I will present recent results on the uniqueness, ergodicity, and attractiveness of stationary solutions to the Kardar-Parisi-Zhang (KPZ) equation on the real line. It is known that this equation admits Brownian motion with a linear drift as a stationary solution. We show that these solutions are attractive, a principle known as one force--one solution (1F1S): the solution to the KPZ equation started in the distant past from an initial condition with a given velocity will converge almost surely to a Brownian motion with that same drift. As a result, we deduce that these stationary measures are in fact totally ergodic. Furthermore, we can couple all these stationary solutions so that the above attractiveness holds simultaneously (i.e. on a single full measure event) for all but a countably infinite (random) set of asymptotic velocities. This is joint work with Chris Janjigian and Timo Seppalainen. Part of the work is also joint with Tom Alberts.

Recent advances in applications such as cryo-electron microscopy have sparked increased interest in the mathematical analysis of multi-reference alignment (MRA) problems, where the goal is to recover a hidden signal from many noisy observations. The simplest model considers observations of a 1-d hidden signal which have been randomly translated and corrupted by high additive noise. This talk generalizes this classic problem by incorporating random dilations into the data model in addition to random translations and additive noise, and explores multiple approaches to its solution based on translation invariant representations. Random dilations cause large perturbations in the high frequencies, making this a challenging model. When the dilation distribution is unknown, the power spectrum of the hidden signal can be approximated by applying a nonlinear unbiasing procedure to a wavelet-based, translation invariant representation and then solving an optimization problem. When the dilation distribution is known, a more accurate unbiasing procedure can be applied directly to the empirical Fourier invariants to obtain an unbiased estimator of the Fourier invariants of the hidden signal, and the convergence rate of the estimator can be precisely quantified in terms of the sample size and noise levels. Theoretical results are supported by extensive numerical experiments on a wide range of signals. Time permitting, we will also see how these signal processing tools can be applied in the novel context of distribution learning from biased, sparse batches.

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Local times of stochastic processes are closely related to sample path irregularities, and fractal and geometric properties of level sets. I will discuss two basic approaches for studying local times: one via Fourier transform and the other via chaos expansion. In particular, I will focus on the first approach in the case of Gaussian random fields and present some local time and related results which can be applied to the solutions of systems of linear SPDEs such as stochastic heat and wave equations. If time permits, I will briefly explain our research direction on local times of non-Gaussian processes with the second approach.

Many two-dimensional random growth models, including first-passage and last-passage percolation, are conjectured to fall within the KPZ universality class under mild assumptions on the underlying noise. In recent years, researchers have focused on a subset of exactly solvable models, where these conjectures can be rigorously verified. A wide array of methods has been employed, encompassing integrable probability, Gibbsian line ensemble, percolation arguments, and coupling techniques. This talk discusses a specific line of research that combines percolation arguments and coupling techniques to gain insights into the random geometry and space-time profiles of such growth models in the positive-temperature setting.

In random geometry, a recurring theme is that all geodesics emanating from a typical point merge into each other close to their starting point, and we call such points as 1-stars. However, the measure zero set of atypical stars, the points where such coalescence fails, is typically uncountable and the corresponding Hausdorff dimensions of these sets have been heavily investigated for a variety of models including the directed landscape, Liouville quantum gravity and the Brownian map. In this talk, we will consider the directed landscape -- the scaling limit of last passage percolation as constructed in the work Dauvergne-Ortmann-Virág and look into the Hausdorff dimension of the set of atypical stars lying on a geodesic. The main result that we will discuss is that the above dimension is almost surely equal to 1/3. This is in contrast to Ganguly-Zhang where it was shown that the set of atypical stars on the vertical line {x=0} has dimension 2/3. This reduction of the dimension from 2/3 to 1/3 yields a quantitative manifestation of the smoothing of the environment around a geodesic with regard to exceptional behaviour.

The spherical Sherrington--Kirkpatrick (SSK) model and its bipartite analog both exhibit the phenomenon that their free energy fluctuations are asymptotically Gaussian at high temperature but asymptotically Tracy--Widom at low temperature. This was proved in two papers by Baik and Lee, for all non-critical temperatures. The case of critical temperature was recently computed for the SSK model in two separate papers, one by Landon and the other by Johnstone, Klochkov, Onatski and Pavlyshyn. In this talk, we will discuss the critical temperature result for the bipartite SSK model. In particular, we study the free energy fluctuations when the temperature is in a window of size $n^{-1/3}\sqrt{\log n}$ around the critical temperature, the same window for the SSK model. Within this transitional window, the asymptotic fluctuations of the free energy are the sum of independent Gaussian and Tracy--Widom random variables. The talk is based on joint work with Elizabeth Collins-Woodfin.

We consider a class of interacting particle systems where particles perform independent random walks on and spread an infection according to a susceptible-infected-recovered model. I will discuss a new method for understanding this model and some variants. A highlight of this method is that if recovery rate is low, then the infection survives forever with positive probability, and spreads outwards linearly leaving a herd immunity region in its wake. Based on joint work with Allan Sly.

The Kardar-Parisi-Zhang (KPZ) universality class of models is characterized by non Gaussian asymptotic fluctuations coming from random matrices. In this talk, I will define the stochastic six vertex model, a specialization of the classical six vertex model (S6V) which is known to lie in the KPZ class. This model can be viewed as a discrete time version of the asymmetric simple exclusion process. Indeed, it converges to ASEP under a certain limit.

In dimension 1, the directed polymer model is in the celebrated KPZ universality class, and for all positive temperatures, a typical polymer path shows non-Brownian KPZ scaling behavior. In dimensions 3 or larger, it is a classical fact that the polymer has two phases: Brownian behavior at high temperature, and non-Brownian behavior at low temperature. We consider the response of the polymer to an external field or tilt, and show that at fixed temperature, the polymer has Brownian behavior for some fields and non-Brownian behavior for others. In other words, the external field can *induce* the phase transition in the directed polymer model. (joint work with S. Mkrtchyan and S. Neville)