International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library

Primordial non-Gaussianity and Bispectrum Measurements in the Cosmic Microwave Background and Large-Scale Structure

The most direct probe of non-Gaussian initial conditions has come from bispectrum measurements of temperature fluctuations in the Cosmic Microwave Background and of the matter and galaxy distribution at large scales. Such bispectrum estimators are expected to continue to provide the best constraints on the non-Gaussian parameters in future observations. We review and compare the theoretical and observational problems, current results and future prospects for the detection of a non-vanishing primordial component in the bispectrum of the Cosmic Microwave Background and large-scale structure, and the relation to specific predictions from different inflationary models.Comment: 82 pages, 23 figures; Invited Review for the special issue "Testing the Gaussianity and Statistical Isotropy of the Universe" for Advances in Astronom

The non-equilibrium statistical physics of stochastic search, foraging and clustering

This dissertation explores two themes central to the field of non-equilibrium statistical physics. The first is centered around the use of random walks, first-passage processes, and Brownian motion to model basic stochastic search processes found in biology and ecological systems. The second is centered around clustered networks: how clustering modifies the nature of transition in the appearance of various graph motifs and their use in modeling social networks. In the first part of this dissertation, we start by investigating properties of intermediate crossings of Brownian paths. We develop simple analytical tools to obtain probability distributions of intermediate crossing positions and intermediate crossing times of Brownian paths. We find that the distribution of intermediate crossing times can be unimodal or bimodal. Next, we develop analytical and numerical methods to solve a system of diffusive searchers which are reset to the origin at stochastic or periodic intervals. We obtain the optimal criteria to search for a fixed target in one, two and three dimensions. For these two systems, we also develop efficient ways to simulate Brownian paths, where the simulation kernel makes maximal use of first-passage ideas. Finally we develop a model to understand foraging in a resource-rich environment. Specifically, we investigate the role of greed on the lifetime of a diffusive forager. This lifetime shows non-monotonic dependence on greed in one and two dimensions, and surprisingly, a peak for negative greed in 1d. In the second part of this dissertation, we develop simple models to capture the non-tree-like (clustering) aspects of random networks that arise in the real world. By 'clustered networks', we specifically mean networks where the probability of links between neighbors of a node (i.e., 'friends of friends') is positive. We discuss three simple and related models. We find a series of transitions in the density of graph motifs such as triangles (3-cliques), 4-cliques etc as a function of the clustering probability. We also find that giant 3-cores emerge through first- or second-order, or even mixed transitions in clustered networks

Probabilistic Methods for Model Validation

This dissertation develops a probabilistic method for validation and verification (V&V) of uncertain nonlinear systems. Existing systems-control literature on model and controller V&V either deal with linear systems with norm-bounded uncertainties,or consider nonlinear systems in set-based and moment based framework. These existing methods deal with model invalidation or falsification, rather than assessing the quality of a model with respect to measured data. In this dissertation, an axiomatic framework for model validation is proposed in probabilistically relaxed sense, that instead of simply invalidating a model, seeks to quantify the "degree of validation". To develop this framework, novel algorithms for uncertainty propagation have been proposed for both deterministic and stochastic nonlinear systems in continuous time. For the deterministic flow, we compute the time-varying joint probability density functions over the state space, by solving the Liouville equation via method-of-characteristics. For the stochastic flow, we propose an approximation algorithm that combines the method-of-characteristics solution of Liouville equation with the Karhunen-Lo eve expansion of process noise, thus enabling an indirect solution of Fokker-Planck equation, governing the evolution of joint probability density functions. The efficacy of these algorithms are demonstrated for risk assessment in Mars entry-descent-landing, and for nonlinear estimation. Next, the V&V problem is formulated in terms of Monge-Kantorovich optimal transport, naturally giving rise to a metric, called Wasserstein metric, on the space of probability densities. It is shown that the resulting computation leads to solving a linear program at each time of measurement availability, and computational complexity results for the same are derived. Probabilistic guarantees in average and worst case sense, are given for the validation oracle resulting from the proposed method. The framework is demonstrated for nonlinear robustness veri cation of F-16 flight controllers, subject to probabilistic uncertainties. Frequency domain interpretations for the proposed framework are derived for linear systems, and its connections with existing nonlinear model validation methods are pointed out. In particular, we show that the asymptotic Wasserstein gap between two single-output linear time invariant systems excited by Gaussian white noise, is the difference between their average gains, up to a scaling by the strength of the input noise. A geometric interpretation of this result allows us to propose an intrinsic normalization of the Wasserstein gap, which in turn allows us to compare it with classical systems-theoretic metrics like v-gap. Next, it is shown that the optimal transport map can be used to automatically refine the model. This model refinement formulation leads to solving a non-smooth convex optimization problem. Examples are given to demonstrate how proximal operator splitting based computation enables numerically solving the same. This method is applied for nite-time feedback control of probability density functions, and for data driven modeling of dynamical systems