students to make use of IDE debuggers.

For the beginning of 2024, I am running a workshop to promote tech literacy among mathematical researchers.

Tutoring

I haved been involved with tutoring in the School of Mathematics at the University of Edinburgh, where I received an Associate Fellowship of the Higher Education Academy

  • Jan-Apr 2021-22
Symmetry and Geometry

course level: undergraduate (yr 4)

Course about symmetry groups and tilings using Conway’s approach with orbifolds instead of going deep into group theory

  • Jan-Apr 2022-23
  • Jan-Apr 2021-22
  • Jan-Apr 2020-21
Computing and Numerics

course level: undergraduate (yr 2)

Introductory Python course applied to numerical methods (such as integral estimates) alongside teaching good coding practices.

  • Sept-Dec 2022-23
  • Sept-Dec 2022-23
  • Sept-Dec 2021-22
Python Programming

course level: masters

Introductory Python course touching on coding practices and applications around data analysis (numpy/scipy/pandas...). Concluding in open-ended group project analysing open datasets.

  • Sept-Dec 2021-22
Honours Analysis Skills

course level: undergraduate (yr 3)

Presentation skills workshop with topics around real analysis

  • Sept-Dec 2022-23
Facets of Mathematics

course level: undergraduate (yr 2)

Course driven by 3 project-oriented topics making use of a variety of mathematical related soft skills including latex typesetting, poster making & presentation, and numeric & symbolic computing. Mathematical theory involved in course included statistics, finite difference methods, and projective geometry.

  • Jan-Apr 2022-23
  • Jan-Apr 2020-21
Maths Base

course level: undergraduate (yr 2)

Drop in session for students to ask for help in any of their courses