Tom Kojar website

Category: KPZ Universality and SPDEs

  • An SDE perspective on stochastic convex optimization and Rough paths

    In the work “An SDE perspective on stochastic convex optimization”, they study the SDE version of gradient descent

    $latex dX_{t}=-\nabla f(X_{t})+\sigma(X_{t},t)dW_{t},$

    for convex $latex f$. One interesting question is to study this same question but for possibly more irregular potential $latex U(x)$ and volatility coefficient $latex \sigma(X_{t},t)$

    $latex dX_{t}=-U(X_{t})+\sigma(X_{t},t)dW_{t},$

    using the technology of Rough paths. For example, can we relax the regularity of $latex \sigma(X_{t},t)$ from Lipschitz to $latex \alpha$-Hölder for some $latex \alpha \in (\frac{1}{3},\frac{1}{2})$?

  • Microscopic regularity structures

    One of the main questions here is extending the BPHZ renormalization to the situation of discrete regularity structures.

  • Modeling SLE

    Based on Tom Kennedy’s C++ code for Fast SLE. The code was only available in C++ for speed purposes, so now it can run SLE algorithms in Matlab.

    https://www.mathworks.com/matlabcentral/fileexchange/63415-fast-sle-algorithm

  • Periodic TASEP

    KPZ Fixed point

      •  Finding a transition probability formula that is amenable to taking limit as the number of TASEP particles goes to infinity.
        • Is there an analogous backwards heat equation problem as in the KPZ fixed point for Z-TASEP?
      • Identifying the KPZ fixed point in the subcritical region

    .

    RSK correspondence formula

    In “Determinantal transition kernels for some interacting particles on the line” Dieker and Warren derive the transition probability formula for various particle processes on the infinite line. For example, they rederived Schutz’s formula for Z-TASEP. Can the Baik-Liu formula be rederived in the same spirit of techniques.

    • What will be the periodic Gelfand-Tsetlin and Young Tableaus that are in bijection with the dynamics of periodic TASEP?