Friday, 8 February 2019

Tensor clustering of breast cancer data for network construction - Oxford Mathematics Research

Oxford Mathematicians Anna Seigal, Heather Harrington, Mariano Beguerisse Diaz and colleagues talk about their work on trying to find cancer cell lines with similar responses by clustering them with structural constraints.

"Modern data analysis centres around the comparison of multiple different changing factors or variables. For example, we want to understand how different cells respond to different experimental conditions, under a range of doses, and for various different output measurements, across different timepoints. The structure in such data sets guide the design of new drugs in personalized medicine. 

A key way to find structure in data is by clustering: partitioning the data into subsets within which the data share some similarity. As multi-dimensional data sets become more prevalent, the question of how to cluster them becomes more important. Usual clustering algorithms can be used, but they do not conserve the multi-dimensional structure of the original data, and this flattens the insights that can be made, and hampers the interpretability of the results. 

In our paper, we introduce a method to cluster multi-dimensional data while respecting constraints on the composition of each cluster, designed to attribute differences between clusters to interpretable differences for the application at hand. In our method, a high similarity is not enough to cluster two data points together. We also require that their similarity is compatible with a shared explanation. We do this by placing algebraic constraints on the shapes of the clusters. This method allows for better control of spurious associations in the data than other approaches, by constraining the associations to only retain those with a consistent basis for similarity.

We apply our method on an extensive experimental dataset detailing the temporal phosphorylation response of signaling molecules in genetically diverse breast cancer cell lines in response to different ligands (or experimental conditions).  In this setting, we aim to find sets of experiments whose responses are similar, and to interpret these similarities in terms of the unknown underlying signaling mechanisms.  In our data set each experiment is given by a cell line and a ligand. One example of a mechanistic interpretation we could make from a similarity is that the cell lines in a cluster share a mutation, and the ligands are those whose effect is altered by the mutation. 

We constrain the clusters to be rectangular, i.e. to match a subset of cell lines with a subset of ligands.  The constraints only keep experimental measurements that are compatible with a mechanistic interpretation. This facilitates biological insights to be gleaned from the clusters obtained. In our paper, we present two variations of the algorithm:

1. The method can be applied directly to a dataset (i.e. as a standalone clustering tool).  Similarities between data points are encoded in a similarity tensor, the higher-dimensional analogue of a similarity matrix, and constraints about which clusters are allowed form the equations and inequalities in the entries of an unknown tensor, which encodes the clustering assignment.

2. The method can be applied in combination with other clustering methods, to impose constraints onto pre-existing clusters.  The distance between partitions is given by the number of experiments whose clustering assignment changes.  Hence this method can be used in conjunction with any other state-of-the-art method and preserve the features of an initial clustering that are compatible with the constraints.

In both implementations, the interpretability constraints can be encoded as algebraic inequalities in the entries of the clustering assignment, which gives an integer linear program. This can be solved to optimality, using the branch-and-bound algorithm, to find the best clustering assignment. 

Any other constraints that give linear inequalities can also be used. There are possibilities to apply the methodology to problems with other constraints, such as restricting on the size of cluster, imposing certain combinations of data, or finding communities in networks with quotas."

Wednesday, 6 February 2019

Sharp rates of energy decay for damped waves - Oxford Mathematics Research

Differential equations arising in physics and elsewhere often describe the evolution in time of quantities which also depend on other (typically spatial) variables. Well known examples of such evolution equations include the heat equation and the wave equation. A rigorous, functional analytic approach to the study of linear autonomous evolution equations begins by considering the associated abstract Cauchy problem, \begin{equation}\label{eq:ACP} \hspace{100pt}\left\{\begin{aligned} \dot{u}(t)&=Au(t),\quad t\ge0,\hspace{200pt} (1)\\ u(0)&=x\in X. \end{aligned}\right. \end{equation} Here $A$ is a linear operator (typically unbounded) acting on a suitably chosen Banach space $X$, which is usually a space of functions or a product of such spaces. For instance, in the case of the heat equation on a domain $\Omega$ we might choose $X=L^2(\Omega)$ and let $A$ be the Laplace operator with suitable boundary conditions, and for the wave equation we would take $X$ to be a product of two function spaces corresponding, respectively, to the displacement and the velocity of the wave. Assuming the abstract Cauchy problem to be well posed, there exists a family $(T(t))_{t\ge0}$ of bounded linear operators (a so-called $C_0$-semigroup) acting on (1) such that the solution of (1) is given by $u(t)=T(t)x$, $t\ge0$. Of course, we cannot normally hope to solve (1) exactly, so the operators $T(t)$, $t\ge0$, are in general unknown. The main task is to deduce useful information about the semigroup $(T(t))_{t\ge0}$ from what is known about $A$, in particular its spectral properties.

In concrete applications, the norm on the space $X$ often admits a physical interpretation. An important example of this kind is the wave equation, where $X$ is a Hilbert space with the property that the induced norm of the solution $u(t)=T(t)x$ is related in a very natural way to the energy of the solution at time $t\ge0$. Thus we may study energy decay of waves, a fundamental problem in mathematical physics, by investigating the asymptotic behaviour of the norms $\|u(t)\|$ as $t\to\infty$. In the classical (undamped) wave equation the operators $T(t)$, $t\ge0$, are isometries (even unitary operators), so energy is conserved. On the other hand, as soon as there is some sort of damping, for instance due to air resistance or other frictional forces, we expect the energy of any solution to decay over time. But at what rate? As it turns out, we may associate with any damped wave equation an increasing continuous function $M\colon[0,\infty)\to(0,\infty)$, which captures important spectral properties of the operator $A$ and in all cases of interest will satisfy $M(s)\to\infty$ as $s\to\infty$. In practice, obtaining good estimates on the function $M$ may itself be a non-trivial problem (the precise behaviour of $M$ is determined by the nature of the damping), but at least in principle the function $M$ is part of what one knows about the problem at hand. The question becomes: given the function $M$, what can we say about the rate of energy decay of (sufficiently regular) solutions of our damped wave equation?

It is known that the best result one may hope for is an estimate of the form \begin{equation}\label{eq:opt}\hspace{100pt} \|u(t)\|\le \frac{C}{M^{-1}(ct)} \hspace{200pt} (2)\end{equation} for all sufficiently large values of $t>0$, where $C,c$ are positive constants. It is also known that this best possible rate does not hold in all cases, and that sometimes a certain correction factor is needed. On the other hand, a celebrated result from 2010 due to A. Borichev and Y. Tomilov shows that if we consider the damped wave equation (or more generally any abstract Cauchy problem in which $X$ is a Hilbert space) and if $M(s)$ is proportional to $s^\alpha$ for some $\alpha>0$ and all sufficiently large $s>0$, then we do obtain the best possible rate given by (2). This result has been applied extensively throughout the recent literature on energy decay of damped waves and similar systems. A natural question, then, is whether the best possible estimate in (2) holds only for functions $M$ of this special polynomial type or for other functions as well.

In a recent paper (to appear in Advances in Mathematics) Oxford Mathematician David Seifert and his collaborators proved that one in fact obtains the optimal estimate in (2) for a much larger class of functions $M$, known in the literature as functions with positive increase. This class includes all functions of sufficiently rapid and regular growth, and in particular it includes functions $M(s)$ which are eventually proportional to $s^\alpha\log(s)^\beta$, where $C,\alpha >0$ and $\beta\in\mathbb{R}$. Such functions arise naturally in models of sound waves subject to viscoelastic damping at the boundary. Furthermore, the class of functions with positive increase is in a certain sense the largest possible class for which one could hope to obtain the estimate in (2), as is also shown in the paper. The proofs of these results combine techniques from operator theory and Fourier analysis. One particularly important ingredient is the famous Plancherel theorem, which states that the Fourier transform (suitably scaled) is a unitary operator on the space of square-integrable functions taking values in a Hilbert space. In future work, David and his collaborators hope to extend their results to the setting of more general Banach spaces. In such cases, however, the Plancherel theorem is known not to hold, so new ideas based on the finer geometric properties of Banach spaces are likely to be needed. 

Tuesday, 5 February 2019

Mathematics of Random Systems - a new Centre for Doctoral Training. Applications open.

The Mathematics of Random Systems: Analysis, Modelling and Algorithms is our new EPSRC Centre for Doctoral Training (CDT), and a partnership between three world-class departments in the area of probabilistic modelling, stochastic analysis and their applications: the Mathematical Institute, Oxford, the Department of Statistics in Oxford and the Dept of Mathematics, Imperial College London. Its ambition is to train the next generation of academic and industry experts in stochastic modelling, advanced computational methods and data science.

The CDT offers a 4-year comprehensive training programme at the frontier of scientific research in probability, stochastic analysis, stochastic modelling, stochastic computational methods and applications in physics, finance, biology, healthcare and data science.

For further information and instructions on how to apply click here.

Monday, 4 February 2019

Classification of geometric spaces in F-theory

Oxford Mathematician Yinan Wang talks about his and colleagues' work on classification of elliptic Calabi-Yau manifolds and geometric solutions of F-theory.

"In the past century, the unification of gravitational force and particle physics was the ultimate dream for many theoretical physicists. String theory is currently the most well-established example of such a grand unification theory. In the past decades, research in this field has produced many fruitful applications in quantum field theory, condensed matter physics, quantum information theory and pure mathematics.

Nonetheless, the string theory framework is complicated since there are many different versions of string theory: I, IIA, IIB, heterotic, M-theory. Moreover, string theory lives in a very high dimensional spacetime: 10 or 11 dimensions including the time direction. To get a description of our real world four-dimensional physics, we need to put this higher dimensional theory on a very small space (this procedure is called "compactification''). There could be a zillion of such geometric spaces, and their total number was completely unknown.

In our recent work, we studied the compactification of F-theory, which is a geometric description of IIB string theory. This framework unifies the M-theory solutions in many cases as well. In particular, the geometric spaces in this approach are elliptic Calabi-Yau manifolds. They can be thought as having an additional torus over each point on a "base'' space.


Figure 1: A picture of elliptic Calabi-Yau manifold

We partially classified the sets of four-dimensional and six-dimensional bases. They equivalently have two or three complex dimensions if one describes them using complex numbers. In particular, we probed the huge connected network of complex 3D bases, which was estimated to contain more than $10^{3,000}$ nodes. The F-theory compactification on an elliptic Calabi-Yau manifold over such bases will give rise to different 4D physics. Interestingly, we found that the 4D physical model on a typical geometric space is quite different from our known particle physics. The gauge groups in the F-theory models are usually $SU(2)$, $F_4$, $G_2$ and $E_8$ in term of Lie algebra classification, while our real world particle physics has $SU(3)\times SU(2)\times U(1)$ gauge group. Moreover, there are a number of mysterious "strongly coupled'' sectors in a typical F-theory model, without any known gauge theory description. There are many things to be explored about these strongly coupled sectors in the future, which requires novel quantum field theory and geometric techniques. Finally, we hope to figure out whether our particle physics standard model can be realized on such a typical F-theory construction."


Figure 2: A part of the network of complex 3D base geometries

For more on the probing of networks click here
For the work on strongly coupled sector of a typical F-theory model click here

Friday, 1 February 2019

Urban Geometry: Looking for shapes and patterns in an urban setting. Photography Exhibition 4-21 February

Looking for shapes and patterns isn't only a mathematical pursuit of course. Artists are also drawn to geometry. Our latest Oxford Mathematics photography exhibition is 'Urban Geometry' by Ania Ready & Magda Wolna. Ania and Magda describe their work: 

"Human eyes are naturally drawn to shapes and patterns, regardless of whether they look at modern buildings or vast landscapes. We decided to focus on the geometrical beauty of the urban environment. We explore various aspects of it, or to borrow from the shared photographic and geometrical vocabulary, various “angles” of it. We play with lines, focal points, repetitions and also with our Polish heritage."

Ania and Magda are Oxfordshire-based photographers and members of the Oxford Photographic Society. The exhibition runs from 4-21 February 2019.

Thursday, 31 January 2019

Reconstructing the number of edges from a partial deck

Oxford Mathematician Carla Groenland talks about her and Oxford colleagues' work on graph reconstruction.

A graph $G$ consists of a set of vertices $V(G)$ and a set of edges $E(G)$ which may connect two (distinct) vertices. (There are no self-loops or multiple edges.)

A very basic question about graphs is: Is a graph determined by its induced subgraphs? For a graph $G$ and a vertex $v$, a card of the graph is an induced subgraph $G-v$ obtained by removing the vertex $v$ and all adjacent edges. The deck of the graph is the collection of cards $\{G-v:v\in V(G)\}$, allowing multiples. An example of a deck of card is given below.

Below the cards, we see how the cards can be obtained by removing vertices from a single graph, removing one at a time and removing each vertex exactly once. Can two non-isomorphic graphs have the same deck of cards? Yes, if the graphs have two vertices.

If we see a single vertex twice, then we know the original graph had two vertices but there is no way for us to know whether there is an edge between them.

More than 60 years ago, Kelly and Ulam made the beautiful conjecture that if two graphs on at least three vertices have the same deck, they must be isomorphic. In 1977, Bondy and Hemminger wrote the following about this conjecture:

"The Reconstruction Conjecture is generally regarded as one of the foremost unsolved problems in graph theory. Indeed, Harary (1969) has even classified it as a "graphical disease'' because of its contagious nature. According to reliable sources, it was discovered in Wisconsin in 1941 by Kelly and Ulam, and claimed its first victim (P. J. Kelly) in 1942. There are now more than sixty recorded cases, and relapses occur frequently (this article being a case in point)."

There are many subtleties in the problem which might not be apparent at first sight; for example, if two vertices $u$ and $v$ have the same card (i.e. $G-u\cong G-v$) then they are not necessarily "similar'' in the graph, that is, there does not necessarily exist a graph automorphism mapping $u$ to $v$.

Rather than reconstructing the entire graph, can we at least read off some information about the graph? If a graph has vertices $v_1,\dots,v_n$ then \[ \sum_{i=1}^n |E(G-v_i)|=(n-2)|E(G)| \] since each edge appears in all cards except for the two corresponding to vertices it is adjacent to. So we can read off the number of edges of the graph (the size) if we are given a full deck of cards. What if we have only access to a subset of the cards? In Size reconstructibility of graphs, Alex Scott, Hannah Guggiari and I describe a way to reconstruct the size of the graph if at most $\frac1{20}\sqrt{n}$ cards are missing from the deck (for $n$ large). The best previous result in this direction was the case in which two cards are missing.

Our proof works as follows:

* We first note that if we have been given the cards $G-v_1,\dots,G-v_{n-k}$, we can still compute \[ \widetilde{|E(G)|}=\frac{\sum_{i=1}^{n-k} |E(G-v_i)|}{n-2-k}\approx|E(G)| \] which is an (over)estimate on the number of edges.

* The degree $d(v)$ of a vertex is the number of edges adjacent to it. Note that on the card $G-v$, exactly the edges adjacent to $v$ are missing. Hence $|E(G)|-|E(G-v)|=d(v)$. We will try to approximate the degree sequence $(d_t)$, where $d_t$ gives the number of vertices of degree $t$, in two different ways.

* Firstly, using our estimate on the number of edges, we can still approximate the degree of the vertices of the cards that have been given: \[ \widetilde{d(v)}=\widetilde{|E(G)|}-|E(G-v)|.\] Since we overestimate the number of edges, we also overestimate $d(v)$ for every vertex, but always by the same amount $\alpha = \widetilde{|E(G)|}-|E(G)|.$ This means our approximated degree sequence $(\widetilde{d}_t)$ is the actual degree sequence $(d_t),$ but shifted to the right, and moreover some values have been underestimated (since some of the cards are missing). If we could recover the shift $\alpha,$ we could find $|E(G)|$ since we know $\widetilde{|E(G)|}.$ 

* Secondly, we discover many small values $d_t$ exactly. We use these to either reconstruct the entire degree sequence (then we can read off the number of edges from this) or discover some large values. Suppose that we find a segment as in the picture below: one large value, and to one side of it a bunch of known values, many of which are small. We "match up'' the known values to various shifts of $(\widetilde{d}_t).$ For the correct shift, the error will be small, whereas for any other shift the error is lower bounded by the difference between large and small in the figure below. This allows us to recover $\alpha$ and then the number of edges.

Some interesting open problems include:

* The original graph reconstruction conjecture, for which the edge variant is also unknown. (The directed graph, infinite graph and hypergraph analogues are false.) This is also unknown for "nice'' graph classes such as bipartite graphs or planar graphs.

* Improving our result beyond $\sqrt{n}$ or extending it to recovering different information about the graph, such as the degree sequence or the number of triangles.

Thursday, 31 January 2019

Jon Keating appointed to the Sedleian Professorship of Natural Philosophy

Oxford Mathematics is delighted to announce that Prof. Jon Keating FRS, the Henry Overton Wills Professor of Mathematics in Bristol, and Chair of the Heilbronn Institute for Mathematical Research, has been appointed to the Sedleian Professorship of Natural Philosophy in the University of Oxford.

Jon has wide-ranging interests but is best known for his research in random matrix theory and its applications to quantum chaos, number theory and the Riemann zeta function.  In November, he will be the next President of the LMS.

The Sedleian is regarded as the oldest of Oxford's scientific chairs and holders are simultaneously elected to fellowships at Queen's College, Oxford. Recent holders have included Brooke Benjamin (1979-1995) who did highly influential work in the areas of mathematical analysis and fluid mechanics and most recently Sir John Ball (1996-2018), who is distinguished for his work in the mathematical theory of elasticity, materials science, the calculus of variations, and infinite-dimensional dynamical systems.

Sunday, 27 January 2019

Music & Mathematics - Villiers Quartet in concert in the Mathematical Institute

We often need mathematics and science to understand our lives. But we also need the Arts. And especially music. In fact they often work best together.

The Villiers Quartet are Quartet in Residence at Oxford University and on February 8th we welcome them for the first time to the Andrew Wiles Building, home of Oxford Mathematics for an evening of Haydn, Beethoven and Mozart. 

Haydn - Quartet in G, Op. 77 No.1

Mozart -  Quartet G, K. 387

Beethoven - Quartet in C# minor, Op. 131

For more information and how to book click here.

Thursday, 17 January 2019

Multiple zeta values in deformation quantization

Oxford Mathematician Erik Panzer talks about his and colleagues' work on devising an algorithm to compute Kontsevich's star-product formula explicitly, solving a problem open for more than 20 years.

"The transition from classical mechanics to quantum mechanics is marked by the introduction of non-commutativity. For example, let us consider the case of a particle moving on the real line.

From commutative classical mechanics...

Classically, the state of the particle is described by its position $x$ and its momentum $p$. These coordinates parametrize the phase space, which is the tangent space $M=T^1 \mathbb{R} \cong \mathbb{R}^2$. One can view $x,p\colon M \rightarrow \mathbb{R}$ as smooth functions on the phase space, and the set $A=C^{\infty}(M)$ of all smooth functions on the phase space is an algebra with respect to the (commutative) multiplication of functions, e.g. $x\cdot p = p \cdot x$. The dynamics of the system is determined by a Hamiltonian $\mathcal{H} \in A$, which dictates the time evolution of a state according to \begin{equation*} x'(t) = \{x(t), \mathcal{H}\} \quad\text{and}\quad p'(t) = \{p(t), \mathcal{H}\}, \qquad\qquad(1) \end{equation*} where the Poisson bracket on the phase space is given by \begin{equation*} \{\cdot,\cdot\}\colon M\times M \longrightarrow M, \qquad \{f,g\} = \frac{\partial f}{\partial x} \frac{\partial g}{\partial p} - \frac{\partial f}{\partial p} \frac{\partial g}{\partial x}. \end{equation*} non-commutative quantum mechanics.

In the quantum world, the state is described by a wave function $\psi$ that lives in a Hilbert space $L^2(\mathbb{R})$ of square-integrable functions on $\mathbb{R}$. Position and momentum now are described by operators $\hat{x},\hat{p}$ that act on this Hilbert space, namely \begin{equation*} \hat{x} \psi(x) = x \cdot \psi(x) \quad\text{and}\quad \hat{p} \psi(x) = -\mathrm{i}\hbar\frac{\partial}{\partial x} \psi(x) \end{equation*} where $\hbar$ is the (very small) reduced Planck constant. Note that these operators on the Hilbert space do not commute, $\hat{x}\hat{p} \neq \hat{p}\hat{x}$, and the precise commutator turns out to be \begin{equation*} [\hat{x},\hat{p}] = \hat{x}\hat{p}-\hat{p}\hat{x} = \mathrm{i}\hbar = \mathrm{i}\hbar \{x,p\}. \end{equation*} The observable quantity associated to an operator $\hat{f}$ is its expection value $\langle {\psi}| \hat{f} |\psi \rangle$, which is common notation in physics for the scalar product of $\psi$ with $\hat{f}\psi$. The time evolution of such an observable is determined by the Hamiltonian operator $\hat{\mathcal{H}}$ in a way very similar to (1): \begin{equation*} \mathrm{i}\hbar \frac{\partial \langle \psi(t) | \hat{f}| \psi(t)\rangle}{\partial t} = \langle \psi(t)| \big[\hat{f},\hat{\mathcal{H}} \big] |\psi(t)\rangle. \qquad\qquad(2) \end{equation*}

Deformation quantization...

Let us now generalise the example: For each classical quantity $f\in A=C^{\infty}(M)$, there should be a quantum analogue $\hat{f}$ that acts on some Hilbert space. We can think of this quantization $f\mapsto \hat{f}$ as an embedding of the commutative algebra $A$ into a non-commutative algebra of operators. Comparing (1) and (2), we see that quantization identifies the commutator $[\cdot,\cdot]$ with $i\hbar$ times the Poisson bracket $\{\cdot,\cdot\}$. The goal of deformation quantization is to describe this non-commutative structure algebraically on $A$, without reference to the Hilbert space. Namely, can we find a non-commutative product $f\star g$ of smooth functions such that \begin{equation*} \hat{f} \hat{g} = \widehat{f\star g} \quad\text{?} \end{equation*} Since the non-commutativity comes in only at order $\hbar$, it follows from the above that we should have \begin{equation*} f \star g = f\cdot g + \frac{\mathrm{i}\hbar}{2} \{f,g\} + \mathcal{O}(\hbar^2), \qquad\qquad(3)\end{equation*} i.e. the star-product is commutative to leading order (and therefore called a deformation of the commutative product), and the first order correction in $\hbar$ is determined by the Poisson structure.

...has a universal solution. 

Maxim Kontsevich proved in 1997 that such quantizations always exist: We can allow for an arbitrary smooth manifold $M$ with a Poisson structure - that means a Lie bracket $\{\cdot,\cdot\}$ on $A=C^{\infty}(M)$ which acts in both slots as a derivation (Leibniz rule).

Theorem (Kontsevich). Given an arbitrary Poisson manifold $(M,\{\cdot,\cdot\})$, there does indeed exist an associative product $\star$ on the algebra $A[[\hbar]]$ of formal power series in $\hbar$, such that equation (3) holds.

In fact, Kontsevich gives an explicit formula for this star-product in the form \begin{equation*} f \star g = f \cdot g + \sum_{n=1}^{\infty} \frac{(\mathrm{i}\hbar)^n}{n!} \sum_{\Gamma \in G_n} c(\Gamma) \cdot B_{\Gamma}(f,g), \end{equation*} where at each order $n$ in $\hbar$, the sum is over a finite set $G_n$ of certain directed graphs, like

and so on. The term $B_{\Gamma}(f,g)$ is a bidifferential operator acting on $f$ and $g$, which is defined in terms of the Poisson structure. It can be written down very easily directly in terms of the graph. The remaining ingredients in the formula are the real numbers $c(\Gamma) \in \mathbb{R}$. These are universal constants, which means that they do not depend on the Poisson structure or $f$ or $g$. Once these constants are known, the star-product for any given Poisson structure can be written down explicitly.

What are the weights?

These weights are defined as integrals over configurations of points in the upper half-plane $\mathbb{H}$. The simplest example is the unique graph in $G_1$, which gives (exercise!)

This is precisely the factor $1/2$ in front of the Poisson bracket in (3). But for higher orders, these integrals become much more complicated, and until very recently it was not known how they could be computed. Due to this problem, the star product was explicitly known only up to order $\hbar^{\leq 3}$.

It was conjectured, however, that the weights should be expressible in terms of multiple zeta values, which are sums of the form
\sum_{1\leq k_1<\cdots<k_d} \frac{1}{k_1^{n_1} \cdots k_d^{n_d}} \in \mathbb{R},
indexed by integers $n_1,\ldots,n_d$. They generalize the Riemann zeta function $\zeta(n)$, and they play an important role in the theory of periods and motives. In particular, there is a (conjectural) Galois theory for multiple zeta values.

In recent work with Peter Banks and Brent Pym, we developed an algorithm that can evaluate the weight integrals $c(\Gamma)$ for arbitrary graphs $\Gamma$:

Theorem (Banks, Panzer, Pym). For an arbitrary graph $\Gamma\in G_n$, the number $c(\Gamma)$ is a rational linear combination of multiple zeta values, normalized by $(2\mathrm{i}\pi)^{n_1+\ldots+n_d}$.

We calculated and tabulated the weights of all 50821 graphs that appear in $G_n$ for $n\leq 6$, which gives the star-product for all Poisson structures up to order $\hbar^{\leq 6}$. This allows us to study the quantizations in explicit examples and compare them to other quantization methods. Furthermore, our result shows that the Galois group of multiple zeta values acts on the space of star-products.

This research started off with a project by Peter Banks over the summer 2016, when Brent Pym was still in Oxford. Since then we have developed our proof and implemented all steps in computer programs, which are starproducts (publicly available} and automatize the computation of the star product and Kontsevich weights. The underlying techniques make heavy use of the theory of multiple polylogarithms on the moduli space $\mathfrak{M}_{0,n}$ of marked genus 0 curves, a generalization of Stokes' theorem to manifolds with corners and a theory of single-valued integration, influenced by work of Francis Brown (Oxford) and Oliver Schnetz (Erlangen-Nürnberg)."

Related links:

Tuesday, 15 January 2019

Oxford Mathematics to LIVE STREAM an Undergraduate lecture

For the first time, on February 14th at 10am Oxford Mathematics will be LIVE STREAMING a 1st Year undergraduate lecture. In addition we will film (not live) a real tutorial based on that lecture.

After the huge success of making an undergraduate lecture widely available via social media last term, we know there is an appetite to better understand Oxford teaching. In turn we want to demystify what we do, showing that it is both familiar but also distinctive.

The details:
LIVE Oxford Mathematics Student Lecture - James Sparks: 1st Year Undergraduate lecture on 'Dynamics', the mathematics of how things change with time
14th February, 10am-11am UK time

Watch live and ask questions of our mathematicians as you watch

For more information about the 'Dynamics' course:

The lecture will remain available if you can't watch live.

Interviews with students:
We shall also be filming short interviews with the students as they leave the lecture, asking them to explain what happens next. These will be posted on our social media pages.

Watch a Tutorial:
The real tutorial based on the lecture (with a tutor and two students) will be filmed the following week and made available shortly afterwards

For more information and updates: