Abstract:  In the study of (1+1)-dimensional random growth processes, a foundational model is the longest-increasing-subsequence problem: from n independent uniform points in the unit square, find the longest chain whose horizontal and vertical coordinates are both increasing. It is known that the maximal length is asymptotically 2n^(1/2), and the longest chain concentrates near the diagonal line y=x. In this talk, we add a geometric constraint: the chain must be convex. I will discuss the history of this problem, as well as recent joint work with Arnab Sen.

Abstract: The four-corner Cantor set is a planar analogue of the classical Cantor set and arises in several areas of analysis, including the study of Kakeya sets and removable singularities for analytic functions. A central problem is to understand how this set behaves when projected onto lines. This turns out to be a very difficult question, so we study a random variant of the Cantor set, where we are able to obtain sharp estimates. This is joint work with Pablo Shmerkin and Ville Suomala. 

Abstract:  In directed LPP on the square lattice with iid exponential weights, the coalescence time of two semi-infinite geodesics with a fixed asymptotic direction has a heavy tail with KPZ exponent 2/3, and hence infinite expectation. We show that three geodesics behave differently. In some models of directed first- and last-passage percolation with general iid weights, suppose we are given a stationary, non-crossing family of semi-infinite geodesics (such as those arising from Busemann functions). Consider three geodesics started from colinear points spaced distance k apart. We show that for every k, the first time at which some pair among the three coalesces has finite expectation. This confirms a prediction of a heuristic that says that coalescence times of pairs of non-crossing geodesics must have negative association. (joint with Firas Rassoul-Agha and Timo Seppäläinen)

Abstract:  We study the spatial covariance of the one-dimensional KPZ equation started from the flat initial profile, with the height function h(t,x) = log Z(t,x) given by the Cole-Hopf transform of the stochastic heat equation with constant initial data. At a fixed time t, we prove a sharp asymptotic for Cov(h(t,x), h(t,0)) as |x| tends to infinity: the covariance decays like a heat-kernel Gaussian envelope exp(-x^2/(4t)) times an explicit prefactor of order t^{3/2} |x|^{-2}. This is in sharp contrast with the narrow-wedge initial condition, where Gu and Pu recently proved a purely polynomial decay of order t/|x|. The far-field covariance of KPZ is therefore qualitatively, not just quantitatively, sensitive to the initial profile. The proof combines a Clark-Ocone representation, a boundary-layer reduction near the midpoint between the two points, and shear invariance and shear mixing of the white noise; along the way we obtain a closed-form second moment for the normalized continuum directed random polymer partition function.

In the second part we pass to two times. For the flat KPZ fixed point we prove quantitative two-time spatial decorrelation: for fixed times s and t, the covariance of the heights at (t,x) and (s,0) is bounded by C exp(-c|x|^3) for |x| at least 1. Unlike the fixed-time covariance, which is governed by the Airy_1 process, the two-time covariance involves the nonlinear variational evolution of the whole earlier profile; the proof combines cubic-exponential mixing of the Airy_1 process with a uniform localization estimate for intermediate optimizers in the directed landscape. As a consequence, centered spatial averages normalized by the square root of the window size converge to a Gaussian process whose covariance is the space-integrated two-time correlation.

Based on joint work with Juan J. Jimenez (Auburn University) and with Fei Pu (Beijing Normal University).

References:

- L. Chen, J. J. Jiménez, Spatial covariance of KPZ from flat initial profile, arXiv:2603.14174.
- L. Chen, F. Pu, Two-time spatial decorrelation for the flat KPZ fixed point, arXiv:2607.17113.
- Y. Gu, F. Pu, Spatial decorrelation of KPZ from narrow wedge, arXiv:2506.23065.

Abstract:  We study the boundedness of a bilinear singular integral operator arising from rough averages of directional variants of the triangular Hilbert transform. This operator may also be viewed as a twisted paraproduct associated with a rough homogeneous kernel. After briefly reviewing the development of multilinear singular integrals with rough kernels and related results on the triangular Hilbert transform, we discuss the main ideas behind the boundedness argument and the harmonic-analytic tools used to control the resulting rough bilinear operator.

Abstract:  In this talk, we will recall the classical central limit theorem (CLT) and talk about a generalization of CLT for a stationary sequence X_k = f(G_k),  where G is a centered stationary Gaussian sequence and f is square-integrable with respect to the standard Gaussian measure on reals. Motivated by the non-CLT results by Dobrushin, Major, Rosenblatt, and Taqqu, Peter Breuer and Peter Major, in their 1983 celebrated paper, established the asymptotic normality of the partial sum when the Gaussian covariance function has sufficient decay.

Their work is on the convergence of finite-dimensional distributions, while the tightness problem (that would lead to a functional CLT) had remained open in its generality.

Recent work by Nourdin and Nualart established tightness under the extra (2+) moment assumption by using Malliavin-Meyer argument. In a recent work (arXiv:2607.11469) with Mansanarez and Poly, we introduced a Gaussian prediction-theoretic framework to allow us to remove the extra moment assumption. The results in MPZ paper are applicable for `most’ Gaussian sequence and for all test function f in the finite variance setting.  In a work (arXiv:2610.09404) with Dionysis Milesis, we finally remove all unnecessary assumptions.

Abstract:  The Critical Stochastic Heat Flow (SHF) was originally constructed by Caravenna, Sun, and Zygouras (2023), as the scaling limit of partition functions of two-dimensional directed polymers in the critical disorder regime. Its moments are closely related to the two-dimensional delta Bose gas. I will talk about this connection in some detail, including the explicit description of the delta Bose semigroup obtained by Gu, Quastel, and Tsai (2019) through a resolvent expansion.

Recently, Tsai (2024) gave an axiomatic characterization of the critical SHF based on its moments together with a few additional properties. I will briefly explain the main ideas behind the proof of this characterization. As an application, I will also discuss a recent result proving convergence of partition functions to the critical SHF for a class of polymers in spatially correlated Brownian environments.

Abstract: