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This review motivates and synthesizes research efforts in neuroscience-inspired artificial intelligence and biomimetic computing in terms of mortal computation. Specifically, we characterize the notion of mortality by recasting ideas in…
The view of infinity as a metaphor, a basic premise of modern cognitive theory of embodied knowledge, suggests in particular that there may be alternative ways in which one could formalize mathematical ideas about infinity. We discuss the…
These personal reminiscences of the great Russian mathematician Sergey K. Godunov (1929-2023) arose from a request by his daughter, Ekaterina, to contribute a piece to a book she is writing about her father's life. I was honoured to accept…
We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include…
The Turing Test is no longer adequate for distinguishing human and machine intelligence. With advanced artificial intelligence systems already passing the original Turing Test and contributing to serious ethical and environmental concerns,…
This essay examines the relationship between artificial intelligence and the historical evolution of modern Mathematics. Rather than viewing AI as an external rupture, we argue that its effectiveness reveals a structural tendency already…
Since the work [13] by Guo [Invent. Math. 153 (2003), no. 3, 593--630], how to establish the global existence of perturbative classical solutions around a global Maxwellian to the Vlasov-Maxwell-Boltzmann system with the whole range of soft…
This paper investigates the problem of bounding possible output from a counterfactual query given a set of observational data. While various works of literature have described methodologies to generate efficient algorithms that provide an…
The rational base number system, introduced by Akiyama, Frougny, and Sakarovitch in 2008, is a generalization of the classical integer base number system. Within this framework two interesting families of infinite words emerge, called…
We consider the problem of attaining either the maximal increase or reduction of the robustness of a complex network by means of a bounded modification of a subset of the edge weights. We propose two novel strategies combining Krylov…
Kolmogorovs axiomatic framework is the best-known approach to describing probabilities and, due to its use of the Lebesgue integral, leads to remarkably strong continuity properties. However, it relies on the specification of a probability…
The main objective of this paper is the following two results. (1) There exists a computable bi-orderable group that does not have a computable bi-ordering; (2) There exists a bi-orderable, two-generated recursively presented solvable group…
We develop a general game-theoretic framework for reasoning about strategic agents performing possibly costly computation. In this framework, many traditional game-theoretic results (such as the existence of a Nash equilibrium) no longer…
We describe a very general abstract form of sieve based on a large sieve inequality which generalizes both the classical sieve inequality of Montgomery (and its higher-dimensional variants), and our recent sieve for Frobenius over function…
This note complements the paper "One-Way Ticket to Las Vegas and the Quantum Adversary" (arxiv:2301.02003). I develop the ideas behind the adversary bound - universal algorithm duality therein in a different form, using the same perspective…
This paper concerns the ethics and morality of algorithms and computational systems, and has been circulating internally at Facebook for the past couple years. The paper reviews many Nobel laureates' work, as well as the work of other…
The by now standard argument put forth by Yudkowsky, Bostrom and others for why the possibility of a carelessly handled AI breakthrough poses an existential threat to humanity is shown through careful conceptual analysis to be very much…
A definition of entropy via the Kolmogorov algorithmic complexity is discussed. As examples, we show how the meanfield theory for the Ising model, and the entropy of a perfect gas can be recovered. The connection with computations are…
When we want to predict the future, we compute it from what we know about the present. Specifically, we take a mathematical representation of observed reality, plug it into some dynamical equations, and then map the time-evolved result back…
In our previous articles ("The Information and the Matter", v1, v5; more systematically the informational conception is presented in arXiv paper "The Information as Absolute", 2010) it was rigorously shown that the Matter is some…