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Many classical results for the Brownian motion rely on its special properties such as independent increments and Markov property. When moving away from the Brownian motion, strong local nondeterminism (SLND) becomes a useful tool for studying Gaussian random fields including solutions to stochastic PDEs (SPDEs) with additive Gaussian noise. In this talk, I will present some results on SLND of SPDEs and their applications.

Symmetries such as self-similarity are fundamental concepts that appear across mathematics, from probability and operator theory to mathematical physics. In this talk, I will explore how a novel and constructive algebraic perspective can deepen our understanding of these phenomena. I will begin by discussing how an alternative approach to the renormalization group method provides new insights into scaling limits and universality. I will proceed by presenting a unified framework for understanding self-similarity and Lie symmetries through the lens of group representation theory, operator algebras and spectral theory. This approach reveals the central role of the spectral representation in the Hilbert space of stochastic objects and offers a constructive strategy rooted in the celebrated Stone-Von Neumann-Mackey theorem. Within this framework, I will highlight the surprising roles of the Bessel operator and the Laplacian, which provide unexpected connections and insights.

Two core objects in the KPZ universality class are the KPZ fixed point (KPZFP), a Markov process with state space consisting of upper semi-continuous functions, and the directed landscape (DL). These serve as the scaling limits of random growth processes and random planar geometry, respectively. I will talk about a recent work with Duncan Dauvergne, in which we establish an unexpected new characterization of the DL as the unique natural random field driving the KPZFP. This effectively provides a new construction of the DL from the KPZFP without relying on exactly solvable structures. Building on this, we further establish convergence to the DL for a range of models, including some that lack exact solvability, such as general 1D exclusion processes, various couplings of ASEPs (e.g., the colored ASEP), the Brownian web and random walk web distances, and directed polymers.

I shall discuss first passage percolation on Cayley graphs of Gromov hyperbolic groups under mild conditions on the passage time distribution. Appealing to deep geometric and topological facts about hyperbolic groups and their boundaries, several questions become more tractable in this set up compared to their counterparts in Euclidean lattices. In particular, i shall describe several results about time constants, fluctuations, coalescence of geodesics, and exceptional directions where coalescence fails. Some of the results are parallel to what are expected in Euclidean background geometry, while substantially different features are exhibited in other aspects. Based on joint works with Mahan Mj.

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We show that a multiple radial SLE(\kappa) system is characterized by a conformally covariant partition function satisfying the null vector PDEs. Using the screening method, we construct conformally covariant solutions to the null vector equations. By heuristically taking the classical limit of the partition functions, we construct the multiple radial SLE(0) systems through stationary relations. By constructing the field integral of motion for the Loewner flow, we show that the traces of the multiple radial SLE(0) system are the horizontal trajectories of an equivalence class of quadratic differentials. The stationary relations connect the classification of multiple radial SLE(0) systems to the enumeration of critical points of the master function of trigonometric Knizhnik-Zamolodchikov (KZ) equations. From a Hamiltonian perspective, we prove that the Loewner dynamics with a common parametrization of capacity in multiple radial SLE(0) systems are a special type of classical Calogero-Sutherland system.

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