Tuesday, 6 November 2018

Conformal Cyclic Cosmology. Roger Penrose and Hannah Fry - Oxford Mathematics London Public Lecture now online

He calls it a "crazy idea." Then again, he points out, so is the idea of inflation as a way of explaining the beginnings of our Universe.

In our Oxford Mathematics London Public Lecture at the Science Museum in London, Roger Penrose revealed his latest research. In both his talk and his subsequent conversation with fellow mathematician and broadcaster Hannah Fry, Roger speculated on a veritable chain reaction of universes, which he says has been backed by evidence of events that took place before the Big Bang. With Conformal Cyclic Cosmology he argues that, instead of a single Big Bang, the universe cycles from one aeon to the next. Each universe leaves subtle imprints on the next when it pops into being.  Energy can 'burst through' from one universe to the next, at what he calls ‘Hawking points.’

In addition to his latest research Roger also reflects on his own approach to his subject ("big-headedness") and his own time at school where he was actually dropped down a maths class. So we are not alone, universally or personally speaking.

The Oxford Mathematics Public Lectures are generously supported by XTX Markets.

Photos courtesy of the Science Museum Group.



Tuesday, 6 November 2018

Improving techniques for optimising noisy functions

The problem of optimisation – that is, finding the maximum or minimum of an ‘objective’ function – is one of the most important problems in computational mathematics. Optimisation problems are ubiquitous: traders might optimise their portfolio to maximise (expected) revenue, engineers may optimise the design of a product to maximise efficiency, data scientists minimise the prediction error of machine learning models, and scientists may want to estimate parameters using experimental data. In real-world settings, uncertainties and errors are unavoidable, and this can cause stochastic noise to be present in the objective.

Most methods for optimisation rely on being able to evaluate both the objective and its derivatives.  Access to first derivatives is important for finding uphill or downhill directions, which tell us where to search next for optima, and when to terminate the method. However, when the objective has stochastic noise, it is no longer differentiable, and standard optimisation methods do not work. Instead, we must develop ‘derivative-free’ optimisation methods; that is, we have to answer the question “how do you get to the top of a hill when you don’t know which way is up?”. We achieve this by constructing models of the landscape based on sampling objective values – this approach is based on rigorous mathematical principles, and has provable guarantees of success. The figure above shows a noisy landscape, and the points tested by a derivative-free method searching for the true minimum (bottom centre, in green).

Oxford Mathematicians Lindon Roberts and Coralia Cartis, together with Jan Fiala and Benjamin Marteau from Numerical Algorithms Group Ltd (NAG), a British scientific computing company, have developed a new derivative-free method for optimising noisy and expensive objectives. The method automatically detects when the information in the objective value is overwhelmed by noise, and kick-starts the method to bring more information into the models of the landscape. This approach requires fewer evaluations of the (possibly expensive) objective, runs faster and is more scalable, but produces as good solutions as other state-of-the-art methods. Their ideas are being commercialised by NAG and will soon be available in their widely-used software library. This technique is also being applied to parameter estimation for noisy climate simulations, to help scientists find optimal parameters that fit observational climate data, thus helping quantify the sensitivity of our climate to CO2 emissions.

This work is supported by the EPSRC Centre for Doctoral Training in Industrially-Focused Mathematical Modelling. 

Monday, 5 November 2018

Structure or randomness in metric diophantine approximation?

Diophantine approximation is about how well real numbers can be approximated by rationals. Say I give you a real number $\alpha$, and I ask you to approximate it by a rational number $a/q$, where $q$ is not too large. A naive strategy would be to first choose $q$ arbitrarily, and to then choose the nearest integer $a$ to $q \alpha$. This would give $| \alpha - a/q| \le 1/(2q)$, and $\pi \approx 3.14$. Dirichlet, introducing the pigeonhole principle, showed non-constructively that there are infinitely many solutions to $| \alpha - a/q| \le 1/q^2$, and one can use continued fractions to find such approximations, for instance $\pi \approx 22/7$. 

Metric diophantine approximation is about the typical rate of approximation. There are values of $\alpha$, such as the golden ratio, for which one can't do much better than Dirichlet's theorem. However, for all $\alpha$ away from a set of Lebesgue measure zero, one can beat it by a factor of $\log q$ and more. Khintchine's theorem is prototypical, asserting that if $\psi: \mathbb N \to [0, \infty)$ is decreasing then \[ \mathrm{meas} \{ \alpha \in [0,1]: \exists^\infty (q,a) \in \mathbb N \times \mathbb Z \quad | \alpha - a/q| < \psi(q)/q \} = \begin{cases} 1, & \text{if } \sum_{q=1}^\infty \psi(q) = \infty \\ 0,&\text{if } \sum_{q=1}^\infty \psi(q) < \infty. \end{cases} \] One can prove these sorts of results using the Borel-Cantelli lemmas, from probability theory: making a ball of radius $\psi(q)/q$ around each $a/q$, and grouping together the ones with the same $q$, the idea is to show that pairs of groups overlap more or less independently.

According to my mathematical upbringing, all phenomena are explained by the dichotomy between structure and randomness: either there is structure present, or else there is (pseudo)randomness. The probabilistic considerations above had initially led me to believe that randomness was the key to understanding metric diophantine approximation, but after working in the area for a while my opinion is closer to the opposite! The denominators of the good approximations to $\alpha$ lie in Bohr sets (after Harald Bohr, brother of the eminent physicist Niels Bohr) \[ B_N(\alpha, \delta) := \{ n \le N: \| n \alpha \| \le \delta \} \subset \mathbb N, \] where $\| \cdot \|$ denotes distance to the nearest integer. A central tenet of additive combinatorics is that Bohr sets look like generalised arithmetic progressions (GAPs).

I built the GAPs using continued fractions, enabling me to make progress towards the infamous Littlewood (c. 1930) and Duffin-Schaeffer (1941) conjectures. The former is about approximating two numbers at once in a multiplicative sense, that is to find approximations $a/q, b/q$ to $\alpha,\beta$ for which \[ \Bigl | \alpha - \frac a q \Bigr | \cdot \Bigl |\beta - \frac b q \Bigr| < \frac {10^{-100}}{q^3}, \] and the latter is about approximation by reduced fractions. With Niclas Technau, we have since developed a higher-dimensional structural theory using the geometry of numbers. Going forward, I hope to establish a Khintchine-type law for multiplicative approximation on planar curves.

Sam Chow, Oxford Mathematics

Monday, 29 October 2018

Nick Trefethen awarded honorary degrees by Fribourg and Stellenbosch Universities

Oxford Mathematician Professor Nick Trefethen, Professor of Numerical Analysis and Head of Oxford's Numerical Analysis Group has been awarded honorary degrees by the University of Fribourg in Switzerland and Stellenbosch University in South Africa where Nick was cited for his work in helping to cultivate a new generation of mathematical scientists on the African continent.

Nick's research spans a wide range within numerical analysis and applied mathematics, in particular the numerical solution of differential equations, fluid mechanics and numerical linear algebra. He is also the author of several very successful books which, as the Fribourg award acknowledges, have widened interest and nourished scientific discussion well beyond mathematics. 

Thursday, 25 October 2018

The Geometry of Differential Equations - Oxford Mathematics Research by Paul Tod

If you type fundamental anagram of calculus into Google you will be led eventually to the string of symbols 6accdæ13eff7i3l9n4o4qrr4s8t12ux, probably accompanied by an explanation more or less as follows: this is a recipe for an anagram - take six copies of a, two of c, one of d, one of æ and so on, then rearrange these letters into a chunk of Latin. The accepted solution is Data æquatione quotcunque fluentes quantitates involvente fluxiones invenire et vice versa although this has nine copies of t rather than eight, and this lack of a t is assumed to be an error of the author, who was Isaac Newton and who recorded the solution in a notebook. The string occurs in a letter of Newton from 1676, sent to the mathematician Leibniz via Henry Oldenburg, who was the first Secretary of the Royal Society and a much-used conduit for savants. Newton was seeking to establish priority in the invention of calculus without giving anything away. The string is preceded in Newton's letter by the sentence "...[B]ecause I cannot proceed with the explanation of it now, I have preferred to conceal it thus:". In other words he meant to hide his knowledge but still establish priority. Online translations give the meaning of the Latin as Given an equation involving any number of fluent quantities to find the fluxions, and vice versa, which would suggest it is disguising the fundamental theorem of calculus, 'fluents' and 'fluxions' being Newton's terms for time-varying quantities and their derivatives, but one can find looser translations online, along the lines of Differential equations rock or rule or are the way to go, and I'm voting for one of those here.

Newton was a great geometer and phrased his Principia Mathematica in the language of geometry, when a greater use of differential equations might have made it more accessible. Newton would probably have been pleased (if he was ever pleased) to see geometry re-emerging from the study of differential equations and that is what I want to describe here. Given an ordinary differential equation of some order, say $N$, and written as \begin{equation}\label{1} \frac{d^Ny}{dx^N}=F\left(x,y,\frac{dy}{dx},\ldots,\frac{d^{N-1}y}{dx^{N-1}}\right), \;\;\;\;\;\;\;\;\;(1)\end{equation} one can contemplate the set of all possible solutions. Call this $M$ (for moduli space, a common usage) then $M$ has $N$ dimensions, as one sees as follows: choose a value of $x$, say zero for simplicity, and specify the $N$ numbers $\left(y(0),dy/dx|_0,\ldots,d^{N-1}y/dx^{N-1}|_0\right)$, where the notation '$|_0$' should be read as 'evaluated at $x=0$', then, subject to reasonable conditions on $F$, there is one and only one solution of (1) with these values. Thus a point of $M$ is specified by $N$ numbers, which is what you mean by '$M$ having $N$ dimensions'.

If the function $F$ is linear, that is of the form \[F(x,y_0,y_1,\ldots,y_{N-1})=\alpha_0(x)y_0+\alpha_1(x)y_1+\ldots+\alpha_{N-1}(x)y_{N-1}\] then $M$ is a vector space and, for our purposes, flat and dull; but if $F$ is nonlinear then $M$ can be quite interesting.

I'll illustrate this with an example that goes back to the French mathematician Elie Cartan in 1943, [1], but which had a new lease of life from the 1980s, (see e.g. [2], [3]). Take $N=3$, so we are interested in nonlinear, third-order differential equations with corresponding three-dimensional moduli space $M$ of solutions. One thinks of $M$ as a manifold, imagine something like a smooth surface in some higher-dimensional Euclidean space, on which one can move around smoothly. Can one give $M$ a conformal metric? This is a rule for measuring angles between directions at any point, and is a slightly weaker notion than measuring lengths of vectors at any point (an angle being essentially obtained from a ratio of lengths), and it turns out that one can provided $F$ satisfies a certain condition. The condition is a bit messy to write down, so I won't, but really it has to be because it must still hold if we make new choices of $x$ and $y$ according to the change \[(x,y)\rightarrow(\hat{x},\hat{y})=(f(x,y),g(x,y)),\] for any $f$ or $g$, which certainly makes a big change in $F$, but not much change in M (for the expert: there is more freedom than this, of contact transformations rather than just point transformations).

Now one asks another question: can one also give $M$ a conformal connection (which can be thought of as a way of moving angles around and comparing them at different places)? Again there is a single condition on $F$ that allows this, and the upshot, noticed by Cartan, is that given these two conditions on $F$, the moduli space $M$ of solutions of $F$, with its conformal metric and connection, automatically satisfies the most natural conditions in three-dimensions which generalise the Einstein equations of general relativity in four-dimensions. This is really like a-rabbit-out-of-a-hat, and is typical of what happens in this study of the geometry of differential equations.

My second example is more complicated to describe - things get harder quickly now. Here we're following [4] but slightly rephrased as in [5], [6]. Consider then the following quite specific fifth-order differential equation: \begin{equation}\label{2} \left(\frac{d^2y}{dx^2}\right)^{\!2}\frac{d^5y}{dx^5}+\frac{40}{9}\left(\frac{d^3y}{dx^3}\right)^{\!3}-5\frac{d^2y}{dx^2}\frac{d^3y}{dx^3}\frac{d^4y}{dx^4}=0.\;\;\;\;\;\;\;\;(2) \end{equation} Of course this can be rearranged to look like (1) but I've written it like this to avoid denominators. It isn't too hard to solve (2), especially if you notice first that it is equivalent to \[\frac{d^3}{dx^3}\left[\left(\frac{d^2y}{dx^2}\right)^{\!-2/3}\right]=0.\] One rapidly finds that the general solution can be written implicitly as \begin{equation}\label{3}(\begin{array}{ccc} x & y & 1\\ \end{array})\left(\begin{array}{ccc} a_1&a_2&a_3\\ a_2&a_4&a_5\\ a_3&a_5&a_6\\ \end{array}\right)\left(\begin{array}{c}x\\ y\\ 1\\ \end{array}\right)=0, \;\;\;\;\;\;\;\;(3)\end{equation} where $a_1,\ldots,a_6$ are constants (there are six of these and we expect the solution to depend on only five constants, but there is freedom to rescale all six by yet another constant, which reduces the six to an effective five). For convenience we'll allow all quantities $x,y,a_1,\ldots,a_6$ to be complex now, and then (3) can be interpreted as a conic in the complex projective space ${\mathbb{CP}}^2$ (this is like a familiar conic in two-dimensions but complexified and with points added 'at infinity'). Thus $M$, the moduli space of solutions of the fifth-order differential equation (2) can be regarded as the five-complex-dimensional space of such conics. This $M$ is a symmetric space and as such has a metric satisfying the Einstein equations in this dimension. We can say a bit more about this metric: a vector at a point $p\in M$ can be thought of as two infinitesimally-separated conics (one at each end of the vector, thinking of the vector as a tiny arrow), and two conics meet in four points (at least in the complex and with points at infinity included). Thus a vector tangent to $M$ factorises into a product of four two-component 'vectors' (which are in fact spinors). In an index notation a vector $V$ can be written in terms of its components as \[V^a=V^{ABCD}=\alpha^{(A}\beta^B\gamma^C\delta^{D)},\] where $a$ runs from 0 to 4, $A,B,C,\ldots$ from 0 to 1, $\alpha,\beta,\gamma,\delta$ correspond to the four intersections of the infinitesimally-separated conics, and the round brackets imply symmetrisation of the indices contained. In a corresponding way the metric or Levi-Civita covariant derivative factorises: \[\nabla_a=\nabla_{ABCD}.\]Consequently the (linear, second-order) differential equation \begin{equation}\label{4}\nabla_{PQ(AB}\nabla_{CD)}^{\;\;\;\;PQ}\Phi=\lambda\nabla_{ABCD}\Phi,\;\;\;\;\;\;\;\;(4)\end{equation} for a scalar function $\Phi$ on $M$ and a constant $\lambda$ makes sense (the spinor indices are raised or lowered with the spinor 'metric' $\epsilon_{AB}$ or $\epsilon^{AB}$ from which the metric of $M$ can be constructed).

Now here comes the rabbit out of the hat: one fixes a specific $\lambda$ and chooses a solution $\Phi$ of (4) with a non-degeneracy property \begin{equation}\label{5}\Phi_{ABCD}\Phi^{AB}_{\;\;\;PQ}\Phi^{CDPQ}\neq0\mbox{ on }\Phi=0,\;\;\;\;\;\;\;\;(5)\end{equation} (and here $\Phi_{ABCD}=\nabla_{ABCD}\Phi$); we are going to define a new metric on the four-dimensional surface $N=\{\Phi=0\}\subset M$ and we accomplish this by choosing two linearly independent solutions $h_{(1)}^{ABC},h_{(2)}^{ABC}$ of \[\Phi_{ABCD}h_{(i)}^{BCD}=0\mbox{ on }\Phi=0,\] an equation which has a two-dimensional vector space of solutions by virtue of the nondegeneracy condition (5) that we've imposed; then one carefully chooses a linearly independent pair of spinors (or spinor dyad) $(o^A,\iota^A)$ for $M$ and constructs the set of four linearly-independent vectors (or tetrad) \[e_1=o^{(A}h_{(1)}^{BCD)},\;e_2=\iota^{(A}h_{(1)}^{BCD)},\;e_3=o^{(A}h_{(2)}^{BCD)},\;e_4=\iota^{(A}h_{(2)}^{BCD)};\] this is a tetrad of vectors tangent to $N$ and determines a metric on $N$ by \[g=e_1\odot e_4-e_2\odot e_3.\] The rabbit is that this metric automatically has special curvature, in fact it has anti-self-dual Weyl curvature, and a bit more besides: see [5] for details.

For more on Paul's work click here.

[1] E. Cartan, Sur une classe d'espaces de Weyl, Annales scientifiques de l'E.N.S. 3e série, tome 60 (1943), 1-16

[2] N.J.Hitchin, Complex manifolds and Einstein's equations. Twistor geometry and nonlinear systems (Primorsko, 1980), 73-99, Lecture Notes in Math., 970, Springer, Berlin-New York, 1982.

[3] P.Tod, Einstein-Weyl spaces and third-order differential equations. J. Math. Phys. 41 (2000) 5572-5581.

[4] D Moraru, A new construction of anti-self-dual four-manifolds. Ann. Glob. Anal. Geom. 38, (2010), 77-92.

[5] M. Dunajski and P. Tod, Conics, Twistors, and anti-self-dual tri-Kahler metrics

[6] M. Dunajski and P. Tod, An example of the geometry of a 5th-order ODE: the metric on the space of conics 

Thursday, 25 October 2018

An Introduction to Complex Numbers - Oxford Mathematics first year student lecture now online for the first time

Much is written about life as an undergraduate at Oxford but what is it really like? As Oxford Mathematics's new first-year students arrive (273 of them, comprising 33 nationalities) we thought we would take the opportunity to go behind the scenes and share some of their experiences.

Our starting point is a first week lecture. In this case the second lecture from 'An Introduction to Complex Numbers' by Dr. Vicky Neale. Whether you are a past student, an aspiring student or just curious as to how teaching works, come and take a seat. We have already featured snippets from the lecture on social media where comments have ranged from a debate about whiteboards to discussions abut standards. However, there has also been appreciation of the fact that we are giving an insight in to a system that is sometimes seem as unnecessarily mysterious. In fact there is no mystery, just an opportunity to see how we present the subject and how that differs from the school experience, as much in presentation as content though of course that stiffens as the weeks go by.

So take your seat and let us know what you think.



Thursday, 25 October 2018

Bach and the Cosmos - James Sparks and City of London Sinfonia. Oxford Mathematics Public Lecture now online

According to John Eliot Gardiner in his biography of Johann Sebastian Bach, nothing in Bach's rigid Lutheran schooling explains the scientific precision of his work. However, that precision has attracted scientists and mathematicians in particular to the composer's work, not least as its search for structure and beauty seems to chime with their own approach to their subject.

In this Oxford Mathematics Public Lecture Oxford Mathematician James Sparks, himself a former organ scholar at Selwyn College Cambridge, demonstrates just how explicit Bach's mathematical framing is and City of London Sinfonia elucidate with excerpts from the Goldberg Variations. This was one of our most successful Public Lectures, an evening where the Sciences and the Humanities really were in harmony.

Please note this film does not include the full concert performance of the Goldberg Variations.




Wednesday, 24 October 2018

How to sweep up your fusion reaction

Fusion energy may hold the key to a sustainable future of electricity production. However some technical stumbling blocks remain to be overcome. One central challenge of the fusion enterprise is how to effectively withstand the high heat load emanating from the core plasma. Even the sturdiest solid solutions suffer damage over time, which could be avoided by adding a thin liquid coating. This thin liquid-metal layer must be maintained in the presence of the strong magnetic field needed to confine the core plasma and under the influence of the hot particles that escape the confinement and strike the liquid. To reduce the effective heat load it has been proposed to sweep these incoming particles up and down the liquid layer. However it is unclear how this sweeping motion will affect the liquid.

In this work Oxford Mathematicians Davin Lunz and Peter Howell in collaboration with Tokamak Energy examine how the sweeping motion affects the deflections of the liquid layer. They find that there are effectively three regimes: one in which the sweeping motion's influence is neutral as it has little effect on the deflections, one in which the deflections are minimised (that is a positive outcome in the fusion context as the liquid provides a more uniform covering), and one in which the deflections are dangerously amplified (that is a negative outcome in this context, as large deflections can leave the underlying solid exposed and result in liquid particles detaching from the layer and impurities reaching the core plasma). To uncover this, they focus on the (appropriately normalised) governing equation, namely, \begin{align} \frac{\partial h}{\partial t} + \frac \partial{\partial x} \left[ Q(h) \left( 1 - \frac{\partial p}{\partial x} + \frac{\partial ^3 h}{\partial x^3} \right) \right] = 0, \end{align} where $x$ is a horizontal coordinate, $t$ denotes time, $h(x,t)$ is the liquid deflection, $p(x,t)$ is the oscillating pressure exerted on the layer from the impinging plasma, and $Q$ is a flux function. This is a thin film equation where the fluid is driven by gravity and the applied pressure while being modulated by surface tension. 

One key observation is that there are two independent time scales in the problem: the first is the time scale at which the surface equilibrates, and the second is the time scale at which the pressure completes one oscillation. Their work shows that if the pressure time scale is much longer than the deflection time scale (that is, if the sweeping is sufficiently slow) then the deflections are largely unaffected by the motion. In the opposite case - the moving pressure time scale is much shorter than the deflection time scale, that is, the sweeping motion is very fast - the load of the impacting particles is effectively averaged out and this more even distribution minimises deflections. When the two time scales are of similar order, the pressure can oscillate at a speed similar to the natural speed that deflections propagate along the free surface. This has the effect of dangerously amplifying the free-surface deflections and should be avoided in the context of confined fusion.

For more detali on the research please click here.

Monday, 22 October 2018

D-modules on rigid analytic spaces - where algebra and geometry meet number theory

Oxford Mathematician Andreas Bode talks about his work in representation theory and its lesson for the interconnectness of mathematics.

"In many cases, the borders between different areas of mathematical research are much less rigid, much more permeable than one might make out from a modular curriculum. A number theorist might borrow tools from analysis and adapt them for their purposes, a geometer might discover that structures from theoretical physics offer a more intuitive explaination for an interesting phenomenon. These connections, intersections and border-crossings are exciting not just because they are useful (having a more varied arsenal of tools can help you to prove more things), but primarily because these argumentative movements carry enormous potential for insight. A sudden shift of perspective, a reinterpretation of the original question in terms of another field can solve a problem by transforming it, crucially enriching how we view our objects of study.

Representation theory is the study of algebraic systems by way of their actions on other objects: if we consider a group $G$, we might ask how we can represent group elements by automorphisms of some finite-dimensional vector space (and thus by matrices), i.e. we study homomorphisms $G\to \text{GL}_n(\mathbb{C})$. In the same way, if $A$ is an algebra over some field $k$, an action of $A$ on some $k$-vector space $V$ is given by an algebra morphism $A\to \text{End}_k(V)$. The slogan is always the same: we understand groups, rings, algebras, ... by seeing what they do to other objects (usually vector spaces).

While a representation is by definition an algebraic notion, it turns out that many of the most common and interesting groups, rings, algebras, ... also have a very rich geometric structure - and this can often help us to get a handle on their representations. For example, matrix groups like $\text{GL}_n$, $\text{SL}_n$, $\text{SO}_n$ can be viewed as differentiable manifolds, or as algebraic varieties in algebraic geometry. They act on various other varieties in natural ways, and standard geometric procedures (cohomology) then prove to be a rich source of representations. Exploiting these geometric structures in representation theory is what we call, not surprisingly, geometric representation theory.

Here is an example: For groups $G$ like $\text{GL}_n(\mathbb{C})$, we can study the representation theory of $G$ by its 'linear approximation', the associated Lie algebra $\mathfrak{g}$. This is competely analogous to how we might approximate a function by the first terms in its Taylor expansion, or some geometric object by its tangent space (in fact, to say that this is an analogy is an understatement).

One can then show that representations of $\mathfrak{g}$ can be (essentially) identified with a category of geometric objects: $\mathcal{D}$-modules on the flag variety associated to $G$ (we ignore certain twists here). This equivalence, originally due to Beilinson-Bernstein and Brylinski-Kashiwara, is important not because it is an equivalence (mathematics is not an exercise in expressing the same thing in as many different ways as possible), but because it is a particularly illuminating equivalence. Using the language of $\mathcal{D}$-modules and viewing representations through this prism, we can translate some really hard algebraic questions into relatively straightforward geometric ones! The proof of the Kazhdan-Lusztig conjectures, describing the character of simple $\mathfrak{g}$-representations, is a prime example: Replace $\mathfrak{g}$-representations by $\mathcal{D}$-modules, replace $\mathcal{D}$-modules by perverse sheaves via the Riemann-Hilbert correspondence (see below), and everything follows relatively explicitly from the geometry of the flag variety.

So what is a $\mathcal{D}$-module? It is a module over the sheaf of differential operators $\mathcal{D}_X$ on a variety $X$. If you know about $\mathcal{O}$-modules (modules over the sheaf of functions), you can think of $\mathcal{D}$-modules as a non-commutative, differential analogue. Instead of dwelling on a more formal definition, two points are worth making here: Firstly, there are conceptual similarities between $\mathcal{D}_X$ and the enveloping algebra $U(\mathfrak{g})$ of a Lie algebra (both clearly encode infinitesimal, tangential information, the tangent sheaf on $X$ playing very much the same role as $\mathfrak{g}$ in $U(\mathfrak{g})$). In fact, the globally defined differential operators on the flag variety are isomorphic to a quotient of $U(\mathfrak{g})$ so the Beilinson-Bernstein equivalence we mentioned is not an abstract identification, but displays an intrinsic connection between the two concepts. Secondly, the term 'differential operators' rightly suggests a connection to differential equations on $X$: given a system of differential equations, we can form an associated $\mathcal{D}$-module, and there is even an operation of 'solving' this $\mathcal{D}$-module by taking what is called the solution complex. If the differential equation is sufficiently nice, we end up with rather special complexes called perverse sheaves. This is what the Riemann-Hilbert correspondence mentioned above is about.

Thus the theory of $\mathcal{D}_X$-modules establishes valuable connections between algebraic, representation-theoretic questions and much more geometric objects, encoding subtle topological information of the space $X$ (in this case the flag variety). We now also bring number theory into the mix. One main aim of my research is to study $\mathcal{D}$-modules in the setting of nonarchimedean analytic geometry, using what are called rigid analytic varieties (developed by Tate). Can we understand how $\text{GL}_n(\mathbb{Q}_p)$ acts on $\mathbb{Q}_p$-vector spaces (imposing suitable restrictions of continuity or analyticity) by considering $\mathcal{D}$-modules on a rigid analytic flag variety? Can we use this geometric picture to make further progress towards a $p$-adic local Langlands correspondence (which is so far only settled for $\text{GL}_2(\mathbb{Q}_p)$)? What can we say about the relation between $\mathcal{D}$-modules in this $p$-adic setting and $p$-adic differential equations? What form would a $p$-adic Riemann-Hilbert correspondence take?

Konstantin Ardakov, with whom I am working here in Oxford, has initiated this project together with Simon Wadsley, by establishing a well-behaved notion of $\stackrel{\frown}{\mathcal{D}}$-modules on rigid analytic spaces and proving an analogue of the Beilinson-Bernstein equivalence. So far, much of my own work has concentrated on parallels between this new $p$-adic picture and the classical setting: just as $\mathcal{D}$ shares many features with a polynomial algebra, so $\stackrel{\frown}{\mathcal{D}}$ is rather similiar to the algebra of analytic functions on a vector space (but of course non-commutative). Moreover, one can prove a finiteness result for the pushforward along proper morphisms, heavily inspired both by the complex and the commutative non-archimedean case (Kiehl's Proper Mapping Theorem). As Ardakov's equivalence can be realized as a pushforward from the flag variety to a point, this generalizes some of the previous results and places them in a broader geometric context.

But this is not the end of the story. In particular when it comes to regular holonomic $\mathcal{D}$-modules (corresponding to the 'nice' differential equations in the classical picture), recent work suggests that the subtleties of nonarchimedean analysis allow for rather unexpected behaviour, making the theory both more complicated and potentially richer than in the complex case. For example, even for innocent equations like $xf'=\lambda f$ for some scalar $\lambda$, the behaviour of the associated $\stackrel{\frown}{\mathcal{D}}$-module on a punctured $p$-adic disc depends crucially on certain arithmetic properties of the parameter $\lambda$. Much more work is needed to fully develop a language of $\mathcal{D}$-modules as powerful as in the classical case, and to let their flexibility bear on questions from representation theory, number theory and $p$-adic geometry alike."

For more detailed discussion of the work please click here and here.

Wednesday, 17 October 2018

Categorification and Quantum Field Theories

Oxford Mathematician Elena Gal talks about her recently published research.

"Categorification is an area of pure mathematics that attempts to uncover additional structure hidden in existing mathematical objects. A simplest example is replacing a natural number $n$ by a set with $n$ elements: when we attempt to prove a numerical equation, we can think about it in terms of sets, subsets and one-to-one correspondences rather than just use algebraic manipulations. Indeed this is the natural way to think about the natural numbers - this is how children first grasp rules of arithmetic by counting stones, coins or sea shells. In modern mathematics we often encounter far more complicated objects which intuitively we suspect to be "shadows" of some richer structure - but we don't immediately know what this structure is or how to construct or describe it. Uncovering such a structure gives us access to hidden symmetries of known mathematical entities. There is reasonable hope that this will eventually enrich our understanding of the physical world. Indeed the object our present research is concerned with originates in quantum physics.

The term "categorification" was introduced about 15 years ago by Crane and Frenkel in an attempt to construct an example of 4-dimensional Topological Quantum Field Theory (TQFT for short). TQFTs are a fascinating example of the interaction between modern physics and mathematics. The concept of quantum field theory was gradually developed by theoretical physicists to describe particle behavior. It was later understood that it can also be used to classify intrinsic properties of geometric objects. The state of the art for now is that mathematicians have a way of constructing TQFTs in 3 dimensions, but not higher. The key to rigorous mathematical constructions of 4-dimensional TQFTs is thought to lay in categorification.

The kind of structure that sets (rather than numbers!) form is called a category. This concept was first introduced in 1945 by Eilenberg and Mac Lane. It is now ubiquitous in pure mathematics and appears to find its way into more "applied sciences", like computer science and biology. A category is a collection of objects and arrows between them, satisfying certain simple axioms. For example the objects of the category $\mathbf{Sets}$ are sets and the arrows are the maps between them. The arrows constitute the additional level of information that we obtain by considering $\mathbf{Sets}$ instead of numbers.

Suppose we are given two sets: $A$ with $n$ elements and $B$ with $m$ elements - what is the number of different ways to combine them into one set within the category Sets? We must take a set with $n+m$  elements $C$ and count the number of arrows $A \hookrightarrow C$ that realize $A$ as a subset of $C$. The number of such arrows is given by a binomial coefficient $n+m\choose n$. We can now consider a new operation $\diamond$ given by the rule $n\diamond m = {n+m\choose m}(n+m)$. In this way given a sufficiently nice category $\mathcal{C}$ we can define a mathematical object called the Hall algebra of $\mathcal{C}$. Remarkably, if instead of the category $\mathbf{Sets}$ we consider the category $\mathbf{Vect}$ of vector spaces, we obtain a quantization of this operation where the binomial coefficient $n+m\choose m$ is replaced by its deformation with a parameter $q$. The result is the simplest example of a mathematical object called a quantum group. Quantum groups were introduced by Drinfeld and Jimbo 30 years ago, and as it later turned out they (or more precisely the symmetries of vector spaces that they encode their categories of representations) are precisely what is needed to construct 3-dimensional TQFTs. The upshot is that to move one dimension up we need to categorify Hall algebras.

In our joint project Adam GalKobi Kremnitzer and I use a geometric approach to quantum groups pioneered by Lusztig. The categorification of quantum groups is given by performing the Hall algebra construction we saw above for mathematical objects called sheaves. Using sheaves allows us to remember more information about the category. In our recent work we define an algebra structure on the dg-category of constructible sheaves. This recovers the existing work on categorification of quantum groups by Khovanov, Lauda and Rouquier in a different language. The advantage of our approach is that it incorporates the categorification of the "dual" algebra structure that every Hall algebra has. This should lead to an understanding of the class of symmetries that it generates (in mathematical language, to the construction of the monoidal category of its representations). 

This is the key step to constructing 4-dimensional TQFTs. For more details and updates please click here."

This work is funded by an Engineering and Physical Sciences Research Council (EPSRC) grant.