I will discuss some recent progress connecting different quiver gauge theories and some applications of these results.
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.
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.
In many fields of science and engineering, such as fluid or structural mechanics and electric circuit design, large scale dynamical systems need to be simulated, optimized or controlled. They are often described by discretizations of systems of nonlinear partial differential equations yielding high-dimensional discrete phase spaces. For this reason, in recent decades, research has mainly focused on the development of sophisticated analytical and numerical tools to help understand the overall system behavior. During this time meshless methods have enjoyed significant interest in the research community and in some commercial simulators (e.g., LS-DYNA). In this talk I will describe a normalized-corrected meshless method which ensures linear completeness and improved accuracy. The resulting scheme not only provides first order consistency O(h) but also alleviates the particle deficiency (kernel support incompleteness) problem at the boundary. Furthermore, a number of improvements to the kernel derivative approximation are proposed.
To illustrate the performance of the meshless method, we present results for different problems from various fields of science and engineering (i.e. nano-tubes modelling, solid mechanics, damage mechanics, fluid mechanics, coupled interactions of solids and fluids). Special attention is paid to fluid flow in porous media. The primary attraction of the present approach is that it provides a weak formulation for Darcy's law which can be used in further development of meshless methods.
Special Lagrangian submanifolds are area minimizing Lagrangian submanifolds discovered by Harvey and Lawson. There is no obstruction to deforming compact special Lagrangian
submanifolds by a theorem of Mclean. It is however difficult to understand singularities of
special Lagrangian submanifolds (varifolds). Joyce has studied isolated singularities with multiplicity one smooth tangent cones. Suppose that there exists a compact special Lagrangian submanifold M of dimension three with one point singularity modelled on the Clliford torus cone. We may apply the gluing technique to M by a theorem of Joyce.
We obtain then a compact non-singular special Lagrangian submanifold sufficiently close to M as varifolds in Geometric Measure Theory. The main result of this talk is as follows: all special Lagrangian varifolds sufficiently close to M are obtained by the gluing technique.
The proof is similar to that of a theorem of Donaldson in the Yang-Mills theory.
One first proves an analogue of Uhlenbeck's removable singularities theorem in the Yang-Mills theory. One uses here an idea of a theorem of Simon, who proved the uniqueness of multiplicity one tangent cones of minimal surfaces. One proves next the uniqueness of local models for desingularizing M (see above) using symmetry of the Clifford torus cone.
These are the main part of the proof.
We discuss new symmetry results for nonconstant entire local minimizers of the standard Ginzburg-Landau functional for maps in ${H}^{1}_{\rm{loc}}(\mathbb{R}^3;\mathbb{R}^3)$ satisfying a natural energy bound.
Up to translations and rotations, such solutions of the Ginzburg-Landau system are given by an explicit map equivariant under the action of the orthogonal group.
More generally, for any $N\geq 3$ we characterize the $O(N)-$equivariant vortex solution for Ginzburg-Landau type equations in the $N-$dimensional Euclidean space and we prove its local energy minimality for the corresponding energy functional.
In the operation of high frequency resonators in micro electromechanical systems (MEMS)there is a strong need to be able to accurately determine the energy loss rates or alternativelythe quality of the resonance. The resonance quality is directly related to a designer’s abilityto assemble high fidelity system response for signal filtering, for example. This hasimplications on robustness and quality of electronic communication and also stronglyinfluences overall rates of power consumption in such devices – i.e. battery life. Pastdesign work was highly focused on the design of single resonators; this arena of work hasnow given way to active efforts at the design and construction of arrays of coupledresonators. The behavior of such systems in the laboratory shows un-necessarily largespread in operational characteristics, which are thought to be the result of manufacturingvariations. However, such statements are difficult to prove due to a lack of availablemethods for predicting resonator damping – even the single resonator problem is difficult.The physical problem requires the modeling of the behavior of a resonant structure (or setof structures) supported by an elastic half-space. The half-space (chip) serves as a physicalsupport for the structure but also as a path for energy loss. Other loss mechanisms can ofcourse be important but in the regime of interest for us, loss of energy through theanchoring support of the structure to the chip is the dominant effect.
The construction of a basic discretized model of such a system leads to a system ofequations with complex-symmetric (not Hermitian) structure. The complex-symmetryarises from the introduction of a radiation boundary conditions to handle the semi-infinitecharacter of the half-space region. Requirements of physical accuracy dictate rather finediscretization and, thusly, large systems of equations. The core to the extraction of relevantphysical performance parameters is dependent upon the underlying modeling framework.In three dimensional settings of practical interest, such systems are too large to be handleddirectly and must be solved iteratively. In this talk, I will cover the physical background ofthe problem class of interest, how such systems can be modeled, and then solved. Particularinterest will be paid to the radiation boundary conditions (perfectly matched layers versushigher order absorbing boundary conditions), issues associated with frequency domainversus time domain methods, and how these choices interact with iterative solvertechnologies in sometimes unexpected ways. Time permitting I will also touch upon the issue of harmonic inversion methods of this class of problems.
Probability does not exist. At least no more so than "mass" "spin" or "charm" exist. Yet probability forecasts are common, and there are fine reasons for deprecating point forecasts, as they require an unscientific certainty in exactly what the future holds. What roles do our physical understanding and laws of physics play in the construction of probability forecasts to support of decision making and science-based policy? Will probability forecasting more likely accelerate or retard the advancement of our scientific understanding?
Model-based probability forecasts can vary significantly with alterations in the method of data assimilation, ensemble formation, ensemble interpretation, and forecast evaluation, not to mention questions of model structure, parameter selection and the available forecast-outcome archive. The role of each of these aspects of forecasting, in the context of interpreting the forecast as a real-world probability, is considered and contrasted in the cases of weather forecasting, climate forecasting, and economic forecasting. The notion of what makes a probability forecast "good" will be discussed, including the goals of "sharpness given calibration" and "value".
For a probability forecast to be decision-relevant as such, it must be reasonably interpreted as a basis for rational action through the reflection of the probability of the outcomes forecast. This rather obvious sounding requirement proves to be the source of major discomfort as the distinct roles of uncertainty (imprecision) and error (structural mathematical "misspecification") are clarified. Probabilistic forecasts can be of value to decision makers even when it is irrational to interpret them as probability forecasts. A similar statement, of course, can be said for point forecasts, or for spin. In this context we explore the question: do decision-relevant probability forecasts exist?
We provide a general construction of time-consistent sublinear expectations on the space of continuous paths. In particular, we construct the conditional G-expectation of a Borel-measurable (rather than quasi-continuous) random variable.
In April 2010 Eguchi--Ooguri--Tachikawa observed a fascinating connection between the elliptic genus of a K3 surface and the largest Mathieu group. We will report on joint work with Miranda Cheng and Jeff Harvey that identifies this connection as one component of a system of surprising relationships between a family of finite groups, their representation theory, and automorphic forms of various kinds Mock modular forms, and particularly their shadows, play a key role in the analysis, and we find several of Ramanujan's mock theta functions appearing as McKay--Thompson series arising from the umbral groups.
In this talk I review the use of the spectral decomposition for understanding the solution of ill-posed inverse problems. It is immediate to see that regularization is needed in order to find stable solutions. These solutions, however, do not typically allow reconstruction of signal features such as edges. Generalized regularization assists but is still insufficient and methods of total variation are commonly suggested as an alternative. In the talk I consider application of standard approaches from Tikhonov regularization for finding appropriate regularization parameters in the total variation augmented Lagrangian implementations. Areas for future research will be considered.
Twistor theory is a technology that can be used to translate analytical problems on Euclidean space $\mathbb R^4$ into problems in complex algebraic geometry, where one can use the powerful methods of complex analysis to solve them. In the first half of the talk we will explain the geometry of the Twistor correspondence, which realises $\mathbb R^4$ , or $S^4$, as the space of certain "real" lines in the (projective) Twistor space $\mathbb{CP}^3$. Our discussion will start from scratch and will assume very little background knowledge. As an application, we will discuss the Twistor description of instantons on $S^4$ as certain holomorphic vector bundles on $\mathbb{CP}^3$ due to Ward.
In this talk, I will present reflected backward stochastic differential equations (reflected BSDEs) and their connection with the pricing of American options. Then I will present a simple perturbative method for studying them. Under the appropriate assumptions on the coefficient, the terminal condition and the lower obstacle, similar to those used by Kobylankski, this method allows to prove the existence of a solution. I will also provide the usual comparison theorem and a new proof for a refined comparison theorem, specific to RBSDEs.
We study the mean-field models describing the evolution of distributions
of particle radii obtained by taking the small volume fraction limit of
the free boundary problem describing the micro phase separation of
diblock copolymer melts, where micro phase separation consists of an
ensemble of small balls of one component. In the dilute case, we
identify all the steady states and show the convergence of solutions.
Next we study the dynamics for a free boundary problem in two dimension,
obtained as a gradient flow of Ohta- Kawasaki free energy, in the case
that one component is a distorted disk with a small volume fraction. We
show the existence of solutions that a small, almost circular interface
moves along a curve determined via a Green’s function of the domain.
This talk is partly based on a joint work with Xiaofeng Ren.
Let $M$ be the Cantor space or an $n$-dimensional manifold with $C(M,M)$ the set of continuous self-maps of $M$. We analyse the behaviour of the generic $f$ in $C(M,M)$ in terms of attractors and some notions of chaos.
The graph realization problem has received a great deal of attention in recent years, due to its importance in applications such as wireless sensor networks and structural biology. We introduce the ASAP algorithm, for the graph realization problem in R^d, given a sparse and noisy set of distance measurements associated to the edges of a globally rigid graph. ASAP is a divide and conquer, non-incremental and non-iterative algorithm, which integrates local distance information into a global structure determination. Our approach starts with identifying, for every node, a subgraph of its 1-hop neighborhood graph, which can be accurately embedded in its own coordinate system. In the noise-free case, the computed coordinates of the sensors in each patch must agree with their global positioning up to some unknown rigid motion, that is, up to translation, rotation and possibly reflection. In other words, to every patch there corresponds an element of the Euclidean group Euc(3) of rigid transformations in R^3, and the goal is to estimate the group elements that will properly align all the patches in a globally consistent way. The reflections and rotations are estimated using a recently developed eigenvector synchronization algorithm, while the translations are estimated by solving an overdetermined linear system. Furthermore, the algorithm successfully incorporates information specific to the molecule problem in structural biology, in particular information on known substructures and their orientation. In addition, we also propose SP-ASAP, a faster version of ASAP, which uses a spectral partitioning algorithm as a preprocessing step for dividing the initial graph into smaller subgraphs. Our extensive numerical simulations show that ASAP and SP-ASAP are very robust to high levels of noise in the measured distances and to sparse connectivity in the measurement graph, and compare favorably to similar state-of-the art localization algorithms. Time permitting, we briefly discuss the analogy between the graph realization and the low-rank matrix completion problems, as well as an application of synchronization over Z_2 and its variations to bipartite multislice networks.
By recent work of Voevodsky and others, type theories are now considered as a candidate
for a homotopical foundations of mathematics. I will explain what are type theories using the language
of (essentially) algebraic theories. This shows that type theories are in the same "family" of algebraic
concepts such as groups and categories. I will also explain what is homotopic in (intensional) type theories.
Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the first talk.
Random geometric graphs have been well studied over the last 50 years or so. These are graphs that
are formed between points randomly allocated on a Euclidean space and any two of them are joined if
they are close enough. However, all this theory has been developed when the underlying space is
equipped with the Euclidean metric. But, what if the underlying space is curved?
The aim of this talk is to initiate the study of such random graphs and lead to the development of
their theory. Our focus will be on the case where the underlying space is a hyperbolic space. We
will discuss some typical structural features of these random graphs as well as some applications,
related to their potential as a model for networks that emerge in social life or in biological
sciences.
Joyce and Song expressed the wall-crossing behaviour of Donaldson-Thomas invariants using a sum over graphs. Joyce expected that these would have something to do with the Feynman diagrams of suitable physical theories. I will show how this can be achieved in the framework for wall-crossing proposed by Gaiotto, Moore and Neitzke. JS diagrams emerge from small corrections to a hyperkahler metric. The basics of GMN theory will be explained during the
first talk.
Large-scale zonal jets are observed in a wide range of geophysical and astrophysical flows; most strikingly in the atmospheres of the Jovian gas giant planets. Jupiter's upper atmosphere is highly turbulent, with many small vortices, and strong westerly winds at the equator. We consider the thermal shallow water equations as a model for Jupiter's upper atmosphere. Originally proposed for the terrestrial atmosphere and tropical oceans, this model extends the conventional shallow water equations by allowing horizontal temperature variations with a modified Newtonian cooling for the temperature field. We perform numerical simulations that reproduce many of the key features of Jupiter’s upper atmosphere. However, the simulations take a long time to run because their time step is severely constrained by the inertia-gravity wave speed. We filter out the inertia-gravity waves by forming the quasigeostrophic limit, which describes the rapidly rotating (small Rossby number) regime. We also show that the quasigeostrophic energy equation is the quasigeostrophic limit of the thermal shallow water pseudo-energy equation, analogous to the derivation of the acoustic energy equation from gas dynamics. We perform numerical simulations of the quasigeostrophic equations, which again reproduce many of the key features of Jupiter’s upper atmosphere. We gain substantial performance increases by running these simulations on graphical processing units (GPUs).
We show how the reduction procedure for generalized Kahler
structures can be used to recover Hitchin's results about the
existence of a generalized Kahler structure on the moduli space of
instantons on bundle over a generalized Kahler manifold. In this setup
the proof follows closely the proof of the same claim for the Kahler
case and clarifies some of the stranger considerations from Hitchin's
proof.
We consider the stationary flow of Prandtl-Eyring fluids in two
dimensions. This model is a good approximation of perfect plasticity.
The corresponding potential is only slightly super linear. Thus, many
severe problems arise in the existence theory of weak solutions. These
problems are overcome by use of a divergence free Lipschitz
truncation. As a second application of this technique, we generalize
the concept of almost harmonic functions to the Stokes system.