In this talk I will discuss a striking duality, T-duality, we discovered between two seemingly unrelated objects: the hypersimplex and the m=2 amplituhedron. We draw novel connections between them and prove many new properties. We exploit T-duality to relate their triangulations and generalised triangles (maximal cells in a triangulation). We subdivide the amplituhedron into chambers as the hypersimplex can be subdivided into simplices - both enumerated by Eulerian numbers. Along the way, we prove several conjectures on the amplituhedron and find novel cluster-algebraic structures, e.g. a generalisation of cluster adjacency.

This is based on the joint work with Lauren Williams and Melissa Sherman-Bennett https://arxiv.org/abs/2104.08254.

Abstract: We study optimal investment, pricing and hedging problems under model uncertainty, when the reference model is a non-Markovian stochastic factor model, comprising a single stock whose drift and volatility are adapted to the filtration generated by a Brownian motion correlated with that driving the stock. We derive explicit characterisations of the robust value processes and optimal solutions (based on a so-called distortion solution for the investment problem under one of the models) and conduct large-scale simulation studies to test the efficacy of robust strategies versus classical ones (with no model uncertainty assumed) in the face of parameter estimation error.

 

Abstract: We study optimal investment, pricing and hedging problems under model uncertainty, when the reference model is a non-Markovian stochastic factor model, comprising a single stock whose drift and volatility are adapted to the filtration generated by a Brownian motion correlated with that driving the stock. We derive explicit characterisations of the robust value processes and optimal solutions (based on a so-called distortion solution for the investment problem under one of the models) and conduct large-scale simulation studies to test the efficacy of robust strategies versus classical ones (with no model uncertainty assumed) in the face of parameter estimation error.