Applied Math

We derive an option-implied valuation of impermanent loss, quantifying the risk to liquidity providers on decentralized exchanges. Our valuation is 1/8 times the variance-swap rate of the tokens' relative price. Options on relative price do not trade, but we impose a distribution for it,…

Sparse Dictionary Learning seeks to represent data as combinations of a small number of basic elements, or atoms, drawn from an overcomplete dictionary. When all observations are considered together, this framework can be viewed as a form of matrix factorization, where the data matrix Y is…

 I will consider long-time effects caused by perturbations of diffusion processes, in particular of dynamical systems. If the system has some conservation laws, they can be broken by the perturbations. Long- time  motion of the perturbed system can be described, under…

The Stochastic Heat Flow (SHF) emerges as the scaling limit of directed polymers in random environments and the noise-mollified stochastic heat equation, specifically at the critical dimension of two and near the critical temperature. I will present an axiomatic formulation of the SHF as well as…

Transport by fluid flow can provide one of the less understood regularization mechanisms in PDE. In this talk, I will focus on the 2D Keller-Segel equation for chemotaxis set on a general domain and coupled via buoyancy with the fluid obeying Darcy's law - a much studied model of the…

We introduce a 1-dimensional PDE model generating turbulent flows, based on the generalized Constantin-Lax-Majda-DeGregorio equation with viscous dissipation subject to a large-scale random external forcing.  Two kinds of random force are considered.  For a force leading to a…

I’ll talk about various limit theorems for iterated sums and integrals. Collections of such iterated sums and integrals were called signatures in papers on the rough paths theory and their applications to data science, machine learning and neural networks were discussed recently. I’ll speak on…

In this talk, we introduce a sampling-based semi-Lagrangian adaptive rank (SLAR) method, which leverages a cross approximation strategy—also known as CUR or pseudo-skeleton decomposition—to efficiently represent low-rank structures in kinetic solutions. The method dynamically adapts the rank of…

Many MCMC methods either rely on gradients – such as the No-U-Turn Sampler (NUTS) – or struggle with multi-scale distributions, where different regions require vastly different exploration strategies.  NURS is a new  locally adaptive MCMC method that overcomes these…

The cochlea is the main organ of the inner ear. It is responsible for converting the incoming mechanical sound signals from the middle ear into neural signals in the auditory nerve. This transformation involves a fascinating sequence of mechanical and hydrodynamical components, responsible for…