Monday, 16 March 2020

How to reduce damage when freezing cells

Oxford Mathematician Mohit Dalwadi talks about his work on the modelling of cryopreservation.

"While Captain America was able to survive being frozen for over six decades in the Marvel Universe, we are a long way from this type of technology in real-life. At the moment, people can only preserve small numbers of cells when freezing them. This technology is known as cryopreservation, and it works because metabolic processes grind to a halt at low temperatures, essentially keeping cells in suspended animation. Cryopreservation has the potential to help protect endangered species, preserve tissue for organ transplants, and improve food security.

The major challenge in cryopreservation is to ensure that cells are not damaged during the freezing or thawing processes. One source of damage is ice formation; larger, spikier ice crystals are more likely to rip the cell apart. The size and shape of ice crystals depends on how quickly you cool the mixture you are trying to freeze, and so this is one key factor that needs to be understood to improve cryopreservation protocols.

Incredibly, some cold-blooded animals are able to naturally survive being frozen. There are certain species of frogs and fish that can survive being frozen overnight. They do this by releasing sugars into their bloodstream, which act as anti-freeze and reduce the ice formation within their bodies. Humans are trying to develop technology that mimics these frogs by adding cryoprotective agents (CPA) into solutions containing cells they want to freeze. While this does help to reduce ice formation, it comes with its own problems – CPA can be toxic to cells that have not evolved to deal with it.

This means that if we want to try and freeze cells, we have to be very careful about getting the right balance of CPA addition and cooling rate – there is a delicate balance between the two. As you might imagine, it takes a lot of time and effort to determine this optimal balance through experiments. And just because something works for one type of cell, there is no guarantee that it will work for another. By developing mathematical models of these cryopreservation procedures, we can quickly evaluate their efficacy, and determine the optimal protocols for given cell characteristics.

As the solution mixture of cells and CPA is cooled, ice will form and decrease the volume of liquid in the system. This will concentrate the CPA and other solutes, causing cells to dehydrate through osmosis. To account for this mathematically, we must track the heat and mass transfer in the system, which are coupled due to the depressing effect of CPA on the freezing temperature, and account for the moving boundaries at the freezing front and the cell membrane. While these moving boundaries can be computationally challenging to simulate, we are able to systematically reduce the complexity of the model for the freezing of a single cell (e.g. a human egg) by exploiting the disparate timescales over which transport mechanisms occur.

Our analysis shows that while the temperature of the liquid settles down to a spatially uniform value over a matter of seconds, the chemical concentrations take several minutes to do the same. Moreover, we find that the motion of the moving boundaries is strongly coupled to the chemical mass transport. While the freezing front motion and the CPA concentration are both forced by the decrease in temperature, over the timescale of a few minutes the motion of the freezing front is slowed down by CPA being pushed out and building up just ahead of the advancing front. The cell membrane motion occurs over a longer timescale of many minutes, and this motion is mainly governed by the concentration difference across the cell membrane. We are able to use these results to provide predictions on what sort of CPA levels and cooling rates cryobiologists should be trying to aim for, given the cell characteristics.

Find out more about this work which was carried out with colleagues Sarah Waters, Helen Byrne and Ian Hewitt."

Image above: Ice forming within a cell

Image above: model predictions for supercooling at the cell centre over time for different cooling rates.




Friday, 13 March 2020

Coronavirus (Covid-19): advice and updates

Andrew Wiles Building

The University has announced numerous steps to prioritise the health and welfare of staff, students and visitors in the light of the UK’s escalating coronavirus situation. This is an unprecedented and challenging time for our university and department community, and I would ask that you please support each other wherever you can, and follow University guidance, which is continuously updated. MI staff and students should also check their emails regularly for further guidance.

Mike Giles, Head of Department

Wednesday, 11 March 2020

Oxford Mathematicians win 2019 PNAS Cozzarelli Prize

Oxford Mathematicians Derek Moulton and Alain Goriely together with their colleague Régis Chirat (University of Lyon) have won the 2019 PNAS Cozzarelli Prize in the Engineering and Applied Sciences category for their paper 'Mechanics unlocks the morphogenetic puzzle of interlocking bivalved shells.'

The paper describes how two groups of animals—brachiopods and bivalve mollusks—sport interlocking shells that help guard against predators and environmental perturbations, and explains how those shells are formed.

The Cozzarelli Prize is awarded annually to six research teams whose PNAS (Proceedings of the National Academy of Sciences of the United States of America) articles have made outstanding contributions to their fields. Each team represents one of the six classes of the National Academy of Sciences.


Thursday, 27 February 2020

Corrections in infinite dimensions

Several well-known formulas involving reflection groups of finite-dimensional algebraic systems break down in infinite dimensions, but there is often a predictable way to correct them. Oxford Mathematician Thomas Oliver talks about his research getting to grips with what structures underlie the mysterious correction process.

"In algebraic language, a reflection is an order two symmetry. What this means in practical terms is that to invert a reflection, one needs to reflect again. Reflection groups are formed by composing reflections in different mirrors, the effect of which can be visualised using a kaleidoscope.

A Weyl group is a special type of reflection group. Weyl groups describe the reflective symmetries of root systems, which are configurations of vectors with prescribed geometric properties. Root systems and their Weyl groups played a pivotal role in a key achievement of modern mathematics, namely the classification of semisimple finite-dimensional Lie algebras.

There are more general theories of infinite-dimensional Lie algebras, which tentatively have applications to arithmetic and physics. The finite-dimensional theory is not exactly valid, but it can often be "corrected" in a predictable way. In fact several important equations need to be adjusted by the same correction factor.

When mathematicians see the same factors appearing in different contexts, they demand an explanation. In the case of the correction factor, their appearance is due to a new phenomenon for root systems of infinite-dimensional Lie algebras. This is the existence of imaginary roots, which have the unlikely sounding property of having negative length. This is nothing more than colourful language, in which "length" is a word for an inner product.

We calculated the correction factor as an infinite sum over the roots and analysed its "support", that is, the roots contributing non-zero terms to the sum. We found that the reflection group swapped roots around within the support, but never took one outside. That is, the support is Weyl group invariant. This corresponds to a key property of the imaginary roots: no matter what combination of reflections you try, you can never pass from an imaginary root to a real one. Because finite-dimensional Lie algebras do not have any imaginary roots, our analysis in fact gives a new proof of several classical formulas.

The research was carried out with Kyu-Hwan Lee and Dongwen Liu."

Image above: the action of reflection groups can be visualised like a kaleidoscope.

Tuesday, 25 February 2020

Quasiconvexity and its role in the Calculus of Variations

Oxford Mathematician Andre Guerra talks about quasiconvexity and its role in the Calculus of Variations:

"Most physical systems can be described through a Lagrangian, which is a function $f\colon \mathbb{R}^{m\times n}\to \mathbb{R}$ that can be thought of as some kind of 'energy.' For our purposes, the Lagrangian induces a functional $\mathcal F$ by $$\mathcal F[u]\equiv \int_\Omega f(\mbox{D} u(x))\mbox{ d}x,$$ where $u\colon \Omega\subset\mathbb R^n\to \mathbb R^m$ is a vector field. Physical configurations of the system correspond to those $u$ which minimise $\mathcal F$ or, more generally, are its critical points. Familiar examples of Lagrangians are the elastic energy of a rubber band, the electrostatic energy of a battery and the gravitational potential energy of an object. The string in a spider web, for instance, attains a configuration that minimizes its gravitational energy.

The only systematic approach to solve minimisation problems is to use the so-called Direct Method: one takes a sequence of fields $u_j$ such that $$\lim_{j\to \infty}\mathcal{F}[u_j]=\inf \mathcal F$$ and one hopes that $u_j$ converges in some sense to a minimiser $u$ of $\mathcal F$. In the 1950s Morrey recognised that the Lagrangians for which the Direct Method works are those possessing a weak type of convexity, known as quasiconvexity, and which reads as $$f(A)\leq \frac{1}{|\Omega|} \int_\Omega f(A+\mbox{D}u) \mbox{ d}x, \mbox{ for all } A\in \mathbb R^{m\times n} \mbox{ and } \varphi \in C^\infty_c(\Omega,\mathbb R^m).$$

In general, critical points of physical systems are expected to be somewhat smooth and so we are led to consider conditions on $f$ ensuring this smoothness. The most natural condition in this direction is known as rank-one convexity, and all quasiconvex Lagrangians are rank-one convex. It is important to keep in mind that when the Lagrangian acts on scalar fields, so that $m=1$, both quasiconvexity and rank-one convexity reduce to the usual notion of convexity, although this is not so whenever $m>1$.

Although 70 years have passed since Morrey's seminal work, quasiconvexity remains very poorly understood, despite formidable efforts from many mathematicians: there are explicit examples of fourth-order polynomials for which we cannot prove nor disprove quasiconvexity! On the other hand, rank-one convexity is an easier-to-verify condition. One of the most challenging open problems in the Calculus of Variations is to decide whether quasiconvexity and rank-one convexity are different. In 1992, Šverák found, for $m> 2$, a remarkable example of a rank-one convex Lagrangian which is not quasiconvex. In particular, the two-dimensional case remains open and this is known in the literature as Morrey's problem.

Morrey's problem asks whether two very large classes of functions are the same, making it a daunting question to answer. Recently I proved that it is enough to solve this problem in a strictly smaller class of functions, the so-called extremal functions, which should be better behaved than the typical rank-one convex function and thus easier to study. Though the set of extremal functions is smaller than the entire class of rank-one convex functions, there is still no satisfying way of characterizing them. To try to gain some intuition, I am interested in obtaining examples of such functions. I have found several, whose extremality had been conjectured by Šverák already in 1992, on my own. Currently, together with Daniel Faraco from Universidad Autónoma de Madrid, I am trying to produce more examples."

Friday, 21 February 2020

Books at Bedtime - new books by Oxford Mathematicians

Oxford Mathematicians occasionally have time to write and edit books. Their range of topics - from Topology and Geometry to Stochastic Methods and Chaos via the International Congress of 1936 and a candid account of a life escaping from poverty and living with polio - is a testament to how much maths reaches in to our lives. Some are for specialists, some for aspiring specialists, but all give you a flavour of the mathematical life.

Here are some of the those works that have already been published or will appear soon. The Oxford Mathematician (and their individual page) is highlighted in blue along with relevant links to the book itself.

Stochastic Modelling of Reaction-Diffusion Processes - Radek Erban, S. Jonathan Chapman
Stochastic methods have been used by researchers in Oxford Mathematics to model a number of biological systems, ranging in size from molecular dynamics simulations of small biomolecules to stochastic modelling of groups of animals.

This book can be used both for self-study and as a supporting text for advanced undergraduate or beginning graduate-level courses in applied mathematics. It discusses the essence of mathematical methods which appear (under different names) in a number of interdisciplinary scientific fields bridging mathematics and computations with biology and chemistry (including mathematical biology, non-equilibrium statistical physics, computational chemistry, soft condensed matter, physical chemistry or biophysics).

Chaos - An Introduction for Applied Mathematicians - Andrew Fowler and Mark McGuinness
This is a textbook on chaos and nonlinear dynamics, written by applied mathematicians for applied mathematicians. It aims to tread a middle ground between the mathematician's rigour and the physicist’s pragmatism.

The Wonder Book of Geometry - David Acheson
David transports us into the world of geometry, a fundamental and ancient branch of mathematics and argues that geometry can provide the quickest route into the whole spirit of mathematics at its best, especially for the young.

Topology: A Very Short Introduction - Richard Earl
The twentieth century was the century of topology – or so said Jean Dieudonné. From a nascent, intuitively understood subject in the time of Riemann, topology would become a significant area of mathematics, influencing the foundations of mathematics, through to applications in physics and data science.

Meeting under the Integral Sign? The Oslo Congress of Mathematicians on the Eve of the Second World War - Christopher D. Hollings and Reinhard Siegmund-Schultze
This book examines the historically unique conditions under which the International Congress of Mathematicians took place in Oslo in 1936. This Congress was the only one on this level to be held during the period of the Nazi regime in Germany (1933–1945) and after the wave of emigrations from it. Relying heavily on unpublished archival sources, the authors consider the different goals of the various participants in the Congress, most notably those of the Norwegian organisers, and the Nazi-led German delegation. They also investigate the reasons for the absence of the proposed Soviet and Italian delegations.

My Gift of Polio ~ An unexpected Life ~ From Scotland’s Rustic Hills to Oxford’s Hallowed Halls & Beyond - James D Murray
James Murray was the youngest of six children born into a poor working-class family in Moffat, a very small isolated town in rural Scotland, during the Depression of the early 1930s. He caught polio as a baby and his future looked bleak. This profusely illustrated memoir describes his early years growing up in poverty and follows his serendipitous life beyond - taking him from degrees at the University of St. Andrews to international renown in the world of academia at Harvard, Oxford, Paris and other universities around the world.

Conformal Maps and GeometryDmitry Belyaev
Geometric function theory is one of the most interesting parts of complex analysis, an area that has become increasingly relevant as a key feature in the theory of Schramm–Loewner evolution. Though Riemann mapping theorem is frequently explored, there are few texts that discuss general theory of univalent maps, conformal invariants, and Loewner evolution. This textbook provides an accessible foundation of the theory of conformal maps and their connections with geometry.

Leading Within Digital Worlds: Strategic Management for Data Science - Peter Grindrod
With rapidly evolving emerging technologies, the business world is entering a phase of reform. Within times of change, transformative and disruptive environments as well as uncertain futures have created a difficult landscape for leaders within business. This book is written for business leaders who want to remain at the forefront of the business world in these times of technological and digital evolution.

Lectures and Surveys on G2-Manifolds and Related Topics - Spiro Karigiannis, Naichung Conan Leung and Jason Lotay (Eds.)
This book, one of the first on G2 manifolds in decades, collects introductory lectures and survey articles largely based on talks given at a workshop held at the Fields Institute in August 2017, as part of the major thematic program on geometric analysis. It provides an accessible introduction to various aspects of the geometry of G2 manifolds, including the construction of examples, as well as the intimate relations with calibrated geometry, Yang-Mills gauge theory, and geometric flows. It also features the inclusion of a survey on the new topological and analytic invariants of G2 manifolds that have been recently discovered. 

Geometric Group TheoryCornelia Drutu and Misha Kapovich
The book contains proofs of several fundamental results of geometric group theory, such as Gromov's theorem on groups of polynomial growth, Tits's alternative, Stallings's theorem on ends of groups, Dunwoody's accessibility theorem, the Mostow Rigidity Theorem, and quasiisometric rigidity theorems of Tukia and Schwartz. This is the first book in which geometric group theory is presented in a form accessible to advanced graduate students and young research mathematicians.

Probability and Random Processes, 4th edition, & One Thousand Exercises in Probability, 3rd edition - Geoffrey Grimmett and David Stirzaker

Probability and Random Processes:
This book provides an extensive introduction to probability and random processes. It is intended for those working in the many and varied applications of the subject as well as for those studying more theoretical aspects.

One Thousand Exercises in Probability:
This volume contains more than 1300 exercises in probability and random processes together with their solutions. The new edition extends the previous edition by the inclusion of numerous new exercises, and several new sections devoted to further topics in aspects of stochastic processes. Since many exercises have multiple parts, the total number of interrogatives exceeds 3000.

Thursday, 20 February 2020

International Women's Day Event in the Mathematical Institute, Friday 6th March 2020

INTERNATIONAL WOMEN’S DAY EVENT - Mathematical Institute, Lecture Theatre 3, Friday 6th March 2020, 12-2pm

In conjunction with the Mathematical, Physical and Life Sciences Equality & Diversity team, Oxford Mathematics, Department of Statistics and Department of Computer Science invite you to attend an event celebrating women in science and showcasing the achievements of women in our University.  Three speakers from across the departments will talk about their research and careers to date.  The presentations will be followed by a networking lunch and poster session. 

Our speakers are:
Ms Klaudia Krawiecka, DPhil student in Cyber Security, Department of Computer Science
Dr Priya Subramanian, Hooke Research Fellow, Mathematical Institute
Dr Cora Mezger, Director of Statistical Consultancy Services, Department of Statistics

The event is free to attend but please register to attend by emailing by midday on 28th February 2020, noting any dietary or access requirements.

There will also be a poster session, at which Early Career Researchers, undergraduates and postgraduates are invited to present posters that showcase their work or work relating to International Women’s Day.   Posters will be judged by a panel of experts and vouchers awarded to the three best entries (£100 for first place; £50 for second place and £25 for third place).  

To apply to present a poster, please email by midday on 21st February 2020 with your poster title and abstract (no more than 150 words).  Posters should be A0 in size.  Funds are available for printing costs. 

Thursday, 20 February 2020

Why does the risk of cancer and infectious diseases increase with age?

Oxford Mathematician Sam Palmer tackles a crucial issue in our understanding of the risks of serious diseases such as cancer.

"Why does the risk of cancer and infectious diseases increase with age? For cancer, there are two main contributers: the declining immune system, and the accumulation of genetic mutations. My research points to immune system decline as the larger factor, in particular for cancers such as chronic myeloid leukemia and brain cancers. These cancers, along with many infectious diseases, rise exponentially with age at the same rate that the thymus shrinks. The thymus is where T-cells are produced and it involutes exponentially, starting from childhood, at a rate of 4% per year, resulting in a decline in T-cell production at the same rate. 

To explain the observation that disease risk is often inversely proportional to T-cell production, we constructed a minimal mathematical model, which implicates rare stochastic growth and T-cell exhaustion as a rate limiting step in disease progression. This fits with the recent cancer immunotherapy renaissance, where targetting T-cell exhaustion has been hugely successful. Unexpectedly, our model can also explain the rise in cancer risk with age for more common cancers, such as colon and skin cancer, where risk does not rise exponentially, but rather as a power law. The fact that common cancers rise as power laws (usually proportional to age to the power of 5 or 6) has traditionally been viewed as evidence for the mutation-accummulation hypothesis; however our model shows that this behaviour also fits with the immune-decline hypothesis.

Now, I am working on thymus regeneration and using bioinformatics and mathematical modelling to try to understand the gene regulatory networks responsible for cell fate decisions in the thymus. "

Monday, 17 February 2020

The 3rd Oxford Maths Festival - Outreaching

Hundreds of kids of all ages and their families, tables full of games, rooms full of creativity and glorious patterns. Sunday in Oxford Mathematics' Andrew Wiles Building and Saturday in Templars Square, Cowley. Yes, it was the 3rd Oxford Maths Festival 2020.

The aim of the festival is to show the beautiful, creative and collaborative side of mathematics - families were able to do hands-on maths activities (provided by NRICH), make craft items to take home, and play board games together. On the first day, activities took place at Templars Square Shopping Centre in East Oxford, reaching a new audience as part of our desire to make Oxford Maths as accessible as possible. The second day took place here in the Andrew Wiles Building and included Barney Maunder-Taylor of House of Maths providing several maths shows and Andrew Jeffrey who uses magic, juggling and balloons to explore mathematical topics. But let the pictures do the talking (and the Maths).

These events fall under the banner of Outreach. But perhaps the word is inadequate. It is as important as Teaching and Research.

Tuesday, 11 February 2020

The potential of Generalized Kähler geometry

Oxford Mathematician Francis Bischoff talks about his recent work on generalized Kähler geometry and the problem of describing its underlying degrees of freedom.

"Kähler manifolds are important geometric structures which play a central role in supersymmetric quantum field theories and mirror symmetry. These areas of research are at the forefront of modern geometry. A Kähler structure is a manifold equipped with a Riemannian metric $g$, and a compatible complex structure $I$. You can think of a metric as a structure on a manifold which allows you to measure lengths and angles, while a complex structure lets you define complex-valued coordinate functions on the manifold in a consistent way.

A striking feature of a Kähler metric is that it involves drastically less information than an ordinary Riemannian metric. To see this, let's describe a general Riemannian metric in a small region surrounding a point on an $n$-dimensional manifold. We can do this by representing the metric as a symmetric $n \times n$ matrix of functions, in the following way: \[ g = \begin{pmatrix} g_{11} & g_{12} & \ldots & g_{1n} \\ g_{21} & g_{22} & \dots & g_{2n} \\ \vdots & \vdots & \ddots& \vdots \\ g_{n1} & g_{n2} & \ldots & g_{nn}\end{pmatrix}, \qquad g_{ij} = g_{ji}. \] If we now count up the independent component functions $g_{ij}$, remembering that $g_{ij} = g_{ji}$, then we see that there are $\frac{1}{2}n(n + 1)$ of them. This number grows quickly with the dimension. For example, on a $4$-dimensional manifold, the metric already involves $10$ independent components.

On the other hand, a Kähler metric can be reduced to a single function $K$. To do this, we start by choosing complex coordinate functions $z_{i}$ using the complex structure $I$. Then, we calculate the components of the metric by taking derivatives of the function, as follows: \[ g_{i \bar\jmath} = \frac{\partial^2 K}{\partial z_{i} \partial \bar{z}_{j} }. \] This function is called the Kähler potential, and it represents a major simplification in our understanding of the geometry, with important practical consequences. For example, the Kähler potential is used in an essential way in the search for metrics with special curvature properties, since it allows these problems to be reduced to differential equations involving a single function.

In 1984, physicists discovered a more general class of geometric structures, now known as generalized Kähler (GK) geometry. These are expected to play a role in quantum field theories which are similar to those which make Kähler geometry so exciting. A GK structure is a Riemannian manifold equipped with 2 distinct complex structures $I_{+}$ and $I_{-}$, which satisfy a modified compatibility condition. A natural question which has existed since their discovery asks whether GK metrics can also be reduced to a single function, or generalized Kähler potential. In fact, this is predicted on physical grounds, and there have been constructions of special examples of GK potentials in the physics literature.

It turns out that a general solution to this problem lies in a reformulation of Kähler geometry put forward by S. Donaldson. In his approach, a Kähler structure is represented by a submanifold $\cal{L}$ inside a larger space $Z$, which is equipped with a holomorphic symplectic structure $\Omega$. The metric $g$ itself corresponds to the submanifold $\cal{L}$, which has the special property of being Lagrangian for the imaginary part of $\Omega$. This is in line with the famous dictum of A. Weinstein that 'everything is a Lagrangian submanifold'. And since Lagrangians are known to be reducible to a single function, this explains the origin of the Kähler potential.

In recent work together with M. Gualtieri and M. Zabzine, we show that Donaldson's approach extends to the setting of generalized Kähler geometry. Namely, a GK structure (of symplectic type) may also be represented by a submanifold $\cal{L}$ inside a holomorphic symplectic manifold $(Z, \Omega)$, with the property that $\cal{L}$ is Lagrangian with respect to the imaginary part of $\Omega$. The spaces $(Z, \Omega)$ that show up in our description are more general than in the Kähler case. They are known as Morita equivalences and we need to rely on the theory of Poisson geometry to construct them. An upshot of this reformulation is that a GK metric corresponds to a Lagrangian submanifold, and therefore, it can be reduced to a single function. In this way, we solve the long standing problem of describing a GK metric in terms of a generalized Kähler potential."                                                            

Read Francis's PhD thesis on this topic.