I will discuss the notion of badly approximable points and recent progress and problems in this area, including Schmidt's conjecture, badly approximable points on manifolds and real numbers badly approximable by algebraic numbers.
Abstract: We consider different notions of rough paths on manifolds and study some of the relations between these definitions. Furthermore, we explore extensions to manifolds modelled along infinite dimensional Banach spaces.
Let $X$ be a smooth cubic hypersurface of dimension $m$ defined over a global field $K$. A conjecture of Colliot-Thelene(02) states that $X$ satisfies the Hasse Principle and Weak approximation as long as $m\geq 3$. We use a global version of Hardy-Littlewood circle method along with the theory of global $L$-functions to establish this for $m\geq 6$, in the case $K=\mathbb{F}_q(t)$, where $\text{char}(\mathbb{F}_{q})> 3$.
I will speak about the of weak (and the existence and uniqueness of strong solutions) to the stochastic
Landau-Lifshitz equations for multi (one)-dimensional spatial domains. I will also describe the corresponding Large Deviations principle and it's applications to a ferromagnetic wire. The talk is based on a joint works with B. Goldys and T. Jegaraj.
We review the representation theory of the integrable model underlying the AdS_5/CFT_4 correspondence. We will discuss short and long multiplets, and their impact on the issue of the universal R-matrix. We will give special emphasis to the role of the so-called 'secret symmetry', which completes the Yangian symmetry of the system to a yet to be understood new type of quantum group.