We present a survey of how the curvature tensor of all known homogeneous 6 dimensional nearly Kähler spaces (both in the definite and in the pseudo Riemannian case) can be expressed in an invariant way using the induced geometric structures on the 6 dimensional nearly Kähler space.

As an application we show how this can be used to study special classes of submanifolds in these spaces. In the latter case we will in particular focus on totally geodesic Lagrangian submanifolds and equivariant Lagrangian immersions.

Venue: MCS2068

For stochastic analysis on the Wasserstein space, it is crucial to construct a diffusion process which plays a role of Brownian motion in finite-dimensions, or the Ornstein-Uhlenbeck process on a separable Hilbert space. This has been a long standing open problem due to the lack of a volume or Gaussian measure on the Wasserstein space, which could serve as an invariant measure. To study diffusion processes on the $p$-Wasserstein space $\mathcal P_p $ for $p\in [1,\infty)$ over a separable, reflexive Banach space $X$, we present a criterion on the quasi-regularity of Dirichlet forms in $L^2(\mathcal P_p,\Lambda)$ for a reference probability $\Lambda$ on $\mathcal P_p$ by using an upper bound condition with the uniform norm of the intrinsic derivative. The condition is easy to check in applications. As a consequence, a class of quasi-regular local Dirichlet forms are constructed on $\mathcal P_p$ by using image of Dirichlet forms on the tangent space $L^p(X\to X,\mu_0)$ at a reference point $\mu_0\in \mathcal P_p$. In particular, the quasi-regularity is confirmed for Ornstein-Uhlenbeck type Dirichlet forms, and an explicit heat kernel estimate is derived based on the eigenvalues of the covariance operator of the underlying Gaussian measure.

Venue: online (streamed into MCS2068)

The talk discusses generalisations of Liouville's theorem to nonlocal translation-invariant operators. It is based on a joint work with D. Berger and R.L. Schilling, and a further joint work with the same co-authors and T. Sharia. We consider operators with continuous but not necessarily infinitely smooth symbols.

It follows from our results that if $\left\{\eta \in \mathbb{R}^n \mid m(\eta) = 0\right\} \subseteq \{0\}$, then, under suitable conditions, every polynomially bounded weak solution $f$ of the equation $m(D)f=0$ is in fact a polynomial, while sub-exponentially growing solutions admit analytic continuation to entire functions on $\mathbb{C}^n$.

Venue: MCS3070

de-Sitter(dS) space allows for a generalized class of vacua, known as α−vacua, described by some parameters. The Bunch-Davies (BD) vacuum is a point in this parameter space. We show that the correlation function in the α−vacua (for rigid dS space) can be related to three-dimensional CFT correlation functions if we relax the requirement of consistency with OPE limit. We then explore inflationary correlators in α−vacua. Working within the leading slow-roll approximation, we compute the four-point scalar correlator (the trispectrum). We check that the conformal Ward identities are met between the three and four-point scalar α-vacua correlators. Surprisingly, this contrasts the previously reported negative result of the Ward identities being violated between the two and the three-point correlators.

Venue: zoom

Zoom: https://durhamuniversity.zoom.us/j/99116644259?pwd=Q0xLa0ZHdkxZeVRxVXZkNFJCa2Y1Zz09

Mathematical models of non-Newtonian fluids play an important role in science and engineering, and their analysis has been an active field of research over the past decade. This talk is concerned with the mathematical analysis of numerical methods for the approximate solution of systems of nonlinear elliptic partial differential equations that arise in models of chemically reacting viscous incompressible non-Newtonian fluids, such as the synovial fluid found in the cavities of synovial joints. The synovial fluid consists of an ultra filtrate of blood plasma that contains hyaluronic acid, whose concentration influences the shear-thinning property and helps to maintain a high viscosity; its function is to reduce friction during movement. The shear-stress appearing in the model involves a power-law type nonlinearity, where, instead of being a fixed constant, the power law-exponent is a function of a spatially varying nonnegative concentration function, which, in turn, solves a nonlinear convection-diffusion equation. In order to prove the convergence of the sequence of numerical approximations to a solution of this coupled system of nonlinear partial differential equations, a uniform Hölder norm bound needs to be derived for the sequence of numerical approximations to the concentration in a setting, where the diffusion coefficient in the convection-diffusion equation satisfied by the concentration is merely an L^∞ function. This necessitates the development of a discrete counterpart of the De Giorgi–Nash–Moser theory. Motivated by an early paper by Aguilera and Caffarelli (1986) in the simpler setting of Laplace’s equation, we derive such uniform Hölder norm bounds on the sequence of continuous piecewise linear finite element approximations to the concentration. We then use these to deduce the convergence of the sequence of approximations to a weak solution of the coupled system of nonlinear partial differential equations under consideration.

Venue: MCS0001

There are two main constructions of Calabi-Yau cones in dimension 3. Firstly, the anticanonical cones over (log) del Pezzo surfaces and secondly via Gorenstein toric singularities. The toric construction automatically comes with the action of a 3-dimensional torus and the Calabi-Yau condition is automatically fulfilled. For the construction from del Pezzo surfaces we only obtain a 1-dimensional torus action and the Kähler-Einstein condition for the del Pezzo surfaces is crucial to obtain a Calabi-Yau cone metric. In my talk I will address the remaining cases with 2-torus action by discussing a combinatorial approach which interpolates between the two previous constructions and also explain how the Calabi-Yau property is reflected in this combinatorial language.

Venue: MCS2068