Graph algorithms
Here is the handout for today’s training session:
Sorting algorithms
Talk at UCL/KCL Junior Geometry Seminar
Here are the slides for a talk I am giving at the UCL/KCL Junior Geometry Seminar on March 4th 2024.
Oxford go tournament 2024
Last weekend, the Oxford go tournament 2024 was held again for the first time after the pandemic. In total 54 players were present, some of which coming from as far away as Nottingham! In attendance were some of the best players in the UK, as well as almost complete beginners. Harry put in a lot of work to organise a side 9×9 tournament that was particularly popular with the youngsters. One of them played 39 (!) games in addition to their 3 tournament (19×19) games.
Thanks to generous sponsorship we were able to hand out quite a few prizes such as vouchers for online courses or Go books by a small German publishing company (Brett&Stein, you should check them out). We were able to bring out the customary Oxford Go club scroll as decoration, and they were much appreciated.
While we had lovely weather on the day, mother nature still prompted us to stay inside, for outside the rain was not quite gone yet.
Organising the tournament was quite a bit of work, but I hope that the work that Harry and I did this year can translate into a more streamlined process next year. That should leave more time to make sure that the clocks are all set correctly (whoops) and that the printer is working properly when it is needed (likely no amount of preparation is going to fix this, did you know that HP need to be connected to the internet EVEN if you just want to use them with a USB cable?). At least we left some room for improvement! See you all next year.
Deformation theory of Cayley submanifolds
I just published my first paper on the arxive here. It’s about the deformation theory of conically singular and asymptotically conical Cayley manifolds, based on the work of McLean and Moore. I prove the usual theorems about the existence of Kuranishi charts and discuss some relations to lower-dimensional geometries, such as special Lagrangian, complex and coassociative geometries. For example, special Lagrangians in Calabi-Yau fourfolds will always be obstructed as Cayleys. It is the first in a series of three papers working towards constructing fibrations of compact Spin(7) manifolds by Cayley submanifolds. The next paper which is about the desingularisation of Cayley submanifolds will follow shortly.
Weak Cayley fibrations
Slides for a talk at the Geometry and Analysis seminar in Oxford can be found here.
Instanton Floer homology
Floer homology is in essence the extension of the usual Morse homology of closed finite-dimensional manifolds to certain infinite-dimensional situations, where the naive Morse index of a critical point is not necessarily finite, and compactness becomes a subtle issue. In these notes we focus on the instanton Floer homology of a homology three-sphere , which is the Morse homology of the Chern-Simons functional . Defined on the infinite dimensional space of gauge equivalence classes of -connections on and being circle-valued, this functional brings in further features not present in classical Morse homology. For one, due to the presence of reducible connections, the gauge equivalence classes of connections only form a manifold away from a collection of singular points. Furthermore, since is circle-valued it can occur that gradient flow lines form loops, making the trajectory spaces more subtle to describe.
In order to address the difficulties arising from the definition of the Morse index, we will introduce the spectral flow of a family of self-adjoint operators between separable Hilbert spaces, a tool common to all Floer-type theories. Using it, we will be able to define a relative index between two critical points of . This index will only be -valued however, due to looping gradient flow lines. To circumvent the issues arising from reducible connections, we will restrict ourselves to homology three-spheres, for which achieving transversality is easier than for other classes of three-manifolds. A feature of which makes it particularly suited for Morse homology is that its critical points and gradient trajectories have geometrical interpretations. The critical points of are flat connections on , and the flow lines are instantons over the Riemannian tube which join two connections. As we will see, instantons are a special case of Yang-Mills connections, which allows us to use Uhlenbeck’s compactness theorem to derive the compactness up to broken trajectories of the trajectory spaces. The invariant we will obtain in the end takes the form of a -graded vector space over the field . For the usual three-sphere and the Poincaré homology sphere they are given by:
Thus the Floer homology groups will be unrelated to the usual homology groups, and provide a stronger invariant. To conclude, we will investigate further properties of the Floer groups, such as their relation to the representation-theoretic Casson invariant, which appears as the Euler characteristic of the Floer groups, and -dimensional topological quantum field theories.
We will move according to the following outline. In the first chapter, we introduce electromagnetism as an example of an abelian gauge theory, familiarise ourselves with the concepts we will later encounter in the non-abelian setting and provide a link between gauge theory and topology of three-manifolds via Hodge theory. In the second chapter, we will then recall principal bundles properly, fix the notation, and derive the relation between three- and four-dimensional -gauge theory. Next, we will introduce the Chern-Simons and Yang-Mills functionals and in the third chapter and analyse the relation between -gradient flow lines and instantons, which are the local minimizers of . We will also investigate the local picture around a critical point of these functionals. In the last chapter we will construct the trajectory spaces and the Floer homology groups from a chain complex generated by flat connections, emphasising the Fredholm analysis. We will show independence of auxiliary data and of the perturbation chosen, where it will be important to restrict to the case of homology three-spheres.
We assume that the reader is familiar with differential geometry (bundles and connections, Riemannian geometry, de Rham cohomology, Morse homology), algebraic topology (especially computational tools in (co)homology theories, Poincaré duality, intersection theory, some homotopy theory), as well as functional analysis (Sobolev spaces on open domains, differential operators on ).
These notes do not contain original work, but merely adapt and combine the literature on the subject, while supplementing explanations where deemed useful. We mostly follow Donaldson’s monograph on the topic, and apply the treatment of Morgan’s lecture notes to give the analytical foundations for the Chern-Simons and Yang-Mills functionals. The first chapter on electromagnetism reshuffles the first part of the excellent lecture notes by Evans on the Yang-Mills equations, and the appendix bundles statements from multiple sources. For references see the PDF.
I would like to thank my supervisor Prof. Will Merry for guiding me through this endeavour with frequent and useful meetings. His (virtual) office door was always open to me, and I appreciated both his technical knowledge in geometry and analysis and his expertise in the wider world of Floer theories, which often provided helpful analogies an additional angles of attack during my study of the instanton flavoured theory.
The entire thesis can be found here.
The Model of Solovay
The objective of measure theory is to make rigorous the intuitive notions of length, surface area, volume, and measurement of quantity. A mathematical model of all of these can be found in the concept of measure on a set , which in its simplest form is a function such that the following hold:
- Monotonicity:
- -additivity:
- Normalization: and
- Atomlessness:
On the one hand, this notion replicates a number of facts about volume in the physical world. For instance, the first condition can be rephrased as “Bigger objects have bigger volume” and the second one translates to “The combined volume of a collection of disjoint objects is the sum of their respective volumes”. On the other hand, we can interpret a measure as an order-preserving map between partially ordered sets. Here the set is ordered by inclusion, and is ordered by . From this perspective, (i) just means that is order-preserving, (ii) restricts the value of on certain joins, and (iii) and (iv) are normalization conditions. In analogy of the representation theory of groups, the structure of as a poset is compared trough to the poset structure of . This interrelates the two structures and allows to analyze the structure of either one.With this definition in mind, there are a few questions that arise naturally. From the practical point of view, me might ask:
- Does euclidean space admit a measure that is translation invariant and hence reflects our experience of volume?
From the set theoretic point of view, one might ask:
- Which sets admit a measure? In particular, since admits a measure exactly when does, which cardinals admit measures? This question is known as the Measure Problem.
As we will show in this semester project, the answer to the first question is no under . This has peculiar consequences such as the possibility to decompose a solid ball into a small number of pieces, rearranging them, and ending up with two exact copies of the initial ball. This mathematical possibility of copying a ball without adding any mass is known as the Banach-Tarski paradox, and would be impossible if a measure existed on all of , for then mass would have to be conserved.
From here, there are essentially two possible directions to proceed. Either one can weaken the requirement that the measure be translation invariant, in which case there may exist a solution, or one can weaken , which is the route we are taking. Robert M. Solovay showed in 1970 that there is a model of set theory in which every set of reals is Lebesgue measurable and where a such a weakening of , the axiom of dependent choice , still holds.
The main goal of this thesis is to construct this model and to discuss how the structure of the real line relates to strong assumptions, such as further axioms of set theory or large cardinal hypotheses.In the first chapter, we introduce essential concepts and tools from descriptive set theory, such as a representation of the reals used in set theory, the Borel sets and alongside Lebesgue measurability two other regularity properties of the reals, the property of Baire and the perfect set property. The second chapter constructs the Model of Solovay and proves that any subset of the real line in it is Lebesgue measurable, has the Property of Baire and the perfect set property. The proof requires the existence of an inaccessible cardinal to define the Lévy collapse, which is a forcing notion that allows to overwrite any bad behaviour the reals might display. With this a possible solution to the measure problem is presented if is not assumed.In the last chapter we present the work of Shelah and Specker, illuminating when the assumption of the inaccessible cardinal is really needed, and in which cases it can be avoided. Finally, we will compare different alternatives to and present their impact on the regularity of the reals.We generally follow the books by Jech and Kanamori.
The rest of this semester project can be found here.
Morse Homology
Example of Morse functions on
Let us assign each critical point a number according to its type, called its :Then we can consider the sum:
If we calculate , we will find that in both cases its equal to . This turns out not to be a coincidence, since no matter which Morse functions we choose on , this number remains . If we consider the -torus , i.e. a donut, and a Morse function on it, such as the one on the cover, we see that
Again, changing the Morse function will not change . Now one can play this game also for genus surfaces and sees that:
Hence, by the classification of -manifolds (which interestingly can also be proven using Morse theory), we see that for any Riemann surface , we have:
(1)
where is the Euler characteristic of . This is again no coincidence! Let us rewrite the formula for as follows:(2)
This compares nicely to a formula from aglebraic topology. Let be a cell complex structure on . Then we have:It turns out that under certain conditions a Morse function will give rise to a cell complex structure with exactly cells of dimension . If we for assume this result for a moment, we immediately get:
for manifolds which admit these sufficiently regular Morse functions. This means that we can express such as the Euler characteristic solely of height functions. In fact, a single Morse function, by giving rise to a cell complex structure, suffices in principle to classify manifolds up to homeomorphism type. What we will look at is the so called of a manifold, which is in fact just the homology of a cell complex obtained by a Morse function, from the classical point of view. But that is not how we are going to define it, as we will take a more abstract modern route. We will look at the so called associated to a manifold with a Morse function on it. The -th chain groups will be free -modules generated by the critical points of index , and the boundary operator will be defined by counting solutions to a differential equation. Both approaches lead to the same result, however modern Morse homology has several advantages over classical Morse homology. One of them is that it lends itself better to our axiomatic treatment for showing its equivalence to singular homology. We will prove that Morse homology satisfies a number of axioms, called the . Then we will invoke a uniqueness theorem that tells us that any homology theory satisfying the ES-axioms has to be isomorphic to singular homology. Let us now take a quick look at the structure of this thesis. The first chapter will focus on defining the Morse homology groups, and in particular show that it is well defined. We will extend our notion of index to general manifolds, and make concrete the expression “counting solutions to equations” with the so called trajectory spaces. Then the second chapter will deal with extending the groups to functors and show that the ES-axioms are satisfied, so that we can conclude via the uniqueness theorem that Morse homology is in fact isomorphic to singular homology. Finally, in the last chapter we will explores a number of theorems from algebraic topology in the spirit of Morse homology. We will cover Morse cohomology and Poincaré duality, which turns out is almost trivial to prove in Morse homology. To conclude, we will give a quick proof of the existence of an Eilenberg-Zilber chain map in by exploiting an additivity property of Morse functions. The rest of the thesis can be found here.