We shall explain how to give substance to an old dream of Tits, to invent exotic new zeta functions, and discover the skeleton of algebraic varieties (toric manifolds and tropical geometry).
Abstract: The valuation of a finite resource, be it acopper mine, timber forest or gas field, has received surprisingly littleattention from the academic literature. The fact that a robust, defensible andaccurate valuation methodology has not been derived is due to a mixture ofdifficulty in modelling the numerous stochastic uncertainties involved and thecomplications with capturing real day-to-day mining operations. The goal ofproducing such valuations is not just for accounting reasons, but also so thatoptimal extraction regimes and procedures can be devised in advance for use atthe coal-face. This paper shows how one can begin to bring all these aspectstogether using contingent claims financial analysis, geology, engineering,computer science and applied mathematics.
The viability of a market impact model is usually considered to be equivalent to the absence of price manipulation strategies in the sense of Huberman & Stanzl (2004). By analyzing a model with linear instantaneous, transient, and permanent impact components, we discover a new class of irregularities, which we call transaction-triggered price manipulation strategies. Transaction-triggered price manipulation is closely related to the non-existence of measure-valued solutions to a Fredholm integral equation of the first kind. We prove that price impact must decay as a convex decreasing function of time to exclude these market irregularities along with standard price manipulation. We also prove some qualitative properties of optimal strategies and provide explicit expressions for the optimal strategy in several special cases of interest. Joint work with Aurélien Alfonsi, Jim Gatheral, and Alla Slynko.
I'll define the category of B-branes in a LG model, and show that for affine models the Hochschild homology of this category is equal to the physically-predicted closed state space. I'll also explain why this is a step towards proving that LG B-models define TCFTs.
I will describe some recent joint work with Davide Gaiotto and Greg Moore, in which we explain the origin of the wall-crossing formula of Kontsevich and Soibelman, in the context of N=2 supersymmetric field theories in four dimensions. The wall-crossing formula gives a recipe for constructing the smooth hyperkahler metric on the moduli space of the field theory reduced on a circle to 3 dimensions. In certain examples this moduli space is actually a moduli space of ramified Higgs bundles, so we obtain a new description of the hyperkahler structure on that space.
A random environment (in Z^d) is a collection of (random) transition probabilities, indexed by sites. Perform now a random walk using these transitions. This model is easy to describe, yet presents significant challenges to analysis. In particular, even elementary questions concerning long term behavior, such as the existence of a law of large numbers, are open. I will review in this talk the model, its history, and recent advance, focusing on examples of unexpected behavior.