L6

The filtration on the infinite symmetric product of spheres by number of

factors provides a sequence of spectra between the sphere spectrum and

the integral Eilenberg-Mac Lane spectrum. This filtration has received a

lot of attention and the subquotients are interesting stable homotopy

types.

In this talk I will discuss the equivariant stable homotopy types, for

finite groups, obtained from this filtration for the infinite symmetric

product of representation spheres. The filtration is more complicated

than in the non-equivariant case, and already on the zeroth homotopy

groups an interesting filtration of the augmentation ideal of the Burnside

rings arises. Our method is by `global' homotopy theory, i.e., we study

the simultaneous behaviour for all finite groups at once. In this context,

the equivariant subquotients are no longer rationally trivial, nor even

concentrated in dimension 0.

Knot Floer homology (introduced by Ozsvath-Szabo and independently by

Rasmussen) is a powerful tool for studying knots and links in the 3-sphere. In

particular, it gives rise to a numerical invariant, which provides a

nontrivial lower bound on the 4-dimensional genus of the knot. By deforming

the definition of knot Floer homology by a real number t from [0,2], we define

a family of homologies, and derive a family of numerical invariants with

similar properties. The resulting invariants provide a family of

homomorphisms on the concordance group. One of these homomorphisms can be

used to estimate the unoriented 4-dimensional genus of the knot. We will

review the basic constructions for knot Floer homology and the deformed

theories and discuss some of the applications. This is joint work with

P. Ozsvath and Z. Szabo.

We consider a large class of three dimensional continuous dynamic fluctuation models, and show that they all rescale and converge to the stochastic Allen-Cahn equation, whose solution should be interpreted after a suitable renormalization procedure. The interesting feature is that, the coefficient of the limiting equation is different from one's naive guess, and the renormalization required to get the correct limit is also different from what one would naturally expect. I will also briefly explain how the recent theory of regularity structures enables one to prove such results. Joint work with Martin Hairer.

In this talk, I will present our new local existence result to the shallow water equations describing the motions of vertically averaged flows, which are closely related to the $2$-D isentropic Navier-Stokes equations for compressible fluids with density-dependent viscosity coefficients. Via introducing the notion of regular solutions, the local existence of classical solutions is established for the case that the viscosity coefficients are degenerate and the initial data are arbitrarily large with vacuum appearing in the far field.

We shall talk our recent works on the well-posedness of the Prandtl boundary layer equations both in two and three space variables. For the two-dimensional problem, we obtain the well-posedness in the Sobolev spaces by using an energy method under the monotonicity assumption of tangential velocity, and for the three-dimensional Prandtl equations, we construct a special solution by using the Corocco transformation, and obtain it is linearly stable with respect to any three-dimensional perturbation. These works are collaborated with R. Alexandre, C. J. Liu, C. Xu and T. Yang.

I will discuss the well-posedness of a new nonlinear model for nematic

elastomers. The main novelty is that the Frank energy penalizes

spatial variations of the nematic director in the deformed, rather

than in the reference configuration, as it is natural in the case of

large deformations.

We show how to reduce the general formulation of the mass-angular momentum inequality, for axisymmetric initial data of the Einstein equations, to the known maximal case whenever a geometrically motivated system of equations admits a solution. This procedure is based on a certain deformation of the initial data which preserves the relevant geometry, while achieving the maximal condition and its implied inequality (in a weak sense) for the scalar curvature; this answers a question posed by R. Schoen. The primary equation involved, bears a strong resemblance to the Jang-type equations studied in the context of the positive mass theorem and the Penrose inequality. Each equation in the system is analyzed in detail individually, and it is shown that appropriate existence/uniqueness results hold with the solution satisfying desired asymptotics. Lastly, it is shown that the same reduction argument applies to the basic inequality yielding a lower bound for the area of black holes in terms of mass and angular momentum.

Peter Grindrod and Stephen Haben (UoOx)

We are dwelling in the Big Data age. The diversity of the uses of Big Data unleashes limitless possibilities. Many people are talking about ways to use Big Data to track the collective human behaviours, monitor electoral popularity, and predict financial fluctuations in stock markets, etc. Big Data reveals both challenges and opportunities, which are not only related to technology but also to human itself. This talk will cover various current topics and trends in Big Data research. The speaker will share his relevant experiences on how to use analytics tools to obtain key metrics on online social networks, as well as present the challenges of Big Data analytics.

Bio: Ning Wang (Ph.D) works as Researcher at the Oxford Internet Institute. His research is driven by a deep interest in analysing a wide range of sociotechnical problems by exploiting Big Data approaches, with the hope that this work could contribute to the intersection of social behavior and computational systems.