The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.
The course covers the standard material on nonlinear wave equations, including local existence, breakdown criterion, global existence for small data for semi-linear equations, and Strichartz estimate if time allows.
Abstract: Option price data are used as inputs for model calibration, risk-neutral density estimation and many other financial applications. The presence of arbitrage in option price data can lead to poor performance or even failure of these tasks, making pre-processing of the data to eliminate arbitrage necessary. Most attention in the relevant literature has been devoted to arbitrage-free smoothing and filtering (i.e. removing) of data. In contrast to smoothing, which typically changes nearly all data, or filtering, which truncates data, we propose to repair data by only necessary and minimal changes. We formulate the data repair as a linear programming (LP) problem, where the no-arbitrage relations are constraints, and the objective is to minimise prices' changes within their bid and ask price bounds. Through empirical studies, we show that the proposed arbitrage repair method gives sparse perturbations on data, and is fast when applied to real world large-scale problems due to the LP formulation. In addition, we show that removing arbitrage from prices data by our repair method can improve model calibration with enhanced robustness and reduced calibration error.
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The cohomology of a manifold classifies geometric structures over it. One instance of this principle is the classification of line bundles via Chern classes. The classifying space BG associated to a (Lie) group G is a simplicial manifold which encodes the group structure. Its cohomology hence classifies geometric objects over G which play well with its multiplication. These are known as characteristic classes, and yield invariants of G-principal bundles.
I will introduce multiplicative gerbes and show how they realise classes in H^4(BG) when G is compact. Along the way, we will meet different versions of Lie group cohomology, smooth 2-groups and a few spectral sequences.
Link: https://teams.microsoft.com/l/meetup-join/19%3ameeting_ZGRiMTM1ZjQtZWNi…
The purpose of this talk is to provide a quick introduction to the buzzwords in the title. Then I'll discuss some (mostly unexplored) conjectures and thoughts.
In the world of finitely generated groups, presentations are a blessing and a curse. They are versatile and compact, but in general tell you very little about the group. Tietze transformations offer much (but deliver little) in terms of understanding the possible presentations of a group. I will introduce a different way of transforming presentations of a group called a Nielsen transformation, and show how topological methods can be used to study Nielsen transformations.
Since its introduction in 1978 the curve complex has become one of the most important objects to study surfaces and their homeomorphisms. The curve complex is defined only using data about curves and their disjointness: a stunning feature of it is the fact that this information is enough to give it a rigid structure, that is every symplicial automorphism is induced topologically. Ivanov conjectured that this rigidity is a feature of most objects naturally associated to surfaces, if their structure is rich enough.
During the talk we will introduce the curve complex, then we will focus on its rigidity, giving a sketch of the topological constructions behind the proof. At last we will talk about generalisations of the curve complex, and highlight some rigidity results which are clues that Ivanov's Metaconjecture, even if it is more of a philosophical statement than a mathematical one, could be "true".