Monday, 30 August 2021

Watch our student lectures

As the new academic year approaches, we're adding to our catalogue of Oxford Mathematics student lectures on our YouTube Channel.

The latest is a lecture from Vicky Neale (pictured) on Monotonic Sequences, part of her first year Analysis 1 course. There are 50 more lectures for you to watch on the Channel covering many aspects of the undergraduate degree, including two full courses. We will add more over the coming weeks, including more lectures from the third and fourth years when students get to specialise.

All first and second year lectures are followed by tutorials where students meet their tutor to go through the lecture and associated problem sheet and to talk and think more about the maths. Third and fourth year lectures are followed by classes.


Tuesday, 24 August 2021

Frances Kirwan awarded the Royal Society's Sylvester Medal for 2021

Oxford Mathematician Frances Kirwan has been awarded the Sylvester Medal 2021 "for her research on quotients in algebraic geometry, including links with symplectic geometry and topology, which has had many applications."

The Sylvester Medal is awarded annually for an outstanding researcher in the field of mathematics. The award was created in memory of the mathematician James Joseph Sylvester who was Savilian Professor of Geometry at the University of Oxford in the 1880s, a post now held by Frances Kirwan (the Savilian celebrated its 400th anniversary in 2019). The Sylvester medal was first awarded in 1901. It is of bronze and is accompanied by a gift of £2,000.

Frances's specialisation is algebraic and symplectic geometry, notably moduli spaces in algebraic geometry, geometric invariant theory (GIT), and the link between GIT and moment maps in symplectic geometry.

Frances has received many honours including being elected a Fellow of the Royal Society in 2001 (only the third female mathematician to attain this honour), and President of the London Mathematical Society from 2003-2005. She was made a Dame Commander of the Order of the British Empire in 2014.

On receiving the award Frances said: "I am honoured to receive this award, especially as it is named after one of my predecessors as Savilian Professor, James Joseph Sylvester, whose work over a hundred years ago on what is today called invariant theory laid the foundations for my own work on geometric invariant theory."

Wednesday, 18 August 2021

Symplectic duality - a research case study from Andrew Dancer

One of the main themes of geometry in recent years has been the appearance of unexpected dualities between different geometric spaces arising from ideas in mathematical physics. One famous such example is mirror symmetry. Another kind of duality, which I am currently investigating with collaborators from Oxford and Imperial College, is symplectic duality.

The spaces here are holomorphic symplectic - so they have a complex structure and a symplectic 2-form which is holomorphic for that complex stucture. In practice there is usually some additional data, a compatible Riemannian metric, that describes a hyperkahler structure. The duality is between different holomorphic symplectic manifolds that arise as the Higgs and Coulomb branches of a physical theory with $N=4$ supersymmetry. One interesting feature is that this duality can change dimensions - the Higgs and Coulomb branches can be of different dimensions.

Holomorphic symplectic spaces often arise via a kind of quotient construction called the hyperkahler quotient, due to Nigel Hitchin and his collaborators Karlhede, Lindstrom and Rocek. The idea is that we start from a hyperkahler space with a symmetry and produce a new one in which the symmetry has been factored out - this is not simply the quotient of the original space, but rather the quotient of a subvariety defined by the vanishing of the so-called momentum map, a concept whose roots lie in Hamiltonian mechanics. The beauty of the construction is that the original space can be very simple, even flat space, but the quotient can be highly nontrivial.

If the symmetry group by which we quotient is an Abelian torus, this leads us to the objects called hypertoric varieties. Like their classical toric variety cousins, their study reduces to combinatorial convex geometry. Symplectic duality is well understood for such spaces, and is in fact a manifestation of a classical concept, Gale duality, that has long been known to convex geometers and linear programmers.

If the group is non-Abelian, the situation is less clear. It is believed that if the Higgs branch is a hyperkahler quotient of a linear space by a group $K$, then the Coulomb branch will be birational to the quotient by the Weyl group of the cotangent bundle of the complexified dual maximal torus of $K$. Physically, the birational transformations can be interpreted as quantum corrections to the classical picture of the Coulomb branch.

Our current work focuses on finding candidates for the symplectic duals of a class of spaces called universal hyperkahler implosions. Extensive numerical checks can be performed using a formula developed by string theorists called the monopole formula. In the case of special unitary and orthogonal groups, we have produced candidate spaces that pass all such numerical tests.

Andrew Dancer is a Professor of Mathematics in Oxford and Tutorial Fellow in Pure Mathematics at Jesus College.

Further reading:
Antoine Bourget, Andrew Dancer, Julius F. Grimminger, Amihay Hanany, Frances Kirwan and Zhenghao Zhong - Orthosymplectic Implosions

Andrew Dancer, Amihay Hannay, Frances Kirwan - Symplectic duality and implosions

Monday, 16 August 2021

Teaching accreditation for Andrew Krause and Vicky Neale

Many of the news stories on our website are about our research, and rightly so as it is an integral part of the academic life. But alongside research, academics are employed to teach, and to teach to the highest standard, providing a pipeline of graduates with the skills needed across the many sectors where they will work.

Oxford's commitment to a professional teaching body is reflected in our new Teaching Recognition Scheme. This is an Oxford-based scheme through which individuals will be able to make a claim for recognition as a Senior Fellow of the Higher Education Academy (SFHEA). In addition, those who completed the Postgraduate Certificate in Teaching and Learning in Higher Education in the first two cohorts have recently had the opportunity to go on to apply for Fellowship of the Higher Education Academy (FHEA).

Two members of the Mathematical Institute, Andrew Krause (who starts at Durham in September) and Vicky Neale, have recently received teaching accreditation through the University's Teaching Recognition Scheme, having both completed the Postgraduate Certificate in Teaching and Learning in Higher Education through the University in the last couple of years. 

Andrew, a Departmental Lecturer in Applied Mathematics, has been awarded Fellowship of the Higher Education Academy showing that he "demonstrates a personal and institutional commitment to professionalism in learning and teaching in higher education" and is "able to engage with a broad understanding of effective approaches to learning and teaching support as a key contribution to high quality student learning."

Vicky, who is the Mathematical Institute's Faculty Teaching Advisor, was awarded Senior Fellowship of the Higher Education Academy for which the criteria are the same as those for Fellowship, with the addition of "successful coordination, support, supervision, management and/or mentoring of others (whether individuals and/or teams) in relation to learning and teaching."

Friday, 13 August 2021

Quantum invariants via the topology of configuration spaces - Cristina Anghel

Knot theory studies embeddings of the circle into the three dimensional space and the first knot invariant was the Alexander polynomial. The world of quantum invariants started with the milestone discovery of the Jones polynomial and was expanded by Reshetikhin and Turaev’s algebraic construction which starts from a quantum group and leads to link invariants. This method gives two important sequences of quantum invariants: coloured Jones polynomials $\{J_N(L,q) \}_{N \in \mathbb N}$ and coloured Alexander polynomials $\{\Phi_N(L, \lambda)\}_{N \in \mathbb N}$, which recover the original Jones and Alexander polynomials at the first terms.

Questions: Topological information encoded by quantum invariants
Up to this moment, the original Jones polynomial and Alexander polynomial have different natures. More precisely, the Alexander polynomial is well-understood in terms of the complement of the knot. However, the connection between the Jones polynomial and the topology of the knot complement is still a deep and mysterious question. Also, predictions from physics state that the limit of coloured Jones polynomials contain topological information of the complement such as the simplicial volume of the complement (Volume Conjecture).

A very powerful procedure called categorification starts with a polynomial invariant and aims to produce a sequence of groups which encode more information. Khovanov constructed a combinatorial categorification for the Jones polynomial and Ozsvath-Szabo and Rasmussen defined geometrical categorifications for the Alexander polynomial, called knot Floer homology. Later on, Seidel and Smith constructed a Floer type model for Khovanov's categorification. Dowlin showed recently that there is a spectral sequence between these two categorifications, but there are still questions about the geometry of such categorifications and spectral sequences.

On the homological side, Lawrence [5] introduced a sequence of representations $H_{n,m}$ of the braid group $B_n$ from the homology of a $\mathbb Z \oplus \mathbb Z$-covering of the configuration space of $m$ points in the $2n$-punctured disc. Related to this subject, Kohno [4], Ito and Martel worked on identifications between quantum and homological representations of braid groups. Based on Lawrence's work Bigelow [3] gave a topological model for the original Jones polynomial, as an intersection pairing between homology classes in coverings of configuration spaces. We call such a description a topological model.

Research program
I am working on a program aiming to obtain topological models for quantum invariants via intersections of Lagrangians in configuration spaces. This creates a new topological framework permitting one to read off topological information encoded by these invariants and having a goal of achieving geometrical categorifications.

Topological models for $U_q(sl(2))$-quantum invariants
My main results [1], [2] show a unified topological model for the $N^{th}$ coloured Jones and $N^{th}$ coloured Alexander invariants. We define a state sum of graded intersections between Lagrangians in a configuration space in the punctured disk (over $3$ variables) recovering the $N^{th}$ coloured Jones and the $N^{th}$ coloured Alexander polynomial. We will use:
1) Lawrence representations $H_{n,m}$ and $H^{\partial}_{n,m}$ which are $\mathbb Z[x^{\pm 1},d^{\pm 1}]$-modules which carry a $B_n$-action 
2) Intersection pairing $\langle , \rangle: H_{n,m} \otimes H^{\partial}_{n,m}\rightarrow \mathbb Z[x^{\pm},d^{\pm}]$ Poincaré-Lefschetz type duality           

This pairing is encoded by the intersections in the base configuration space, graded by the local system.

Homology classes
The construction of the Lagrangians is done by drawing certain curves in the punctured disc, taking their product and quotienting it in the unordered configuration space. Using this, for any indices $i_1,...,i_{n-1} \in \{0,...,N-1\}$ we define two Lagrangians and consider two classes given by their lifts:

                                                                                                            Theorem (Unified model through a state sum of Lagrangian intersections [2]) Let $L$ be an oriented link and $\beta_n \in B_n$ such that $L=\hat{\beta}_n$ (as braid closure). We consider the following state sum of Lagrangian intersections $\Lambda_N(\beta_n)(u,x,d) \in \mathbb Z[u^{\pm1},x^{\pm 1},d^{\pm 1}]$: $$\Lambda_N(\beta_n)(u,x,d):=u^{-w(\beta_n)} u^{-(n-1)} \sum_{i_1,...,i_{n-1}=0}^{N-1} \langle(\beta_{n} \cup {\mathbb I}_{n} ){ \mathscr F_{i_1,...,i_{n-1}}}, { \mathscr L_{i_1,...,i_{n-1}}}\rangle.$$Then, $\Lambda_N$ recovers the $N^{th}$ coloured Jones and $N^{th}$ coloured Alexander invariants: \begin{equation} \begin{aligned} &J_N(L,q)=\Lambda_N(\beta_n)|_{\psi_{1,q,N-1}}\\ &\Phi_{N}(L,\lambda)=\Lambda_N(\beta_n)|_{\psi_{1-N,\xi_N,\lambda}}. \end{aligned} \end{equation} (here $w(\beta_n)$ is the writhe of the braid, $\psi_{c,q,\lambda}$ a specialisation of the coefficients and $\xi_N=e^{\frac{2\pi i}{2N}}$)

We can read off the $N^{th}$ coloured Jones polynomial and the $N^{th}$ coloured Alexander polynomial from the same geometric picture, given by the graded intersection between the above Lagrangians, in the configuration space of $(n-1)(N-1)+1$ particles in the $2n$-punctured disk.


                                                                        Figure 2: Jones and Alexander polynomials of the trefoil knot

Framework for categorifications of quantum invariants
This explicit model sets up a framework for investigating categorification problems for the whole families of coloured Alexander polynomials and coloured Jones polynomials. Moreover, it provides a tool for understanding possible relations between them. 


We think that the state sum $\Lambda_2(\beta_n)$ (for the case $N=2$) leads to knot Floer homology of $L$ (specialised through $\psi_{-1,\xi_2,\lambda}$). We expect that a cousin of this state sum (obtained from immersed Lagrangians given by figure eights as in [2] $\Omega_2(\beta_n)$ leads to a geometrical categorification for the Jones polynomial. Moreover, we expect that these two results will lead to a spectral sequence between categorifications of the Jones polynomial and knot Floer homology.

Cristina Anghel is a Postdoctoral Research Associate in Oxford Mathematics. You can watch a short film introducing her work here.

[1] C. Anghel - Coloured Jones and Alexander polynomials as topological intersections of cycles in configuration spaces, math.GT arXiv:2002.09390, 47 pages, (2020).

[2] C. Anghel - $U_q(sl(2))$−quantum invariants unified via intersections of embedded Lagrangians, math.GT arxiv: 2010.05890, 24 pages, (2020).

[3] S. Bigelow - A homological definition of the Jones polynomial, Invariants of knots and 3-manifolds Kyoto, 2001, Geom. Topol. Monogr. 4, 29-41, (2002).

[4] T.Kohno - Monodromy representations of braid groups and Yang–Baxter equations, Ann. Inst. Fourier (Grenoble) 37, 139–160 (1987).

[5] R. J. Lawrence - A functorial approach to the one-variable Jones polynomial, J. Differential Geom., 37(3):689–710, (1993).


Thursday, 5 August 2021

Generalised Lie algebras and their applications - Lukas Brantner

Oxford Mathematician Lukas Brantner explains how generalised Lie algebras lead to new insights in Galois theory, deformation theory, and the theory of configuration spaces. Lukas has just been awarded a Royal Society University Research Fellowship. These long term fellowships provide outstanding scientists with the opportunity to build independent research careers and give them the freedom to pursue innovative and transformative scientific research.

"Many celebrated developments in algebraic geometry and topology have progressed in two steps. First, a question is resolved in characteristic zero, e.g. over the real or complex numbers. There, our geometric intuition is sound and the tools of calculus, like exponentials and integrals, are at our disposal. Secondly, new ideas and techniques are introduced to prove an analogue in characteristic $p$ - e.g. over the finite field $\mathbb{F}_p$. This regime is significantly more subtle, because the tools of analysis break down and we cannot divide by $p$, and so expressions like $e^x = \sum_n \frac{x^n}{n!}$ do not make sense.

For an example in algebraic geometry, let $X\subset \mathbb{P}^n(\mathbb{C})$ be a smooth complex manifold defined as the vanishing locus $\left\{f_i(x_0, \ldots, x_n) = 0 \right\}$ of homogeneous polynomials $f_1,\ldots, f_k$. The $i^{th}$ Betti number $b^{\mathbb{C}}_i(X) = \dim_\mathbb{C}(\mathrm{H}^i(X,\mathbb{C}))$ is a good measure for the number of $i$-dimensional holes in $X$; however, some holes can be detected only by $\mathbb{F}_p$-Betti numbers $b_i^{\mathbb{F}_p} = \dim_{\mathbb{F}_p}(\mathrm{H}^i(X,\mathbb{F}_p))$ in characteristic $p$. We can compute the usual Betti numbers $b_i^{\mathbb{C}}(X)$ from the polynomials $f_1, \ldots f_k$ using the so-called de Rham complex, by integrating differential forms along cycles. Recent advances in $p$-adic Hodge theory by Bhatt-Morrow-Scholze show that we can even bound the $\mathbb{F}_p$-Betti numbers $b_i^{\mathbb{F}_p}(X)$ using certain de Rham cohomology groups, provided $X$ is defined over $\mathbb{Q}$ and has good reduction at $p$.

My recent work has followed the paradigm of passing from characteristic $0$ to characteristic $p$ with respect to the following three goals:
(1) Set up a Galois correspondence for general field extensions;
(2) Classify infinitesimal deformations of algebro-geometric objects;
(3) Compute invariants of unordered configuration spaces.

A key ingredient is the theory of partition Lie algebras, a new algebraic structure I have discovered while studying the poset of partitions of $\{1,\ldots n\}$ with its natural action by the symmetric group $\Sigma_n$.

Very informally, partition Lie algebras are generalisations of classical Lie algebras where antisymmetry and the Jacobi identity need not hold on the nose, but only up to coherent homotopy. This means that $[x,[y,z]]$ and $ -[y,[z,x]] - [z,[x,y]]$ need not be equal, but there is a path from the former to the latter. 

Formally, partition Lie algebras can either be defined $\infty$-categorically using Goodwillie calculus, which is often helpful in conceptual arguments (cf [BF] [BW]) or $1$-categorically as chain complexes equipped with many operations (cf [BCN]), which has the advantage of being more explicit.

Generalised Lie algebras have several intriguing applications:

(1) Galois Theory. Given a polynomial $f(x)$ with coefficients in a field $K$ of characteristic zero, we can construct a new field $F$ by adding all roots of $f(x)$ to $K$. The Fundamental Theorem of Galois Theory then establishes a one-to-one correspondence between intermediate fields $K\subset E \subset F$ and subgroups $H\subset \mathrm{Gal}(F/K)$ of the group of automorphisms of $F$ fixing $K$. For example, $f(x)= x^4-4x^2+2$ has four real roots $\pm \sqrt{2\pm \sqrt{2}}$, and adding them to $\mathbb{Q}$ gives a new field $F$ for which $\mathrm{Gal}(F/\mathbb{Q})\cong \mathbb{Z}/4\mathbb{Z} $; the subgroup $2\mathbb{Z}/4\mathbb{Z}$ corresponds to the intermediate field $\mathbb{Q}(\sqrt{2})$.

If $f(x)$ has coefficients in a field $K$ of characteristic $p$, the above Galois correspondence often breaks down. For instance, taking the field of rational functions $K=\mathbb{F}_p(T)$ and adjoining a $p^{n}$-th root of $T$, we obtain a nontrivial extension $F/K$ for which the classical Galois group $\mathrm{Gal}(F/K)$ vanishes. This is an example of a purely inseparable extension, meaning that for any $x\in F$, some high power $x^{p^n}$ belongs to $K$. In [BW], Waldron and I generalise the Fundamental Theorem of Galois Theory to the purely inseparable setting, by establishing a correspondence between intermediate fields $K \subset E \subset F$ and subalgebras of the Galois partition Lie algebra $\mathfrak{gal}_{F/K}$ satisfying three simple homological conditions. This generalises an old result of Jacobson from 1944 on extensions $F/K$ satisfying $x^p \in K$ for all $x\in F$.    


                                                                   Figure 1 - A (non-modular) purely inseparable field extension.

(2) Deformation Theory. Given a smooth projective variety $X$ over the complex numbers, we can study the infinitesimal deformations obtained by varying the defining equations of $X$.


                                                            Figure 2 - A sequence of infinitesimal deformations of higher and higher order.

First order deformations of $X$ are classified by the first cohomology group $\mathrm{H}^1(X, T_X)$ of its tangent bundle $T_X$. To formulate a criterion for when first order deformations extend to higher order, we can equip the cohomology $\mathrm{H}^\ast(X,T_X)$ with a Lie bracket, by using the commutator bracket on vector fields. A first order deformation $\widetilde{X} \rightarrow \mathrm{Spec}(\mathbb{C}[\epsilon]/\epsilon^2)$ then extends to higher order precisely if the corresponding class $x\in \mathrm{H}^1(X, T_X)$ satisfies $[x,x] = 0$.

In fact, we can lift the Lie bracket to the so-called Dolbeault complex $$C^\ast(X,{T}_X) = (\ \mathcal{A}^{0,0}({T}_X) \rightarrow \mathcal{A}^{0,1}({T}_X) \rightarrow \mathcal{A}^{0,2}({T}_X) \rightarrow \ \ldots \ ),$$ the homology of which recovers $\mathrm{H}^\ast(X,{T}_X)$. One miracle in Kodaira-Spencer theory is that we can recover all infinitesimal deformations of $X$ from the resulting differential graded Lie algebra, by computing solutions to the Maurer-Cartan equation.

This illustrates a general principle in algebraic geometry, which was discovered by Deligne, Drinfel'd, and Feigin, and later formalised by Lurie and Pridham: infinitesimal deformations of algebro-geometric objects in characteristic zero are controlled by differential graded Lie algebras.

In [BM], Mathew and I generalise this result to characteristic $p$ and beyond, using partition Lie algebras. Infinitesimal deformations in this setting are of particular interest in arithmetic geometry and number theory, for example in the study of deformations of Galois representations. Relying on my work on partition complexes with Arone [AB], we also determine the natural operations acting on the homotopy of partition Lie algebras.

(3) Configuration Spaces. Configuration spaces govern several central objects in mathematics and physics, ranging from loop spaces in topology to Hurwitz spaces in geometry and phase spaces in mechanics. The ordered configuration space of $k$ non-colliding particles in a manifold $M$ is given by $$\mathrm{Conf}_k(M)=\{(x_1,\ldots,x_k) \in M^{k} \ | \ x_i \neq x_j \mbox{ for all } i \neq j\},$$ and the corresponding unordered configuration space is given by the quotient $B_k(M) =\mathrm{Conf}_k(M)/\Sigma_k$.

For example, noting that the location of two non-colliding particles in $\mathbb{R}^3$ is determined by their centre, their distance, and the direction of the line joining them, we find that $\mathrm{Conf}_2(\mathbb{R}^3) = \mathbb{R}^3 \times \mathbb{R}_{>0} \times S^2 \simeq S^2$. Swapping the particles reverses the direction, and so $B_k(\mathbb{R}^3) \simeq S^2/\Sigma_2 \simeq \mathbb{R} P^2$ is the real projective plane.

As we can see in this example, the spaces $B_k(M)$ can be significantly more complicated than the base manifold $M$, and many of their topological properties remain unknown, even in the case of surfaces. The rational cohomology groups $\mathrm{H}^\ast(B_k(M),\mathbb{Q})$, however, are often well-understood.

In my recent paper with Hahn and Knudsen [BHK], we use generalised Lie algebras to access the $\mathbb{F}_p$-cohomology, $K$-theory, and Lubin-Tate theory of configuration spaces, thereby computing several new invariants."


                                                                                 Figure 3: A configuration of points on a surface

Lukas introduces his work in this short film.

[AB] Gregory Arone and Lukas Brantner. The Action of Young Subgroups on the Partition Complex. Publ. Math. Inst. Hautes Etudes Sci., 33:47–156, 2021.

[BCN] Lukas Brantner, Ricardo Campos, and Joost Nuiten. PD Operads and Explicit Partition Lie Algebras. arXiv preprint arXiv:2104.03870, 2021.

[BHK] Lukas Brantner, Jeremy Hahn, and Ben Knudsen. The Lubin–Tate Theory of Configuration Spaces: I. arXiv preprint arXiv:1908.11321, 2020.

[BM] Lukas Brantner and Akhil Mathew. Deformation Theory and Partition Lie Algebras. arXiv preprint arXiv:1904.07352, 2019.

[BW] Lukas Brantner and Joe Waldron. Purely Inseparable Galois Theory I: The Fundamental Theorem., 2020.

Thursday, 29 July 2021

Population network structure of genetic algorithms

Oxford Mathematician Aymeric Vie, first year DPhil student at the Centre for Doctoral Training, Mathematics of Random Systems, describes his work on the population network structure of genetic algorithms.This work identifies new ways to improve the performance of those stochastic algorithms, and has received a Best Paper Award at the Genetic and Evolutionary Computation Conference 2021.

A Genetic Algorithm (GA) is a population-based search algorithm invented in the 70s by John Holland. Inspired by Darwin's theory of evolution, the GA simulates some of the evolution processes: selection, fitness, reproduction, crossover, mutation. Facing a given problem described by an objective function, the algorithm samples at random a population of initial solutions in a given search space. Individual solutions' fitness is measured so that most promising solutions are recombined together, while all can encounter mutations, i.e. applications of noise. These steps are iterated for a large number of generations. Being gradient free and capable of exploring very large or deceptive fitness landscapes, those methods have been quite popular in many domains.

When two solutions are recombined together, they have previously been selected for their fitness, i.e. their ability to solve a given problem instance. This is usually measured using a fitness function, other words for an objective or a reward function. In the standard GA thus, any pair of solutions can be recombined together to give birth to new, potentially better solutions. In network science terms, this is equivalent to assuming that there exists a complete network in the population of solutions, in which everyone is connected to everyone.

However, in nature as in social systems, such complete networks are actually quite rare. Individuals in these contexts usually interact with a tiny subset of the whole population, and these interactions are often not random, but characterised by locations, or similarity: we are more likely to interact with people that are geographically, or on other characteristics, similar to us.

The standard GA, and in fact, most other evolutionary algorithms, carry this implicit assumption of the complete population network. I wondered what would happen to the performance of the GA as an optimisation tool if we were performing this evolution of optimal solutions under alternative network structures. The article explored two main network types: Erdos-Renyi and Albert-Barabasi networks, that both give rise to the high variety of structures shown in Figure 1.                                                               


Figure 1: Varied structures obtained from Erdos Renyi (ER) and Albert-Barabasi (AB) networks. ER networks are described by a probability $p$ of existence of a link between a pair of nodes, showing the empty (top left) and complete (top right) networks. AB networks simulate network growth with preferential attachment (mechanism of the type highly cited papers are more likely to be cited by new incoming papers) strength $m$, giving tree networks (bottom left) or even star networks (bottom right).

The resulting Networked Genetic Algorithm (NGA) constrains recombination between individuals as a function of a network structure drawn once at the beginning of search. To identify the impact of network structure on GA performance, I used the networked GA with various network structures to solve instances of minimisation problems. The optimisation performance of the algorithms was evaluated measuring how close and how fast the solutions evolved by the program matched the known global optimum for the optimisation tasks.

Three standard functions in optimisation were used: the Sphere function, a very smooth, convex, easy to optimise function; and the Rastrigin and Ackley functions, that display many local optima, and very rugged structures that make them difficult for randomised search heuristics. Figure 2 plots the fitness landscape of those functions.


Figure 2: Fitness landscapes of the Rastrigin, Sphere and Ackley functions in two dimensions, for commonly used intervals for the arguments.

The results were richer than expected. They showed a clear impact of network structure over optimisation performance. Networks that were disconnected, i.e., that had sets of nodes not connected to other sets, were struggling to maximise the functions. Networks that were not dense enough, or too centralised, failed similarly. The best results were obtained for networks with intermediate levels of density and low average shortest path length, i.e., an average low number of nodes to cross to go from any node to any other. Those seemingly optimal networks resemble the modular networks frequent in biology.

An unexpected and much welcome finding was that the standard complete GA was actually not the optimal structure. For all the structures of ER and AB networks, the best NGA structure displayed a performance superior by more than 50% to the standard complete network GA.

Assessing further the robustness and generalisability of those findings could place population network structure optimisation as one tool to significantly improve the performance of evolutionary algorithms. The implications of this could be wide reaching, as such complete network structure is implicit in most randomised search heuristics used in Evolutionary Computation. This work also opens the possibility of self-adaptive network structures in evolutionary algorithms, a new paradigm in which the network structure of the algorithm could be continuously optimised during search to further improve performance.

Tuesday, 13 July 2021

Freedom with boundaries

Oxford Mathematician Connor Behan discusses the ways in which a free quantum field can be coupled to a spatial boundary. His recent work with Lorenzo di Pietro, Edoardo Lauria and Balt van Rees sheds light on this question using the non-perturbative bootstrap technique.

"In a material undergoing a second order phase transition, the critical behaviour is described by one of the scale invariant field theories. In the space of quantum field theories (which form the backbone of theoretical physics), these are distinguished points which are under better mathematical control. More precisely, exciting progress has been made in the study of conformal field theories (CFTs) where by conformal transformation, we mean any map which preserves angle. A simple example is a holomorphic function, and indeed, early work on CFTs focused on two dimensions where it is possible to exploit this deep connection.

Much of my research concerns the fact that CFTs in all dimensions posess an operator algebra - a product of two fields $\phi(x_1) \phi(x_2)$ can be written as a convergent sum of local operators $\mathcal{O}(x)$ at a third point. This allows CFTs to be characterized by a discrete set of numbers called the spins and scaling dimensions $(\ell_i, \Delta_i)$ which we are interested in constraining. To demonstrate a key technique, a correlation function with four copies of $\phi$ can be computed in different ways depending on which pairs are "fused'' together. This leads to the equation \begin{equation} \sum_{\Delta, \ell} c^2_{\Delta, \ell} G_{\Delta, \ell}(x_1, x_2, x_3, x_4) = \sum_{\Delta, \ell} c^2_{\Delta, \ell} G_{\Delta, \ell}(x_3, x_2, x_1, x_4), \quad\quad (1) \end{equation} where $G_{\Delta, \ell}$ is a known special function. Crucially, the coefficients are non-negative which means we can be ignorant of their precise values and still investigate the conditions for (1) to have a solution. With the help of convex optimization, this leads to rigorous bounds on $(\ell_i, \Delta_i)$ which must be obeyed by all valid theories.



This story becomes richer when we consider theories on a half space, thus allowing us to talk about a $d$ dimensional bulk and a $d - 1$ dimensional boundary. Boundary correlators obey the same CFT axioms as before but the bulk has new ingredients because some of the conformal symmetry there is broken. As a result, one can already see non-trivial correlation functions with two points. From these, a consistency condition of the form \begin{equation} \sum_{\hat{\Delta}} b^2_{\hat{\Delta}} F^\partial_{\hat{\Delta}}(x_1, x_2) = \sum_{\Delta} a_{\Delta} c_{\Delta} F^B_{\Delta}(x_1, x_2) \quad\quad (2) \end{equation} may be extracted. However, it only has manifestly non-negative coefficients on the left hand side. There have been many attempts to sidestep this problem but most of them require model dependent input or a small parameter in which to expand. Alternatively, if one restricted her attention to the boundary in order to continue working with four-point functions, the bounds obtained would be very weak. In particular, they would need to allow for strange phenomena in which energy seemingly disappears in one place and reappears in another. This process is only revealed to be a local one, where energy moves through the bulk, in the unrestricted system governed by (2).




In work that appeared in 2020, my collaborators and I developed a hybrid approach which combines both points of view. The first step is to pick a conformal field theory which is well understood on the whole space in $d$ dimensions. As a proof of concept, we picked the free massless scalar since it obeys the wave equation \begin{equation} (\Box_x + \partial_y^2) \phi = 0. \label{eom} \end{equation} The second step is to introduce a half space with respect to the $y$ variable, leading to \begin{equation} \phi(x, y) = b_1 \left [ \phi(x, 0) + \dots \right ] + \frac{b_2}{y} \left [ \partial_y \phi(x, 0) + \dots \right ]. \label{sol} \end{equation} We can then use model dependent analogues of (2) to search for exact relations. One of them says that $b_1$ and $b_2$ are fixed in terms of a single parameter in the interval $[-2^{2 - d}, +2^{2 - d}]$ which we call $a_{\phi^2}$. Another one says that at $y = 0$, the three-point functions $\left < \phi \phi \mathcal{O} \right >$, $\left < \phi \partial_y \phi \mathcal{O} \right >$ and $\left < \partial_y \phi \partial_y \phi \mathcal{O} \right >$ are all related to one another by a conformally covariant integral operator. This last statement is especially interesting since it can be phrased entirely in terms of boundary modes. Moreover, after some algebra, it can be put on the same footing as other constraints commonly used in convex optimization problems. This makes it safe to zoom in on the boundary for the last step.



Figure 1: allowed values for the dimension of a spin-2 boundary operator as a function of the bulk one-point function of $\phi^2$. Sub-optimal curves show how the bound converges as numerics are improved.

Our final results are produced by applying (1) to four-point functions on the whole space in $d - 1$ dimensions. This analysis benefits greatly from the exact relations which encode the effect of the bulk. One result which surprised us is an upper bound on the scaling dimension of the simplest spinning operator $\hat{\tau}^{\mu\nu}$ in $d = 4$. The value of 3 on the left hand side can be understood by thinking about the two simplest boundary conditions for the wave equation - the ones realized on either end of a wind instrument. These are Dirichlet and Neumann, which have $b_1 = 0$ and $b_2 = 0$ respectively. Conversely, other points on the curve (provided they are physical) must correspond to more exotic theories which have boundary localized interactions. In other words, the free massless scalar is coupled to additional degrees of freedom which live only on the boundary. Our work is about bootstrapping these boundary localized interactions because it is part of a long tradition of studying CFTs using nothing but the principles of CFT itself. In the past, it has proven fruitful to regard "kinks'' in the bounds as places where a physical theory is likely to sit. This is exactly what appears on the right hand side of the plot, where the dimension of $\hat{\tau}^{\mu\nu}$ jumps almost to 4. We are still having fun thinking about what this newly bootstrapped theory might be."


Friday, 2 July 2021

Oxford Mathematicians awarded LMS Prizes

Four Oxford Mathematicians have been awarded 2021 London Mathematical Society (LMS) Prizes. Ehud Hrushovski is awarded a Pólya Prize, Endre Süli is awarded the Naylor Prize and Lectureship, and Patrick Farrell and Stuart White receive Whitehead Prizes.

Udi Hrushovski's work is concerned with mapping the interactions and interpretations among different mathematical worlds. Endre's research is concerned with the analysis of numerical algorithms for the approximate solution of partial differential equations and the mathematical analysis of nonlinear partial differential equations in continuum mechanics. Patrick works on the numerical solution of partial differential equations, while Stuart's research focuses on operator algebras, a branch of functional analysis with connections to many other branches of pure mathematics.

Find out more about all the prize winners.

Sunday, 27 June 2021

Mfano Africa - Oxford Mathematics Virtual Mentorship Programme

When you watch the Olympics this Summer and admire the achievements of the medalists, ponder on the unknown number of people who may have done just as well if only they'd had the chance. Of course this is true of many fields of human endeavour. Take mathematics for instance. In a world of data overload, pandemics and cyber intrusion never have we needed all the mathematical talent we can muster. Yet many people are denied the opportunity, both at school and, as crucially, at research level.

This problem is especially acute in Sub-Saharan Africa which is where Mfano Africa comes in. Mfano Africa is a mentorship platform launched in August 2020 and which helps pre-PhD mathematics students to improve the quality of their postgraduate applications. And in June 2021, it has come (virtually) to Oxford via the Mfano Africa - Oxford Mathematics Virtual Mentorship Programme. This pilot aims to encourage a small cohort of 11 competitively recruited and promising pre-PhD students, based in Sub-Saharan Africa, who hope to continue PhD study within the mathematical sciences. 

However, the best person to explain the project is Geoffrey Mboya, Founder and Director of Mfano Africa. Geoffrey is a PhD student here in Oxford Mathematics and a student of St Peter's College, Oxford. Over to you Geoffrey.