Past

It is very well known that every graph on $n$ vertices and $m$ edges admits a bipartition of size at least $m/2$. This bound can be improved to $m/2 + (n-1)/4$ for connected graphs, and $m/2 + n/6$ for graphs without isolated vertices, as proved by Edwards, and Erd\"os, Gy\'arf\'as, and Kohayakawa, respectively. A bisection of a graph is a bipartition in which the size of the two parts differ by at most 1. We prove that graphs with maximum degree $o(n)$ in fact admit a bisection which asymptotically achieves the above bounds.These results follow from a more general theorem, which can also be used to answer several questions and conjectures of Bollob\'as and Scott on judicious bisections of graphs.
Joint work with Po-Shen Loh and Benny Sudakov

In 1932 Signorini formulated the first variational inequality as a model of an elastic body laying on a rigid surface. In this talk we will revisit this problem from the point of view of regularity theory.

We will sketch a proof of optimal regularity and regularity of the contact set. Similar result are known for scalar equations. The proofs for scalar equations are however based on maximum principles and thus not applicable to Signorini's problem which is modelled by a system of equations.

Let G be a finitely generated group generated by g_1,..., g_n. Consider the alphabet A(G) consisting of the symbols g_1,..., g_n and the symbols '+' and '-'. The words in this alphabet represent elements of the integral group ring Z[G]. In the talk we will investigate the computational problem of deciding whether a word in the alphabet A(G) determines a zero-divisor in Z[G]. We will see that a version of the Atiyah conjecture (together with some natural assumptions) imply decidability of the zero-divisor problem; however, we'll also see that in the group (Z/2 \wr Z)^4 the zero-divisor problem is not decidable. The technique which allows one to see the last statement involves "embedding" a Turing machine into a group ring.

We consider the approximation of the marginal distribution of solutions of stochastic partial differential equations by splitting schemes. We introduce a functional analytic framework based on weighted spaces where the Feller condition generalises. This allows us to apply the theory of strongly continuous semigroups. The possibility of achieving higher orders of convergence through cubature approximations is discussed.

Applications of these results to problems from mathematical finance (the Heath-Jarrow-Morton equation of interest rate theory) and fluid dynamics (the stochastic Navier-Stokes equations) are considered. Numerical experiments using Quasi-Monte Carlo simulation confirm the practicality of our algorithms.

Parts of this work are joint with J. Teichmann and D. Veluscek.

I will discuss properties of stochastic differential equations and numerical algorithms for sampling Gibbs (i.e smooth) measures. Methods such as Langevin dynamics are reliable and well-studied performers for molecular sampling.   I will show that, when the objective of simulation is sampling of the configurational distribution, it is possible to obtain a superconvergence result (an unexpected increase in order of accuracy) for the invariant distribution.   I will also describe an application of thermostats to the Hamiltonian vortex method in which the energetic interactions with a bath of weak vortices are treated as thermal fluctuations

I will discuss some recent progress connecting different quiver gauge theories and some applications of these results.

• Thomas März - Calculus on surfaces with general closest point functions
• Jay Newby - Modeling rare events in biology
• Hugh McNamara - Stochastic parameterisation and variational multiscale

Sorry, this has been cancelled at short notice!

The talk will survey recent developments concerning the existence and the approximation of global weak solutions to a general class of coupled microscopic-macroscopic bead-spring chain models that arise in the kinetic theory of dilute solutions of polymeric liquids with noninteracting polymer chains. The class of models involves the unsteady incompressible Navier-Stokes equations in a bounded domain for the velocity and the pressure of the fluid, with an elastic extra-stress tensor appearing on the right-hand side of the momentum equation. The extra-stress tensor stems from the random movement of the polymer chains and is defined by the Kramers expression through the associated probability density function that satisfies a Fokker-Planck type parabolic equation. Models of this kind were proposed in work of Hans Kramers in the early 1940's, and the existence of global weak solutions to the model has been a long-standing question in the mathematical analysis of kinetic models of dilute polymers.

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We also discuss computational challenges associated with the numerical approximation of the high-dimensional Fokker-Planck equation featuring in the model.

In this talk I will discuss the refraction of shocks on the interface for 2-d steady compressible flow. Particularly, the class of E-H type regular refraction is defined and its global stability of the wave structure is verified. The 2-d steady potential flow equations is employed to describe the motion of the fluid. The stability problem of the E-H type regular refraction can be reduced to a free boundary problem of nonlinear mixed type equations in an unbounded domain. The corresponding linearized problem has similarities to a generalized Tricomi problem of the linear Lavrentiev-Bitsadze mixed type equation, and it can be reduced to a nonlocal boundary value problem of an elliptic system. The later is finally solved by establishing the bijection of the corresponding nonlocal operator in a weighted H\"older space via careful harmonic analysis.

This is a joint work with CHEN Shuxing and HU Dian.

Atherosclerosis is the leading cause of death, both above and below age 65, in the United States and all Western countries. Its earliest prelesion events appear to be the transmural (across the wall)-pressure (DP)-driven advection of large molecules such as low-density lipoprotein (LDL) cholesterol from the blood into the inner wall layers across the monolayer of endothelial cells that tile the blood-wall interface. This transport occurs through the junctions around rare (~one cell every few thousand) endothelial cells whose junctions are wide enough to allow large molecules to pass. These LDL molecules can bind to extracellular matrix (ECM) in the wall’s thin subendothelial intima (SI) layer and accumulate there. On the other hand, the overall transmural water flow can dilute the local intima LDL concentration, thereby slowing its kinetics of binding to ECM, and flushes unbound lipid from the wall. An understanding of the nature of this water flow is clearly critical.

            We have found that rat aortic endothelial cells express the ubiquitous membrane water-channel protein aquaporin-1 (AQP), and that blocking its water channel or knocking down its expression significantly reduces the apparent hydraulic conductivity Lp of the endothelium and, consequently of the entire wall. This decrease has an unexpected and strong DP -dependence. We present a fluid mechanics theory based on the premise that DP compacts the SI, which, as we show, lowers its Lp. The theory shows that blocking or knocking down AQP flow changes the critical DP at which this compaction occurs and explains our observed dependence of Lp on DP. Such compaction may affect lipid transport and accumulation in vivo. However, AQP’s sharp water selectivity gives rise to an oncotic paradox: the SI should quickly become hypotonic and shut down this AQP flow. The mass transfer problem resolve this paradox. The importance of aquaporin-based, rather than simply junctional water transport is that transport via protein channels allows for the possibility of active control of vessel Lp by up- or down-regulation of protein expression. We show that rat aortic endothelial cells significantly change their AQP numbers in response to chronic hypertension (high blood pressure), which may help explain the as yet poorly-understood fact that hypertension correlates with atherosclerosis. We also consider lowering AQP numbers as a strategy to affect disease progression.

TALK 1 -- social media for OII:

TITLE: Computational Perspectives on the Structure and Information

Flows in On-Line Networks

ABSTRACT:

With an increasing amount of social interaction taking place in on-line settings, we are accumulating massive amounts of data about phenomena that were once essentially invisible to us: the collective behavior and social interactions of hundreds of millions of people Analyzing this massive data computationally offers enormous potential both to address long-standing scientific questions, and also to harness and inform the design of future social computing applications: What are emerging ideas and trends? How is information being created, how it flows and mutates as it is passed from a node to node like an epidemic?

We discuss how computational perspective can be applied to questions involving structure of online networks and the dynamics of information flows through such networks, including analysis of massive data as well as mathematical models that seek to abstract some of the underlying phenomena.

TALK 2 -- Community detection:

TITLE: Networks, Communities and the Ground-Truth

ABSTRACT: Nodes in complex networks organize into communities of nodes that share a common property, role or function, such as social communities, functionally related proteins, or topically related webpages. Identifying such communities is crucial to the understanding of the structural and functional roles of networks.Current work on overlapping community detection (often implicitly) assumes that community overlaps are less densely connected than non-overlapping parts of communities. This is unnatural as it means that the more communities nodes share, the less likely it is they are linked. We validate this assumption on a diverse set of large networks and find an increasing relationship between the number of shared communities of a pair of nodes and the probability of them being connected by an edge, which means that parts of the network where communities overlap tend to be more densely connected than the non-overlapping parts of communities. Existing community detection methods fail to detect communities with such overlaps. We propose a model-based community detection method that builds on bipartite node-community affiliation networks. Our method successfully detects overlapping, non-overlapping and hierarchically nested communities. We accurately identify relevant communities in networks ranging from biological protein-protein interaction networks to social, collaboration and information networks. Our results show that while networks organize into overlapping communities, globally networks also exhibit a nested core-periphery structure, which arises as a consequence of overlapping parts of communities being more densely connected.

Nodes in complex networks organize into communities of nodes that share a common property, role or function, such as social communities, functionally related proteins, or topically related webpages. Identifying such communities is crucial to the understanding of the structural and functional roles of networks.

Current work on overlapping community detection (often implicitly) assumes that community overlaps are less densely connected than non-overlapping parts of communities. This is unnatural as it means that the more communities nodes share, the less likely it is they are linked. We validate this assumption on a diverse set of large networks and find an increasing relationship between the number of shared communities of a pair of nodes and the probability of them being connected by an edge, which means that parts of the network where communities overlap tend to be more densely connected than the non-overlapping parts of communities.

Existing community detection methods fail to detect communities with such overlaps. We propose a model-based community detection method that builds on bipartite node-community affiliation networks. Our method successfully detects overlapping, non-overlapping and hierarchically nested communities. We accurately identify relevant communities in networks ranging from biological protein-protein interaction networks to social, collaboration and information networks. Our results show that while networks organize into overlapping communities, globally networks also exhibit a nested core-periphery structure, which arises as a consequence of overlapping parts of communities being more densely connected.

https://www.maths.ox.ac.uk/groups/occam/events/occam-4th-uk-graduate-modelling-camp-2012

Registration for this event is now closed

Kernel functions are suitable tools for multivariate scattered data approximation. In this talk, we discuss the conditioning and stability of optimal reconstruction schemes from multivariate scattered data by using

(conditionally) positive definite kernel functions. Our discussion first provides basic Riesz-type stability estimates for the utilized reconstruction method, before particular focus is placed on upper and lower bounds of the Lebesgue constants.

If time allows, we will finally draw our attention to relevant aspects concerning the stability of penalized least squares approximation.

Please email @email to enquire about attending this meeting

Biomedical science relies on the parallel development of mathematical and experimental models to progress understanding and develop new approaches to the diagnosis, treatment and monitoring of disease. Recognition is growing that knowledge of the roles of physical and mechanical processes in influencing and controlling biological responses in living systems is critical in order for the great promise of tissue engineering and regenerative therapies to be fulfilled. This field of study of physical and mechanical effects on biology is known as mechanobiology, and requires close collaboration between clinicians, biologists, mathematicians and engineers to advance. The aim of this meeting is to bring together researchers in biomedical engineering, biology, and mathematical biology to discuss the latest developments in this fast moving field, the close collaboration across disciplines ensuring that mathematical and biological models develop in parallel, so plenty of time will be allotted for discussion and for more informal interaction.

https://www.maths.ox.ac.uk/groups/occam/events/occam-hosts-mathematical-and-experimental-models-mechanobiology-conference

https://www.maths.ox.ac.uk/system/files/attachments/website_material.pdf

Topic to be confirmed. (This is the postponed workshop from Michaelmas term!)

String theory derives the features of the quantum field theory describing the gauge interactions between the elementary particles in four spacetime dimensions from the physics of strings propagating on the internal manifold, e.g. a Calabi-Yau threefold. A simplified version of this correspondence relates the SU(2)-equivariant generalization of the Donaldson theory (and its further generalizations involving the non-abelian monopole equations) to the Gromov-Witten (GW) theory of the so-called local Calabi-Yau threefolds, for the SU(2) subgroup of the rotation symmetry group SO(4). In recent years the GW theory was related to the Donaldson-Thomas (DT) theory enumerating the ideal sheaves of curves and points. On the toric local Calabi-Yau manifolds the latter theory is studied using localization, producing the so-called topological vertex formalism (which was originally based on more sophisticated open-closed topological string dualities).

In order to accomodate the full SO(4)-equivariant version of the four dimensional Donaldson theory, the so-called "refined topological vertex" was proposed. Unlike that of the ordinary topological vertex, its relation to the DT theory remained unclear.

In these talks, based on joint work with Andrei Okounkov, this gap will be partially filled by showing that the equivariant K-theoretic version of the DT theory reproduces both the SO(4)-equivariant Donaldson theory in four dimensions, and the refined topologica vertex formalism, for all toric Calabi-Yau's admitting the latter.

String theory derives the features of the quantum field theory describing the gauge interactions between the elementary particles in four spacetime dimensions from the physics of strings propagating on the internal manifold, e.g. a Calabi-Yau threefold. A simplified version of this correspondence relates the SU(2)-equivariant generalization of the Donaldson theory (and its further generalizations involving the non-abelian monopole equations) to the Gromov-Witten (GW) theory of the so-called local Calabi-Yau threefolds, for the SU(2) subgroup of the rotation symmetry group SO(4). In recent years the GW theory was related to the Donaldson-Thomas (DT) theory enumerating the ideal sheaves of curves and points. On the toric local Calabi-Yau manifolds the latter theory is studied using localization, producing the so-called topological vertex formalism (which was originally based on more sophisticated open-closed topological string dualities).

In order to accomodate the full SO(4)-equivariant version of the four dimensional Donaldson theory, the so-called "refined topological vertex" was proposed. Unlike that of the ordinary topological vertex, its relation to the DT theory remained unclear.

In these talks, based on joint work with Andrei Okounkov, this gap will be partially filled by showing that the equivariant K-theoretic version of the DT theory reproduces both the SO(4)-equivariant Donaldson theory in four dimensions, and the refined topological vertex formalism, for all toric Calabi-Yau's admitting the latter.