Speaker: Yevhen Zelenyuk (Wits. University)

Title: Discontinuity of multiplication and left translations in $G^{LUC}$

Abstract: Every locally compact group $G$ has a largest semigroup

compactification, denoted $G^{LUC}$. For every $p\in G$ and $q\in G^{LUC}$,

the multiplication in $G^{LUC}$ is continuous at $(p,q)$ (Ellis 1957), and

for every $p\in G^*=G^{LUC}\setminus G$, the left translation by $p$ in

$G^*$ is discontinuous (Protasov and Pym 2001). We shall discuss the

question whether there are $p,q\in G^*$ such that the multiplication in

$G^{LUC}$ is continuous at $(p,q)$ or the left translation by $p$ in $G^*$

is continuous at $q$.

Title: On stochastic parameterizing manifolds: Pullback characterization and Non-Markovian reduced equations

Abstract: In this talk, a general approach to provide approximate parameterizations of the “small” scales by the “large” ones, will be presented for stochastic partial differential equations (SPDEs) driven by linear multiplicative noise. This is accomplished via the concept of parameterizing manifolds (PMs) that are stochastic manifolds which improve in mean square error the partial knowledge of the full SPDE solution u when compared to the projection of u onto the resolved modes, for a given realization of the noise.

Backward-forward systems are designed to give access to such PMs in practice. The key idea consists of representing the modes with high wave numbers (as parameterized by the sought PM) as a pullback limit depending on the time-history of the modes with low wave numbers.

The resulting manifolds obtained by such a procedure are not subject to a spectral gap condition such as encountered in the classical theory. Instead, certain PMs can be determined under weaker non-resonance conditions.

Non-Markovian stochastic reduced systems are then derived based on such a PM approach. Such reduced systems take the form of SDEs involving random coefficients that convey memory effects via the history of the Wiener process, and arise from the nonlinear interactions between the low modes, embedded in the “noise bath.” These random coefficients follow typically non-Gaussian statistics and exhibit an exponential decay of correlations whose rate depends explicitly on gaps arising in the non-resonances conditions.

It is shown on a stochastic Burgers-type equation, that such PM-based reduced systems can achieve very good performance in reproducing statistical features of the SPDE dynamics projected onto the resolved modes, such as the autocorrelations and probability functions of the corresponding modes amplitude, even when the latter is large and the amount of noise is significant.

Title: Granular Flows and Martian Gullies

Speaker: Prof. Jim McElwaine, Department of Earth Sciences, University of Durham

Abstract:

Dense granular flows occur frequently in both nature and industry,

yet, despite their prevalence, they remain poorly understood. Most

theories are empirically based and are unreliable when applied outside

their narrow range of validity. For terrestrial phenomena this is only

an inconvenience as more detailed experiments always be performed. For

extra-terrestrial phenomena however this is largely impossible and

likely to remain so. For this reason it is essential to develop

physically based theories that can be applied throughout the solar

systems where gravity, air pressure and temperature may have very

different values. I report on chute and drum experiments of granular

flows and explain how these observations are applicable in

interpreting observations. As a case study I focus on the flow of

carbon dioxide blocks down Martian dunes.

Speaker: Markus Pflaum (U. Colorado)

Title: Whitney Functions, the real homotopy type of a semi-analytic set, and a

Hochschild-Kostant-Rosenberg type theorem

Abstract:

In the talk we consider semi-analytic subsets of a real analytic

manifold and their homology and real homotopy type.

It is well-known that de Rham’s Theorem does not hold true in general

for singular spaces such as semi-analytic sets.

We show that to remedy this one can replace the de Rham complex by the

Whitney-de Rham complex to compute the

singular homology of such sets. Beyond that, the Whitney-deRham complex

even determines the real homotopy type of

a semi-analytic set which extends a result by Sullivan for the de Rham

complex on smooth manifolds.

Finally, we comment on a Hochschild-Kostant-Rosenberg type theorem for

Whitney functions.

The talk is based on joint work with B. Chriestenson and J.-P.

Brasselet.

A draft of the thesis is available at http://math.hawaii.edu/home/theses/MA_2013_Brown.pdf.