Past

In micro-fluidics, at small scales where inertial effects become negligible, surface to volume ratios are large and the interfacial processes are extremely important for the overall dynamics. Integral
equation based methods are attractive for the simulations of e.g. droplet-based microfluidics, with tiny water drops dispersed in oil, stabilized by surfactants. In boundary integral formulations for
Stokes flow, jumps in pressure and velocity gradients are naturally taken care of, viscosity ratios enter only in coefficients of the equations, and only the drop surfaces must be discretized and not the volume inside nor in between.

We present numerical methods for drops with insoluble surfactants, both in two and three dimensions. We discretize the integral equations using Nyström methods, and special care is taken in the evaluation of singular and also nearly singular integrals that is needed in the case of close drop interactions. A spectral method is used to solve the advection-diffusion equation on each drop surface that describes the evolution of surfactant concentration. The drop velocity and surfactant concentration couple together through an equation of state for the surface tension coefficient. An adaptive time-stepping strategy is developed for the coupled problem, with the constraint to minimize the number of Stokes solves, since this is the computationally most expensive part.

For high quality discretization of the drops throughout the simulations, a hybrid method is used in two dimensions, offering an arc-length parameterization of the interface. In three dimensions, a
reparameterization procedure is developed to optimize the spherical harmonics representation of the drop, while conserving the drop volume and amount of surfactant.

We present results from some validation tests and illustrate the ability of the numerical methods in different challenging problems.

The basic mathematical models that describe the behavior of fluid flows date back to the eighteenth century, and yet many phenomena observed in experiments are far from being well understood from a theoretical viewpoint. For instance, especially challenging is the study of fundamental stability mechanisms when weak dissipative forces (generated, for example, by molecular friction) interact with advection processes, such as mixing and stirring. The goal of this talk is to have an overview on recent results on a variety of aspects related to hydrodynamic stability, such as the stability of vortices and laminar flows, the enhancement of dissipative force via mixing, and the statistical description of turbulent flows.

We will discuss topological and algebraic aspects of splittings of free groups. In particular we will look at the core of two splittings in terms of CAT(0) cube complexes and systems of surfaces in a doubled handlebody.

Multivariate cryptography is one of a handful of proposals for post-quantum cryptographic schemes, i.e. cryptographic schemes that are secure also against attacks carried on with a quantum computer. Their security relies on the assumption that solving a system of multivariate (quadratic) equations over a finite field is computationally hard. 

Groebner bases allow us to solve systems of polynomial equations. Therefore, one of the key questions in assessing the robustness of multivariate cryptosystems is estimating how long it takes to compute the Groebner basis of a given system of polynomial equations. 

After introducing multivariate cryptography and Groebner bases, I will present a rigorous method to estimate the complexity of computing a Groebner basis. This approach is based on techniques from commutative algebra and is joint work with Alessio Caminata (University of Barcelona).

About fifteen years ago, Thomas Scanlon and I gave a description of sets that arise as the intersection of a subvariety with a finitely generated subgroup inside a semiabelian variety over a finite field. Inspired by later work of Derksen on the positive characteristic Skolem-Mahler-Lech theorem, which turns out to be a special case, Jason Bell and I have recently recast those results in terms of finite automata. I will report on this work, as well as on the work-in-progress it has engendered, also with Bell, on an effective version of the isotrivial Mordell-Lang theorem.

Starting from the seminal paper of Caporaso-Harris-Mazur, it has been proved that if Lang's Conjecture holds in arbitrary dimension, then it implies a uniform bound for the number of rational points in a curve of general type and analogue results in higher dimensions. In joint work with Kenny Ascher we prove analogue statements for integral points (or more specifically stably-integral points) on curves of log general type and we extend these to higher dimensions. The techniques rely on very recent developments in the theory of moduli spaces for stable pairs, a higher dimensional analogue of pointed stable curves.
If time permits we will discuss how very interesting problems arise in dimension 2 that are related to the geometry of the log-cotangent bundle.

Please note that this seminar will take place at the Physical and Theoretical Chemistry Laboratory within the
Department of Chemistry, room, PTCL lecture theatre.

We give an alternative, statistical physics based proof of the Ω(d2^{-d}) lower bound for the maximum sphere packing density in dimension d by showing that a random configuration from the hard sphere model has this density in expectation. While the leading constant we achieve is not the best known, we do obtain additional geometric information: we prove a lower bound on the entropy density of sphere packings at this density, a measure of how plentiful such packings are. This is joint work with Felix Joos and Will Perkins.

Within the wide field of sparse approximation, convolutional sparse coding (CSC) has gained considerable attention in the computer vision and machine learning communities. While several works have been devoted to the practical aspects of this model, a systematic theoretical understanding of CSC seems to have been left aside. In this talk, I will present a novel analysis of the CSC problem based on the observation that, while being global, this model can be characterized and analyzed locally. By imposing only local sparsity conditions, we show that uniqueness of solutions, stability to noise contamination and success of pursuit algorithms are globally guaranteed. I will then present a Multi-Layer extension of this model and show its close relation to Convolutional Neural Networks (CNNs). This connection brings a fresh view to CNNs, as one can attribute to this architecture theoretical claims under local sparse assumptions, which shed light on ways of improving the design and implementation of these networks. Last, but not least, we will derive a learning algorithm for this model and demonstrate its applicability in unsupervised settings.

In this talk, we revisit the cubic regularization (CR) method for solving smooth non-convex optimization problems and study its local convergence behaviour. In their seminal paper, Nesterov and Polyak showed that the sequence of iterates of the CR method converges quadratically a local minimum under a non-degeneracy assumption, which implies that the local minimum is isolated. However, many optimization problems from applications such as phase retrieval and low-rank matrix recovery have non-isolated local minima. In the absence of the non-degeneracy assumption, the result was downgraded to the superlinear convergence of function values. In particular, they showed that the sequence of function values enjoys a superlinear convergence of order 4/3 (resp. 3/2) if the function is gradient dominated (resp. star-convex and globally non-degenerate). To remedy the situation, we propose a unified local error bound (EB) condition and show that the sequence of iterates of the CR method converges quadratically a local minimum under the EB condition. Furthermore, we prove that the EB condition holds if the function is gradient dominated or if it is star-convex and globally non-degenerate, thus improving the results of Nesterov and Polyak in three aspects: weaker assumption, faster rate and iterate instead of function value convergence. Finally, we apply our results to two concrete non-convex optimization problems that arise from phase retrieval and low-rank matrix recovery. For both problems, we prove that with overwhelming probability, the local EB condition is satisfied and the CR method converges quadratically to a global optimizer. We also present some numerical results on these two problems.

We study how the synchronizability of a diffusive network increases (or decreases) when we add some links in its underlying graph. This is of interest in the context of power grids where people want to prevent from having blackouts, or for neural networks where synchronization is responsible of many diseases such as Parkinson. Based on spectral properties for Laplacian matrices, we show some classification results obtained (with Tiago Pereira and Philipp Pade) with respect to the effects of these links.
 

Hardy's axiomatic approach to quantum theory revealed that just one axiom
distinguishes quantum theory from classical probability theory: there should
be continuous reversible transformations between any pair of pure states. It
is the single word `continuous' that gives rise to quantum theory. This
raises the question: Does there exist a finite theory of quantum physics
(FTQP) which can replicate the tested predictions of quantum theory to
experimental accuracy? Here we show that an FTQP based on complex Hilbert
vectors with rational squared amplitudes and rational phase angles is
possible providing the metric of state space is based on p-adic rather than
Euclidean distance. A key number-theoretic result that accounts for the
Uncertainty Principle in this FTQP is the general incommensurateness between
rational $\phi$ and rational $\cos \phi$. As such, what is often referred to
as quantum `weirdness' is simply a manifestation of such number-theoretic
incommensurateness. By contrast, we mostly perceive the world as classical
because such incommensurateness plays no role in day-to-day physics, and
hence we can treat $\phi$ (and hence $\cos \phi$) as if it were a continuum
variable. As such, in this FTQP there are two incommensurate Schr\"{o}dinger
equations based on the rational differential calculus: one for rational
$\phi$ and one for rational $\cos \phi$. Each of these individually has a

simple probabilistic interpretation - it is their merger into one equation
on the complex continuum that has led to such problems over the years. Based
on this splitting of the Schr\"{o}dinger equation, the measurement problem
is trivially solved in terms of a nonlinear clustering of states on $I_U$.
Overall these results suggest we should consider the universe as a causal
deterministic system evolving on a finite fractal-like invariant set $I_U$
in state space, and that the laws of physics in space-time derive from the
geometry of $I_U$. It is claimed that such a  deterministic causal FTQP will
be much easier to synthesise with general relativity theory than is quantum
theory.

We discuss mathematical questions that play a fundamental role in quantitative analysis of incompressible viscous fluids and other incompressible media. Reliable verification of the quality of approximate solutions requires explicit and computable estimates of the distance to the corresponding generalized solution. In the context of this problem, one of the most essential questions is how to estimate the distance (measured in terms of the gradient norm) to the set of divergence free fields. It is closely related to the so-called inf-sup (LBB) condition or stability lemma for the Stokes problem and requires estimates of the LBB constant. We discuss methods of getting computable bounds of the constant and espective estimates of the distance to exact solutions of the Stokes, generalized Oseen, and Navier-Stokes problems.

One of the fundamental themes of geometric group theory is to
view finitely generated groups as geometric objects in their own right,
and to then understand to what extent the geometry of a group determines
its algebra. A theorem of Stallings says that a finitely generated group
has more than one end if and only if it splits over a finite subgroup.
In this talk, I will explain an analogous geometric characterisation of
when a group admits a splitting over certain classes of infinite subgroups.

The study of the Geometry of random nodal domains has attracted a lot of attention in the recent past, in particular due to their connection with famous conjectures such as Yau's conjecture on the nodal volume of eigenfunctions of the Laplacian on compact manifolds, and Berry's conjecture on the relation between the geometry of the nodal sets associated to these eigenfunctions and the geometry of the nodal sets associated to toric random waves.

At first, the randomness involved in the definition of random nodal domains is often chosen of Gaussian nature. This allows in particular the use of explicit techniques, such as Kac--Rice formula, to derive the asymptotics of many observables of interest (nodal volume, number of connected components, Leray's measure etc.). In this talk, we will raise the question of the universality of these asymptotics, which consists in deciding if the asymptotic properties of random nodal domains do or do not depend on the particular nature of the randomness involved. Among other results, we will establish the local and global universality of the asymptotic volume associated to the set of real zeros of random trigonometric polynomials with high degree.

(Joint work with Siva Athreya & Leonid Mytnik).

It is well known from the literature that ordinary differential equations (ODEs) regularize in the presence of noise. Even if an ODE is “very bad” and has no solutions (or has multiple solutions), then the addition of a random noise leads almost surely to a “nice” ODE with a unique solution. The first part of the talk will be devoted to SDEs with distributional drift driven by alpha-stable noise. These equations are not well-posed in the classical sense. We define a natural notion of a solution to this equation and show its existence and uniqueness whenever the drift belongs to a certain negative Besov space. This generalizes results of E. Priola (2012) and extends to the context of stable processes the classical results of A. Zvonkin (1974) as well as the more recent results of R. Bass and Z.-Q. Chen (2001).

In the second part of the talk we investigate the same phenomenon for a 1D heat equation with an irregular drift. We prove existence and uniqueness of the flow of solutions and, as a byproduct of our proof, we also establish path-by-path uniqueness. This extends recent results of A. Davie (2007) to the context of stochastic partial differential equations.

[1] O. Butkovsky, L. Mytnik (2016). Regularization by noise and flows of solutions for a stochastic heat equation. arXiv 1610.02553. To appear in Annal. Probab.

[2] S. Athreya, O. Butkovsky, L. Mytnik (2018). Strong existence and uniqueness for stable stochastic differential equations with distributional drift. arXiv 1801.03473.

I will talk about recent work, joint with M. Gröchenig and D. Wyss, on two related results involving the cohomology of moduli spaces of Higgs bundles. The first is a positive answer to a conjecture of Hausel and Thaddeus which predicts the equality of suitably defined Hodge numbers of moduli spaces of Higgs bundles with SL(n)- and PGL(n)-structure. The second is a new proof of Ngô's geometric stabilization theorem which appears in the proof of the fundamental lemma. I will give an introduction to these theorems and outline our argument, which, inspired by work of Batyrev, proceeds by comparing the number of points of these moduli spaces over finite fields via p-adic integration.

We study the spectrum of local operators living on a defect in a generic conformal field theory, and their coupling to the local bulk operators. We establish the existence of universal accumulation points in the spectrum at large s, s being the charge of the operators under rotations in the space transverse to the defect. Our main result is a formula that inverts the bulk to defect OPE, analogous to the Caron-Huot formula for the four-point function of CFTs without defects.

Yaolong Cao: Gauge Theories on Geometric Spaces
In this talk, I will very briefly discuss gauge theories on various geometric spaces, including Riemann surfaces, 4-manifolds and manifolds with special or exceptional holonomy. More details on Calabi-Yau 4-folds will be mentioned, which are related to my research interests.

Doireann O'Kiely: Dynamic Wrinkling of Elastic Sheets
Our lives contain many scenarios in which thin structures wrinkle: a piece of tin foil or cling film crumples in our hand, and creases form in our skin as we age. In this talk I will discuss experimental and theoretical work by researchers in the Mathematical Institute on wrinkling of elastic sheets.
We study the impact of a solid onto an elastic sheet floating at a liquid-air interface. We observe a wave that is reminiscent of the ripples caused by dropping a stone in a pond, as well as spoke-like wrinkles, whose wavelength evolves in time. We describe these phenomena using a combination of asymptotic analysis, numerical simulations and scaling arguments.
 

Numerical simulation provides an important contribution to the management of oil reservoirs, and the ‘reservoir simulator’ has been an essential tool for reservoir engineers since the 1970’s. I will describe the role of the ‘well model’ in reservoir simulation. Its main purpose is to determine the production and injection flows of the reservoir fluids at the surface under a variety of operating constraints, and to supply source and sink terms to the grid cells of the reservoir model.

Advances in well technology (horizontal, multilateral, and smart wells containing flow control devices) have imposed additional demands on the well model. It must allow the fluid mixture properties to vary with position in the well, and enable different fluid streams to comingle. Friction may make an important contribution to the local pressure gradient. To provide an improved representation of the physics of fluid flow, the well is discretised into a network of segments, where each segment has its own set of variables describing the multiphase flow conditions. Individual segments can be configured to represent flow control devices, accessing lookup tables or built-in correlations to determine the pressure drop across the device as a function of the flow conditions.

The ability to couple the wells to a production facility model such as a pipeline network is a crucial advantage for field development and optimization studies, particularly for offshore fields. I will conclude by comparing two techniques for combining a network model with the reservoir simulation. One method is to extend the simulator’s well model to include the network, providing a fully integrated reservoir/well/network simulation. The other method is to run the reservoir and facility models as separate simulations coupled by a ‘controller’, which periodically balances them by exchanging boundary conditions. The latter approach allows the engineer to use a choice of specialist facility simulators.

This work deals with a class of stochastic optimal control problems in the presence of state constraints. It is well known that for such problems the value function is, in general, discontinuous, and its characterisation by a Hamilton-Jacobi equation requires additional assumptions involving an interplay between the boundary of the set of constraints and the dynamics
of the controlled system. Here, we give a characterization of the epigraph of the value function without assuming the usual controllability assumptions. To this end, the stochastic optimal control problem is first translated into a state-constrained stochastic target problem. Then a level-set approach is used to describe the backward reachable sets of the new target problem. It turns out that these backward reachable sets describe the value function. The main advantage of our approach is that it allows us to easily handle the state constraints by an exact penalisation. However, the target problem involves a new state variable and a new control variable that is unbounded.
 

Serre's uniformity question concerns the possible ways the Galois group of Q can act on the p-torsion of an elliptic curve over Q. In this talk I will survey what is known about this question, and describe two recent results related to the Chabauty-Kim method. The first, which is joint work with Jennifer Balakrishnan, Steffen Muller, Jan Tuitman and Jan Vonk, completes the classification of elliptic curves over Q with split Cartan level structure. The second, which is work in progress with Samuel Le Fourn, Samir Siksek and Jan Vonk, concerns the applicability of the Chabauty-Kim method in determining the elliptic curves with non-split Cartan level structure.
 

I will explain a beautiful link between differential and algebraic geometry, called the Kempf-Ness Theorem, which says that the natural notions of "quotient spaces" in the symplectic and algebraic categories can often be identified. The result will be presented in its most general form where actions are not necessarily free and hence I will also introduce the notion of stratified spaces.

When soft ferromagnetic particles are suspended at air-water interfaces in the presence of a vertical magnetic field, dipole-dipole repulsion competes with capillary attraction such that 2d structures self-assemble. The complex arrangements of such floating bodies are emphasized. The equilibrium distance between particles exhibits hysteresis when the applied magnetic field is modified. Irreversible processes are evidenced. By adding a horizontal and oscillating magnetic field, periodic deformations of the assembly are induced. We show herein that collective particle motions induce locomotion at low Reynolds number. The physical mechanisms and geometrical ingredients behind this cooperative locomotion are identified. These physical mechanisms can be exploited to much smaller scales, offering the possibility to create artificial and versatile microscopic swimmers.

Moreover, we show that it is possible to generate complex structures that are able to capture particles, perform cargo transport, fluid mixing, etc.

An extension of the expected shortfall as well as the value at risk to
model uncertainty has been proposed by P. Shige.
In this talk we will present a systematic extension of the general
class of optimized certainty equivalent that includes the expected
shortfall.
We show that its representation can be simplified in many cases for
efficient computations.
In particular we present some result based on a probability model
uncertainty derived from some Wasserstein metric and provide explicit
solution for it.
We further study the duality and representation of them.

This talk is based on a joint work with Daniel Bartlxe and Ludovic
Tangpi

The Boltzmann equation is a well-studied PDE that describes the statistical evolution of a dilute gas of spherical particles. However, much less is known — both from the physical and mathematical viewpoints — about the Boltzmann equation for non-spherical particles. In this talk, we present some new results on the non-existence and non-uniqueness of weak solutions to the initial-boundary value problem for N non-spherical particles which have importance for the Boltzmann equation.

We present work which was done jointly with L. Saint-Raymond (ENS Lyon), and also with P. Palffy-Muhoray (Kent State), E. Virga (Pavia) and X. Zheng (Kent State).

In this talk Michael Bonsall will explore how we can use mathematics to link between scales of organisation in biology. He will delve in to developmental biology, ecology and neurosciences, all illustrated and explored with real life examples, simple games and, of course, some neat maths.

Michael Bonsall is Professor of Mathematical Biology in Oxford.

7 February 2018, 5pm-6pm, Mathematical Institute, Oxford

Please email @email to register or watch online: https://livestream.com/oxuni/bonsall

CAT(0) spaces are defined as having triangles that are no fatter than Euclidean triangles, so it is no surprise that under special conditions  you find pieces of the Euclidean plane appearing in CAT(0) spaces. What is surprising though is how weak these special conditions seem to be. I will present some well known results of this phenomenon, along with detailed sketch proofs.

Large parts of the cryptography in use today,

key-agreement protocols and digital signatures based on the

hardness of factoring large integers or solving the

discrete-logarithm problem, are not secure against attackers

equipped with a large universal quantum computer. It is not

clear when such a large quantum computer will be built, but

continuous progress by various labs around the world suggests

that it may well be less than two decades until today's

cryptography will become insecure.

To address this issue, NIST started a public competition to

identify suitable replacements for today's cryptosystems. In

my talk, I will describe two of these systems: the

key-encapsulation mechanism Kyber and the digital signature

scheme Dilithium. Both schemes are based on the hardness of

solving problems in module lattices and they together form the

"Cryptographic Suite for Algebraic Lattices -- CRYSTALS".

Let L be a lattice in R^n and let Z in R^(m+n) a parameterized family of subsets Z_T of R^n. Starting from an old result of Davenport and using O-minimal structures, together with Martin Widmer, we proved for fairly general families Z an estimate for the number of points of L in Z_T, which is essentially best possible.
After introducing the problem and stating the result, we will present applications to counting algebraic integers of bounded height and to Manin’s Conjecture.

Let L be a lattice in R^n and let Z in R^(m+n) a parameterized family of subsets Z_T of R^n. Starting from an old result of Davenport and using O-minimal structures, together with Martin Widmer, we proved for fairly general families Z an estimate for the number of points of L in Z_T, which is essentially best possible. 
After introducing the problem and stating the result, we will present applications to counting algebraic integers of bounded height and to Manin’s Conjecture.

The question of what happens to the eigenvalues of a matrix after an additive perturbation has a long history, with notable contributions from Wilkinson, Sorensen, Golub, H\"ormander, Ipsen and Mehrmann, among many others. If the perturbed matrix $C \in \mathbb{C}^{n \times n}$ is given by $C = A + B$, these theorems typically consider the case where $A$ and/or $B$ are symmetric and $B$ has rank one. In this talk we will prove a theorem that bounds the number of distinct eigenvalues of $C$ in terms of the number of distinct eigenvalues of $A$, the diagonalisability of $A$, and the rank of $B$. This new theorem is more general in that it applies to arbitrary matrices $A$ perturbed by matrices of arbitrary rank $B$. We will also discuss various refinements of my bound recently developed by other authors.
 

It is well-known that nilpotent orbits in $\mathfrak{sl}_n(\mathbb C)$ correspond bijectively with the set of partitions of $n$, such that the closure (partial) ordering on orbits is sent to the dominance order on partitions. Taking dual partitions simply turns this poset upside down, so in type $A$ there is an order-reversing involution on the poset of nilpotent orbits. More generally, if $\mathfrak g$ is any simple Lie algebra over $\mathbb C$ then Lusztig-Spaltenstein duality is an order-reversing bijection from the set of special nilpotent orbits in $\mathfrak g$ to the set of special nilpotent orbits in the Langlands dual Lie algebra $\mathfrak g^L$.
It was observed by Kraft and Procesi that the duality in type $A$ is manifested in the geometry of the nullcone. In particular, if two orbits $\mathcal O_1<\mathcal O_2$ are adjacent in the partial order then so are their duals $\mathcal O_1^t>\mathcal O_2^t$, and the isolated singularity attached to the pair $(\mathcal O_1,\mathcal O_2)$ is dual to the singularity attached to $(\mathcal O_2^t,\mathcal O_1^t)$: a Kleinian singularity of type $A_k$ is swapped with the minimal nilpotent orbit closure in $\mathfrak{sl}_{k+1}$ (and vice-versa). Subsequent work of Kraft-Procesi determined singularities associated to such pairs in the remaining classical Lie algebras, but did not specifically touch on duality for pairs of special orbits.
In this talk, I will explain some recent joint research with Fu, Juteau and Sommers on singularities associated to pairs $\mathcal O_1<\mathcal O_2$ of (special) orbits in exceptional Lie algebras. In particular, we (almost always) observe a generalized form of duality for such singularities in any simple Lie algebra.
 

We consider a system of nonlinear partial differential equations modelling the steady motion of an incompressible non-Newtonian fluid, which is chemically reacting. The governing system consists of a steady convection-diffusion equation for the concentration and the generalized steady Navier–Stokes equations, where the viscosity coefficient is a power-law type function of the shear-rate, and the coupling between the equations results from the concentration-dependence of the power-law index. This system of nonlinear partial differential equations arises in mathematical models of the synovial fluid found in the cavities of moving joints. We construct a finite element approximation of the model and perform the mathematical analysis of the numerical method. Key technical tools include discrete counterparts of the Bogovski operator, De Giorgi’s regularity theorem and the Acerbi–Fusco Lipschitz truncation of Sobolev functions, in function spaces with variable integrability exponents.

Assortative mixing in networks is the tendency for nodes with the same attributes, or metadata, to link to each other. It is a property often found in social networks manifesting as a higher tendency of links occurring between people with the same age, race, or political belief. Quantifying the level of assortativity or disassortativity (the preference of linking to nodes with different attributes) can shed light on the factors involved in the formation of links and contagion processes in complex networks. It is common practice to measure the level of assortativity according to the assortativity coefficient, or modularity in the case of discrete-valued metadata. This global value is the average level of assortativity across the network and may not be a representative statistic when mixing patterns are heterogeneous. For example, a social network spanning the globe may exhibit local differences in mixing patterns as a consequence of differences in cultural norms. Here, we introduce an approach to localise this global measure so that we can describe the assortativity, across multiple scales, at the node level. Consequently we are able to capture and qualitatively evaluate the distribution of mixing patterns in the network. We find that for many real-world networks the distribution of assortativity is skewed, overdispersed and multimodal. Our method provides a clearer lens through which we can more closely examine mixing patterns in networks.

Link to arxiv paper:  https://arxiv.org/abs/1708.01236

I will describe how to recast perturbative quantum gravity using non-perturbative techniques from conformal field theory, focussing on the case of N=4 super Yang-Mills theory. By resolving the degeneracy among double trace operators at large N we are able to bootstrap one-loop supergravity corrections from the OPE of the CFT.
 

Numerical methods for SDEs typically use only the discretized increments of the driving Brownian motion. As one would expect, this approach is sensible and very well studied.

In addition to generating increments, it is also straightforward to generate time integrals of Brownian motion. These quantities give extra information about the Brownian path and are known to improve the strong convergence of methods for one-dimensional SDEs. Despite this, numerical methods that use time integrals alongside increments have received less attention in the literature.

In this talk, we will develop some underlying theory for these time integrals and introduce a new numerical approach to SDEs that does not require evaluating vector field derivatives. We shall also discuss the possible implications of this work for multi-dimensional SDEs.

The Z/2-equivariant Heegaard Floer cohomlogy of the double cover of S^3 along a knot, defined by Lipshitz, Hendricks, and Sarkar, 
is an isomorphism class of F_2[\theta]-modules. In this talk, we show that this invariant is natural, and is functorial under based cobordisms. 
Given a transverse knot K in the standard contact 3-sphere, we define an element of the Z/2-equivariant Heegaard Floer cohomology 
that depends only on the tranverse isotopy class of K, and is functorial under certain symplectic cobordisms.

Based on the notion of paracontrolled distributions, existence and uniqueness results are presented for rough convolution equations. In particular, this wide class of equations includes rough differential equations with possible delay, stochastic Volterra equations, and moving average equations driven by Lévy processes. The talk is based on a joint work with Mathias Trabs.

In real dimension two, the symplectic mapping class group of a surface agrees with its `classical' mapping class group, whose properties are well-understood. To what extend do these generalise to higher-dimensions? We consider specific pairs of symplectic manifolds (S, M), where S is a surface, together with collections of Lagrangian spheres in S and in M, say v_1, ...,v_k and V_1, ...,V_k, that have analogous intersection patterns, in a sense that we will make precise. Our main theorem is that any relation between the Dehn twists in the V_i must also hold between Dehn twists in the v_i. Time allowing, we will give some corollaries, such as embeddings of certain interesting groups into auto-equivalence groups of Fukaya categories.

I am interested in the moduli spaces of heterotic vacua. These are closely related to the moduli spaces of stable holomorphic bundles but in which the base and bundle vary simultaneously, together with additional constraints deriving from string theory. I will first summarise some pre-Brexit results we have derived. These include an explicit Kaehler metric and Kaehler potential for both the moduli space and its first cousin, the matter field space. I will secondly describe new, post-Brexit work in which these results are encased within an elegant geometry, which we call a universal heterotic geometry. Beyond compelling aesthetics, the framework is surprisingly useful giving both a concise derivation of our pre-Brexit results as well as some new results. 

Today, everyone knows everything: with only a quick trip through WebMD or Wikipedia, average citizens believe themselves to be on an equal intellectual footing with doctors and diplomats. All voices, even the most ridiculous, demand to be taken with equal seriousness, and any claim to the contrary is dismissed as undemocratic elitism. Tom Nichols argues that in this climate, democratic institutions themselves are in danger of falling either to populism or to technocracy- or in the worst case, a combination of both.

Tom Nichols is Professor of National Security Affairs at the US Naval War College, an adjunct professor at the Harvard Extension School, and a former aide in the U.S. Senate. His latest book is The Death of Expertise: The Campaign Against Established Knowledge and Why it Matters. This lecture is based on that book.

All welcome. No need to book.

In this talk I will discuss the upcoming REF2021 and its significance for early career researchers (research fellows and postdocs) including

• why it is so important to all UK maths departments
• why the timing of it could have important career consequences for ECRs
• publication issues such as quality versus quantity, and choice of journal
• the importance of Impact Case Studies
   

The ability of cells to sense and respond to the mechanical properties of their environments is fundamental to cellular behaviour, with stiffness found to be a key control parameter. The physical mechanisms underpinning mechanosensing are, however, not well understood. I here consider the key physical cellular behaviours of active contractility of the internal cytoskeleton and cell growth, coupling these into mechanical models. These models suggest new distinct mechanisms of mechanotransduction in cells and tissues.

This is the first part of a discussion of  Ezra Miller's paper https://arxiv.org/abs/1709.08155.