Past

Log-Sobolev inequalities with weighted square field are derived from a class of super Poincaré inequalities. As applications, stronger versions of Talagrand's transportation-cost inequality are provided on Riemannian manifolds. Typical examples are constructed to illustrate these results.

Abstract: Twistor-string theory is reformulated as a `half-twisted heterotic' theory with target $CP^3$. This in effect gives a Dolbeault formulation of a theory of holomorphic curves in twistor space and gives a clearer picture of the mathematical structures underlying the theory and how they arise from the original Witten and Berkovits models. It is also explained how space-time physics arises from the model. It intended that the lecture be, to a certain extent, pedagogical.

I will describe how atom interferometry can be used to set limits on beyond the Standard Model forces.

A new primal-dual augmented Lagrangian merit function is proposed that may be minimized with respect to both the primal and dual variables. A benefit of this approach is that each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of classical primal methods are given: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual l1 linearly constrained Lagrangian (pdl1-LCL) method.

This talk will be about the systematic simplification of differential equations.

After giving a geometric reformulation of the concept of a differential equation using prolongations, I will show how we can prolong group actions relatively easily at the level of Lie algebras. I will then discuss group-invariant solutions.

The key example will be the heat equation.

Abstract: Self-dual connections on ALF spaces can be encoded in terms of bow diagrams, which are natural generalizations of quivers. This provides a convenient description of the moduli spaces of these self-dual connections. We make some comments about the associated twistor data. Via the Nahm transform we construct two explicit examples: a single instanton and a single monopole on a Taub-NUT space.

The Lovasz Local Lemma is known to have an extension for cases where the dependency graph requirement is relaxed to negative dependency graph (Erdos-Spencer 1991). The difficulty is to find relevant negative dependency graphs that are not dependency graphs. We provide two generic constructions for negative dependency graphs, in the space of random matchings of complete and complete bipartite graphs. As application, we prove existence results for hypergraph packing and Turan type extremal problems. We strengthen the classic probabilistic proof of Erdos for the existence of graphs with large girth and chromatic number by prescribing the degree sequence, which has to satisfy some mild conditions. A more surprising application is that tight asymptotic lower bounds can be obtained for asymptotic enumeration problems using the Lovasz Local Lemma. This is joint work with Lincoln Lu.

Local Bifurcation Theory (II): Principle of exchange of stability, Lyapunov-Schmidt reduction, Theorem of Ize.