Past

Uncertainty quantification can begin by specifying the initial state of a system as a probability measure. Part of the state (the 'parameters') might not evolve, and might not be directly observable. Many inverse problems are generalisations of uncertainty quantification such that one modifies the probability measure to be consistent with measurements, a forward model and the initial measure. The inverse problem, interpreted as computing the posterior probability measure of the states, including the parameters and the variables, from a sequence of noise-corrupted observations, is reviewed in the talk. Bayesian statistics provides a natural framework for a solution but leads to very challenging computational problems, particularly when the dimension of the state space is very large, as when arising from the discretisation of a partial differential equation theory.

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In this talk we show how the Bayesian framework leads to a new algorithm - the 'Variational Smoothing Filter' - that unifies the leading techniques in use today. In particular the framework provides an interpretation and generalisation of Tikhonov regularisation, a method of forecast verification and a way of quantifying and managing uncertainty. To deal with the problem that a good initial prior may not be Gaussian, as with a general prior intended to describe, for example a geological structure, a Gaussian mixture prior is used. This has many desirable properties, including ease of sampling to make 'numerical rocks' or 'numerical weather' for visualisation purposes and statistical summaries, and in principle can approximate any probability density. Robustness is sought by combining a variational update with this full mixture representation of the conditional posterior density.

I will discuss methods for spatio-temporal modelling in cellular and molecular biology. Three classes of models will be considered: (i) microscopic (molecular-based, individual-based) models which are based on the simulation of trajectories of individual molecules and their localized interactions (for example, reactions); (ii) mesoscopic (lattice-based) models which divide the computational

domain into a finite number of compartments and simulate the time evolution of the numbers of molecules in each compartment; and (iii) macroscopic (deterministic) models which are written in terms of reaction-diffusion-advection PDEs for spatially varying concentrations. In the first part of my talk, I will discuss connections between the modelling frameworks (i)-(iii). I will consider chemical reactions both at a surface and in the bulk. In the second part of my talk, I will present hybrid (multiscale) algorithms which use models with a different level of detail in different parts of the computational domain. The main goal of this multiscale methodology is to use a detailed modelling approach in localized regions of particular interest (in which accuracy and microscopic detail is important) and a less detailed model in other regions in which accuracy may be traded for simulation efficiency. I will also discuss hybrid modelling of chemotaxis where an individual-based model of cells is coupled with PDEs for extracellular chemical signals.

We study minimizers of the Ginzburg-Landau (GL) functional \[E_\epsilon(u):=\frac{1}{2}\int_A |\nabla u|^2 + \frac{1}{4\epsilon^2} \int_A(1-|u|^2)^2\] for a complex-valued order parameter $u$ (with no magnetic field). This functional is of fundamental importance in the theory of superconductivity and superuidity; the development of these theories led to three Nobel prizes. For a $2D$ domain $A$ with holes we consider “semistiff” boundary conditions: a Dirichlet condition for the modulus $|u|$, and a homogeneous Neumann condition for the phase $\phi = \mathrm{arg}(u)$. The principal

result of this work (with V. Rybalko) is a proof of the existence of stable local minimizers with vortices (global minimizers do not exist). These vortices are novel in that they approach the boundary and have bounded energy as $\epsilon\to0$.

In contrast, in the well-studied Dirichlet (“stiff”) problem for the GL PDE, the vortices remain distant from the boundary and their energy blows up as

$\epsilon\to 0$. Also, there are no stable minimizers to the homogeneous Neumann (“soft”) problem with vortices.

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Next, we discuss more recent results (with V. Rybalko and O. Misiats) on global minimizers of the full GL functional (with magnetic field) subject to semi-stiff boundary conditions. Here, we show the existence of global minimizers with vortices for both simply and doubly connected domains and describe the location of their vortices.

The aim of this talk is to give an introduction to Geometric Invariant Theory (GIT) for reductive groups over the complex numbers. Roughly speaking, GIT is concerned with constructing quotients of group actions in the category of algebraic varieties. We begin by discussing what properties we should like quotient varieties to possess, highlighting so-called `good' and `geometric' quotients, and then turn to search for these quotients in the case of affine and projective varieties. Here we shall see that the construction runs most smoothly when we assume our group to be reductive (meaning it can be described as the complexification of a maximal compact subgroup). Finally, we hope to say something about the Hilbert-Mumford criterion regarding semi-stability and stability of points, illustrating it by constructing the rough moduli space of elliptic curves.

Joint work with: Sina Greenwood, Brian Raines and Casey Sherman

Abstract: We say a space $X$ with property $\C P$ is \emph{universal} for orbit spectra of homeomorphisms with property $\C P$ provided that if $Y$ is any space with property $\C P$ and the same cardinality as $X$ and $h:Y\to Y$ is any (auto)homeomorphism then there is a homeomorphism$g:X\to X$ such that the orbit equivalence classes for $h$ and $g$ are isomorphic. We construct a compact metric space $X$ that is universal for homeomorphisms of compact metric spaces of cardinality the continuum. There is no universal space for countable compact metric spaces. In the presence of some set theoretic assumptions we also give a separable metric space of size continuum that is universal for homeomorphisms on separable metric spaces.

Actin polymerization in vivo is regulated spatially and temporally by a web of signalling proteins. We developed detailed physico-chemical, stochastic models of lamellipodia and filopodia, which are projected by eukaryotic cells during cell migration, and contain dynamically remodelling actin meshes and bundles. In a recent work we studied how molecular motors regulate growth dynamics of elongated organelles of living cells. We determined spatial distributions of motors in such organelles, corresponding to a basic scenario when motors only walk along the substrate, bind, unbind, and diffuse. We developed a mean field model, which quantitatively reproduces elaborate stochastic simulation results as well as provides a physical interpretation of experimentally observed distributions of Myosin IIIa in stereocilia and filopodia. The mean field model showed that the jamming of the walking motors is conspicuous, and therefore damps the active motor flux. However, when the motor distributions are coupled to the delivery of actin monomers towards the tip, even the concentration bump of G-actin that they create before they jam is enough to speed up the diffusion to allow for severalfold longer filopodia. We found that the concentration profile of G-actin along the filopodium is rather non-trivial, containing a narrow minimum near the base followed by a broad maximum. For efficient enough actin transport, this non-monotonous shape is expected to occur under a broad set of conditions. We also find that the stationary motor distribution is universal for the given set of model parameters regardless of the organelle length, which follows from the form of the kinetic equations and the boundary conditions.

I’ll report on my recent work (with co-authors Holt and Ciobanu) on Artin

groups of large type, that is groups with presentations of the form

G = hx1, . . . , xn | xixjxi · · · = xjxixj · · · , 8i 3. (In fact, our results still hold when some, but not all

possible, relations with mij = 2 are allowed.)

Recently, Holt and I characterised the geodesic words in these groups, and

described an effective method to reduce any word to geodesic form. That

proves the groups shortlex automatic and gives an effective (at worst quadratic)

solution to the word problem. Using this characterisation of geodesics, Holt,

Ciobanu and I can derive the rapid decay property for most large type

groups, and hence deduce for most of these that the Baum-Connes conjec-

ture holds; this has various consequence, in particular that the Kadison-

Kaplansky conjecture holds for these groups, i.e. that the group ring CG

contains no non-trivial idempotents.

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This talk will be about various ways in which a variety can be "connected by higher genus curves", mimicking the notion of rational connectedness. At least in characteristic zero, the existence of a curve with a large deformation space of morphisms to a variety implies that the variety is in fact rationally connected. Time permitting I will discuss attempts to show this result in positive characteristic.

Cloaking recently attracts a lot of attention from the scientific community due to the progress of advanced technology. There are several ways to do cloaking. Two of them are based on transformation optics and negative index materials. Cloaking based on transformation optics was suggested by Pendry and Leonhardt using transformations which blow up a point into the cloaked regions. The same transformations had previously used by Greenleaf et al. to establish the non-uniqueness for Calderon's inverse problem. These transformations are singular and hence create a lot of difficulty in analysis and practical applications. The second method of cloaking is based on the peculiar properties of negative index materials. It was proposed by Lai et al. and inspired from the concept of complementary media due to Pendry and Ramakrishna. In this talk, I will discuss approximate cloaking using these two methods. Concerning the first one, I will consider the situation, first proposed in the work of Kohn et al., where one uses transformations which blow up a small ball (instead of a point) into cloaked regions. Many interesting issues such as finite energy and resonance will be mentioned. Concerning the second method, I provide the (first) rigorous analysis for cloaking using negative index materials by investigating the situation where the loss (damping) parameter goes to 0. I will also explain how the arguments can be used not only to establish the rigor for other interesting related phenomena using negative index materials such as superlenses and illusion optics but also to enlighten the mechanism of these phenomena.

This talk will consist of two parts. In the first part we will present a motivating application from oil reservoir simulation, namely finding the location and trajectory of an oil producing well which maximises oil production. We will show how such a problem can be tackled through the use of radial basis function (RBF) approximation (also known as Kriging or Gaussian process regression) and a branch and bound global optimization algorithm.

In the second part of the talk we will show how one can improve the branch and bound algorithm through the use of Lipschitz continuity of the RBF approximation. This leads to an entirely new global optimization algorithm for twice differentiable functions with Lipschitz continuous Hessian. The algorithm makes use of recent cubic regularisation techniques from local optimization to obtain the necessary bounds within the branch and bound algorithm.

In this talk, approximations to utility indifference prices for a contingent claim in the large position size limit are provided. Results are valid for general utility functions and semi-martingale models. It is

shown that as the position size approaches infinity, all utility functions with the same rate of decay for large negative wealths yield the same price. Practically, this means an investor should price like an exponential investor. In a sizeable class of diffusion models, the large position limit is seen to arise naturally in conjunction with the limit of a complete model and hence approximations are most appropriate in this setting.

The issue of resource management arises with any sensor which is capable either of sensing only a part of its total field of view at any one time, or which is capable of having a number of operating modes, or both.

A very simple example is a camera with a telephoto lens.  The photographer has to decide what he is going to photograph, and whether to zoom in to get high resolution on a part of the scene, or zoom out to see more of the scene.  Very similar issues apply, of course, to electro-optical sensors (visible light or infra-red 'TV' cameras) and to radars.

The subject has, perhaps, been most extensively studied in relation to multi mode/multi function radars, where approaches such as neural networks, genetic algorithms and auction mechanisms have been proposed as well as more deterministic mangement schemes, but the methods which have actually been implemented have been much more primitive.

The use of multiple, disparate, sensors on multiple mobile, especially airborne, platforms adds further degrees of freedom to the problem - an extension is of growing interest.

The presentation will briefly review the problem for both the single-sensor and the multi-platform cases, and some of the approaches which have been proposed, and will highlight the remaining current problems.

In this seminar we discuss the gas dynamics of chimneys, solar updraft towers and energy towers. The main issue is to discuss simple fluid dynamic models which still describe the main features of the mentioned applications. We focus first on one dimensional compressible models. Then we apply a small Mach number asymptotics to reduce to complexity and to avoid the known problems

of fully compressible models in the small Mach number regime. In case of the energy tower in addition we have to model the evaporation process.

Finally we obtain a much simpler fluid dynamic model which allows robust and very fast numerical simulations. We discuss the qualitative behaviour and the good agreement with expermental data (in cases such data are available).

Mirkovic-Vilonen polytopes are a combinatorial tool for studying
perfect bases for representations of semisimple Lie algebras.  They
were originally introduced using MV cycles in the affine Grassmannian,
but they are also related to the canonical basis.  I will explain how
MV polytopes can also be used to describe components of Lusztig quiver
varieties and how this allows us to generalize the theory of MV
polytopes to the affine case.

This talk will present a computationally efficient method of simulating cardiac electrical propagation using an

adaptive high-order finite element method. The refinement strategy automatically concentrates computational

effort where it is most needed in space on each time-step. We drive the adaptivity using a residual-based error

indicator, and demonstrate using norms of the error that the indicator allows to control it successfully. Our

results using two-dimensional domains of varying complexity demonstrate in that significant improvements in

efficiency are possible over the state-of-the-art, indicating that these methods should be investigated for

implementation in whole-heart scale software.

We present a variational model for the quasi-static evolution of brutal brittle damage for geometrically-linear elastic materials. We

allow for multiple damaged states. Moreover, unlike current formulations, the materials are allowed to be anisotropic and the

deformations are not restricted to anti-plane shear. The model can be formulated either energetically or through a strain threshold. We

explore the relationship between these formulations. This is joint work with Christopher Larsen, Worcester Polytechnic Institute.

Given a manifold $M$ and a basepointed labelling space $X$ the space of unordered finite configurations in $M$ with labels in $X$, $C(M;X)$ is the space of finite unordered tuples of points in $M$, each point with an associated point in $X$. The space is topologised so that particles cannot collide. Given a compact submanifold $M_0\subset M$ we define $C(M,M_0;X)$ to be the space of unordered finite configuration in which points `vanish' in $M_0$. The scanning map is a homotopy equivalence between the configuration space and a section space of a certain $\Sigma^nX$-bundle over $M$. Throughout the 70s and 80s this map has been given several unsatisfactory and convoluted definitions. A natural question to ask is whether the map is equivariant under the diffeomorphism group of the underlying manifold. However, any description of the map relies heavily on `little round $\varepsilon$-balls' and so only actions by isometry have any chance at equivariance. The goal of this talk is to give a more natural definition of the scanning map and show that diffeomorphism equivariance is an easy consequence.

I show how to construct some Fano 3-folds that have the same Hilbert series but different Betti numbers, and so lie on different components of the Hilbert scheme. I would like to show where these fit in to a speculative (indeed fantastical) geography of Fano 3-folds, and how the projection methods I use may apply to other questions in the geography.

 Higher-order transformations are ubiquitous within data management. In relational databases, higher-order queries appear in numerous aspects including query rewriting and query specification. In XML databases, higher-order functions are natural due to the close connection of XML query languages with functional programming. We investigate higher-order query languages that combine higher- order transformations with ordinary database query languages. We define higher-order query languages based on Relational Algebra and XQuery. We also study basic problems for these query languages including evaluation, containment, and type inference. We show that even though evaluating these higher-order query languages is non-elementary, there are subclasses that are polynomially reducible to evaluation for ordinary query languages.

Solitons are localised non-singular lumps of energy which describe particles non perturbatively. Finding the solitons usually involves solving nonlinear differential equations, but I shall show that in some cases the solitons emerge directly from the underlying space-time geometry: certain abelian vortices arise from surfaces of constant mean curvature in Minkowski space, and skyrmions can be constructed from the holonomy of gravitational instantons.

We describe an invariant manifold of the equations of molecular dynamics associated to a given discrete group of isometries. It is a time-dependent manifold, but its dependence on time is explicit. In the case of the translation group, it has dimension 6N, where N is an assignable positive integer. The manifold is independent of the description of the atomic forces within a general framework. Most of continuum mechanics inherits some version of this manifold, as do theories in-between molecular dynamics and continuum mechanics, even though they do not inherit the time reversibility of molecular dynamics on this manifold. The manifold implies a natural statistics of molecular motion, which suggests a simplifying ansatz for the Boltzmann equation which, in turn, leads to new explicit far-from-equilibrium solutions of this equation. In some way the manifold underlies experimental science, i.e., the viscometric flows of fluids and the bending and twisting of beams in solids and the procedures commonly used to measure constitutive relations, this being related to the fact that the form of the manifold can be prescribed independent of the atomic forces.

Which positive integers are the area of a right angled triangle with rational sides? In this talk I will discuss this classical problem, its reformulation in terms of rational points on elliptic curves and Tunnell's theorem which gives a complete solution to this problem assuming the Birch and Swinnerton-Dyer conjecture.

 The unknotting number of a knot is an incredibly difficult invariant to compute.
In fact, there are many knots which are conjectured to have unknotting number 2 but for
which no proof of this is currently available. It therefore remains an unsolved problem to find an
algorithm that determines whether a knot has unknotting number one. In my talk, I will
show that an analogous problem for links is soluble. We say that a link has splitting number
one if some crossing change turns it into a split link. I will give an algorithm that
determines whether a link has splitting number one. (In the case where the link has
two components, we must make a hypothesis on their linking number.) The proof
that the algorithm works uses sutured manifolds and normal surfaces.

abstract:In this talk, we describe how to compute the critical point for various lattice models of planar statistical physics. We will first introduce the percolation, Ising, Potts and random-cluster models on the square lattice. Then, we will discuss how critical points of these different models are related. In a final part, we will compute the critical point of these models. This last part harnesses two main ingredients that we will describe in details: duality and sharp threshold theorems. No background is necessary.

Edge-reinforced random walk (ERRW), introduced by Coppersmith and Diaconis in 1986, is a random process which takes values in the vertex set of a graph G, and is more likely to cross edges it has visited before. We show that it can be represented in terms of a Vertex-reinforced jump process (VRJP) with independent gamma

conductances: the VRJP was conceived by Werner and first studied by Davis and Volkov (2002,2004), and is a continuous-time process favouring sites with more local time. We show that the VRJP is a mixture of time-changed Markov jump processes and calculate the mixing measure. The mixing measure is interpreted as a marginal of the supersymmetric hyperbolic sigma model introduced by Disertori, Spencer and Zirnbauer.

This enables us to deduce that VRJP and ERRW are strongly recurrent in any dimension for large reinforcement (in fact, on graphs of bounded degree), using a localisation result of Disertori and Spencer (2010).

(Joint work with Pierre Tarrès.)

Instantons and W-bosons in 5d N=2 Yang-Mills theory arise from a circle

compactification of the 6d (2,0) theory as Kaluza-Klein modes and winding

self-dual strings, respectively. We study an index which counts BPS

instantons with electric charges in Coulomb and symmetric phases. We first

prove the existence of unique threshold bound state of U(1) instantons for

any instanton number. By studying SU(N) self-dual strings in the Coulomb

phase, we find novel momentum-carrying degrees on the worldsheet. The total

number of these degrees equals the anomaly coefficient of SU(N) (2,0) theory.

We finally propose that our index can be used to study the symmetric phase of

this theory, and provide an interpretation as the superconformal index of the

sigma model on instanton moduli space. 

The geometric Lévy model (GLM) is a natural generalisation of the geometric Brownian motion (GBM) model. The theory of such models simplifies considerably if one takes a pricing kernel approach. In one dimension, once the underlying Lévy process has been specified, the GLM has four parameters: the initial price, the interest rate, the volatility and the risk aversion. The pricing kernel is the product of a discount factor and a risk aversion martingale. For GBM, the risk aversion parameter is the market price of risk. In this talk I show that for a GLM, this interpretation is not valid: the excess rate of return above the interest rate is a nonlinear function of the volatility and the risk aversion such that it is positive, and is increasing with respect to these variables. In the case of foreign exchange, Siegel’s paradox implies that one can construct foreign exchange models for which the excess rate of return is positive for both the exchange rate and the inverse exchange rate. Examples are worked out for a range of Lévy processes. (The talk is based on a recent paper: Brody, Hughston & Mackie, Proceedings of the Royal Society London, to appear in May 2012).  

Please note that this is a joint seminar with the William Dunn School of Pathology and will be held in the EPA Seminar Room

Please note the unusual start-time.

In order to run accurate electrochemical models of batteries (and other devices) it is necessary to know a priori the values of many geometric, electrical and electrochemical parameters (10-100 parameters) e.g. diffusion coefficients, electrode thicknesses etc. However a basic difficulty is that the only external measurements that can be made on cells without deconstructing and destroying them are surface temperature plus electrical measurements (voltage, current, impedance) at the terminals. An interesting research challenge therefore is the accurate, robust estimation of physically realistic model parameters based only on external measurements of complete cells. System identification techniques (from control engineering) including ‘electrochemical impedance spectroscopy’ (EIS) may be applied here – i.e. small signal frequency response measurement. However It is not clear exactly why and how impedance correlates to SOC/ SOH and temperature for each battery chemistry due to the complex interaction between impedance, degradation and temperature.

I will give a brief overview of some of the recent work in this area and try to explain some of the challenges in the hope that this will lead to a fruitful discussion about whether this problem can be solved or not and how best to tackle it.

I will explain how to define a notion of stable-independence in NIP

theories, which is an attempt to capture the "stable part" of types.

The displacement of a liquid by an air finger is a generic two-phase flow that

underpins applications as diverse as microfluidics, thin-film coating, enhanced

oil recovery, and biomechanics of the lungs. I will present two intriguing

examples of such flows where, firstly, oscillations in the shape of propagating

bubbles are induced by a simple change in tube geometry, and secondly, flexible

vessel boundaries suppress viscous fingering instability.

1) A simple change in pore geometry can radically alter the behaviour of a

fluid displacing air finger, indicating that models based on idealized pore

geometries fail to capture key features of complex practical flows. In

particular, partial occlusion of a rectangular cross-section can force a

transition from a steadily-propagating centred finger to a state that exhibits

spatial oscillations via periodic sideways motion of the interface at a fixed

location behind the finger tip. We characterize the dynamics of the

oscillations and show that they arise from a global homoclinic connection

between the stable and unstable manifolds of a steady, symmetry-broken

solution.

2) Growth of complex dendritic fingers at the interface of air and a viscous

fluid in the narrow gap between two parallel plates is an archetypical problem

of pattern formation. We find a surprisingly effective means of suppressing

this instability by replacing one of the plates with an elastic membrane. The

resulting fluid-structure interaction fundamentally alters the interfacial

patterns that develop and considerably delays the onset of fingering. We

analyse the dependence of the instability on the parameters of the system and

present scaling arguments to explain the experimentally observed behaviour.

The Arnoldi method for standard eigenvalue problems possesses several

attractive properties making it robust, reliable and efficient for

many problems. We will present here a new algorithm equivalent to the

Arnoldi method, but designed for nonlinear eigenvalue problems

corresponding to the problem associated with a matrix depending on a

parameter in a nonlinear but analytic way. As a first result we show

that the reciprocal eigenvalues of an infinite dimensional operator.

We consider the Arnoldi method for this and show that with a

particular choice of starting function and a particular choice of

scalar product, the structure of the operator can be exploited in a

very effective way. The structure of the operator is such that when

the Arnoldi method is started with a constant function, the iterates

will be polynomials. For a large class of NEPs, we show that we can

carry out the infinite dimensional Arnoldi algorithm for the operator

in arithmetic based on standard linear algebra operations on vectors

and matrices of finite size. This is achieved by representing the

polynomials by vector coefficients. The resulting algorithm is by

construction such that it is completely equivalent to the standard

Arnoldi method and also inherits many of its attractive properties,

which are illustrated with examples.

We discuss shock reflection problem for compressible gas dynamics, and von Neumann conjectures on transition between regular and Mach reflections. Then we will talk about some recent results on existence, regularity and geometric properties of regular reflection solutions for potential flow equation. In particular, we discuss optimal regularity of solutions near sonic curve, and stability of the normal reflection soluiton. Open problems will also

be discussed. The talk will be based on the joint work with Gui-Qiang Chen, and with Myoungjean Bae.

The mid 1980s saw a shift in the nature of the relationship between mathematics and physics. Differential equations and geometry applied in a classical setting were no longer the principal players; in the quantum world topology and algebra had come to the fore. In this talk we discuss a method of classifying 2-dim invertible Klein topological quantum field theories (KTQFTs). A key object of study will be the unoriented cobordism category $\mathscr{K}$, whose objects are closed 1-manifolds and whose morphisms are surfaces (a KTQFT is a functor $\mathscr{K}\rightarrow\operatorname{Vect}_{\mathbb{C}}$). Time permitting, the open-closed version of the category will be considered, yielding some surprising results.

Particle-based stochastic reaction-diffusion models have recently been used to study a number of problems in cell biology. These methods are of interest when both noise in the chemical reaction process and the explicit motion of molecules are important. Several different mathematical models have been used, some spatially-continuous and others lattice-based. In the former molecules usually move by Brownian Motion, and may react when approaching each other. For the latter molecules undergo continuous time random-walks, and usually react with fixed probabilities per unit time when located at the same lattice site.

As motivation, we will begin with a brief discussion of the types of biological problems we are studying and how we have used stochastic reaction-diffusion models to gain insight into these systems. We will then introduce several of the stochastic reaction-diffusion models, including the spatially continuous Smoluchowski diffusion limited reaction model and the lattice-based reaction-diffusion master equation. Our work studying the rigorous relationships between these models will be presented. Time permitting, we may also discuss some of our efforts to develop improved numerical methods for solving several of the models.

A perfect obstruction theory for a commutative ring is a morphism from a perfect complex to the cotangent complex of the ring

satisfying some further conditions. In this talk I will present work in progress on how to associate in a functorial manner commutative

differential graded algebras to such a perfect obstruction theory. The key property of the differential graded algebra is that its zeroth homology

is the ring equipped with the perfect obstruction theory. I will also indicate how the method introduced can be globalized to work on schemes

without encountering gluing issues.

We call a graph $H$ \emph{Ramsey-unsaturated} if there is an edge in the

complement of $H$ such that the Ramsey number $r(H)$ of $H$ does not

change upon adding it to $H$. This notion was introduced by Balister,

Lehel and Schelp who also showed that cycles (except for $C_4$) are

Ramsey-unsaturated, and conjectured that, moreover, one may add {\em

any} chord without changing the Ramsey number of the cycle $C_n$, unless

$n$ is even and adding the chord creates an odd cycle.

We prove this conjecture for large cycles by showing a stronger

statement: If a graph $H$ is obtained by adding a linear number of

chords to a cycle $C_n$, then $r(H)=r(C_n)$, as long as the maximum

degree of $H$ is bounded, $H$ is either bipartite (for even $n$) or

almost bipartite (for odd $n$), and $n$ is large.

This motivates us to call cycles \emph{strongly} Ramsey-unsaturated.

Our proof uses the regularity method.