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Tuesday, 19 February 2019

Fano Manifolds Old and New

Oxford Mathematician Thomas Prince talks about his work on the construction of Fano manifolds in dimension four and their connection with Calabi-Yau geometry.

"Classical algebraic geometry studies the vanishing loci of finite collections of polynomial equations; usually under some conditions that ensure this locus has some desirable properties. The first objects studied in this subject (in its modern history) were Riemann surfaces, one dimensional objects over the complex numbers, topologically equivalent to $n$-holed tori. The attempt to replicate the classification of curves in the context of algebraic surfaces made by the 'Italian school' led by Castelnuovo and Enriques in the early 20th century led to a fundamental insight: the classification divides naturally into two distinct problems. First one studies a courser 'birational' classification of surfaces, before analysing the surfaces within each birational class. In two dimensions the second problem has a simple solution: these surfaces are related by 'blowing up' and 'blowing down', explicit operations first described by Noether. This became the model and prototype for the modern subject of birational geometry, which developed rapidly in the later 20th century, with fundamental contributions made by Hironaka, Mori, Shafarevich, and others.

In the contemporary treatment of the subject, a particularly privileged role is given to two classes of algebraic (or complex analytic) objects. The Calabi-Yau manifolds, generalising elliptic curves; as well as a particular 'minimal' class of surfaces called K$3$ surfaces, and Fano manifolds. Fano manifolds play a key role in Mori's minimal model program, itself a sweeping higher-dimensional generalisation of key methods used in the classification of surfaces. In particular this program led to the spectacular Mori-Mukai classification of Fano manifolds in dimension three, building on work on Iskovskikh.

The construction of Fano manifolds in dimension four is thus a central open problem, and the focus of ongoing research. My own interest relates to their connection with Calabi-Yau geometry: a very rough analogy would say that a Fano manifold is to a Calabi-Yau manifold what a manifold with boundary is to a manifold. Recent ideas from string theory - in particular from the field of mirror symmetry - have introduced a number of new tools to the study of Calabi-Yau manifolds, particularly following Kontsevich, Strominger-Yau-Zaslow, and Gross-Siebert. Following Givental and Kontsevich the subject of mirror symmetry has also been extended to incorporate Fano manifolds, and suggests an approach to their construction via toric degeneration. The focus of my own research is to develop these insights to produce systematic constructions of Fano manifolds along quite a different line from that taken in birational geometry. Recent progress includes a new construction of surfaces with certain classes of singularities and the classification of 527 'new' Fano fourfolds - obtained in joint work with Coates and Kasprzyk - as complete intersections in 8-dimensional toric manifolds."

Tuesday, 19 February 2019

Oxford Mathematics Student Lecture live streamed for the first time

One of our aims in Oxford Mathematics is to show what it is like to be an Oxford Mathematics student. With that in mind we have started to make student course materials available and last Autumn we filmed and made available a first year lecture on Complex Numbers. And last week, as we promised, we went a step further and livestreamed a first year lecture. James Sparks was our lecturer and Dynamics his subject. In addition, we interviewed students as they left the lecture in preparation for filming a tutorial which will also be made available later this week. 

It has taken over 800 years to get here, but we are delighted to be able to share what we do and show that it is both familiar and challenging. The lecture is below together with the interviews. We welcome your thoughts. The tutorial will follow.


 

 


 

 

 

 

 

Tuesday, 12 February 2019

Love and Maths - first ever live streaming of student lecture this Thursday, 14th Feb,10am

Lecture theatre 1

It's Valentine's Day this Thursday (14th February in case you've forgotten) and Love AND Maths are in the air. For the first time, 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.

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

https://www.facebook.com/OxfordMathematics
https://livestream.com/oxuni/undergraduate-lecture

For more information about the 'Dynamics' course: https://courses.maths.ox.ac.uk/node/37555

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
https://www.youtube.com/channel/UCLnGGRG__uGSPLBLzyhg8dQ

For more information and updates:
https://twitter.com/OxUniMaths
https://facebook.com/OxfordMathematics

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.

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