Georgia Institute of Technology College of Sciences

In 1970 Matiyasevich, building on earlier work of Davis--Putnam--Robinson, proved that every enumerable subset of Z is Diophantine, thus showing that Hilbert's 10th problem is undecidable for $\mathbb{Z}$. The problem of extending this result to the ring of integers of number fields (and more generally to finitely generated infinite rings) has attracted significant attention and, thanks to the efforts of many mathematicians, the task has been reduced to the problem of constructing, for certain quadratic extensions of number fields $L/K$, an elliptic curve $E/K$ with $rk(E(L))=rk(E(K))>0$. 

In this talk I will explain joint work with Peter Koymans, where we use Green--Tao to construct the desired elliptic curves, settling Hilbert 10 for every finitely generated infinite ring.

This presentation will expound the challenges involved in the generation of digital twins (DT) as valuable tools for supporting innovation and providing informed decision support for the optimization of properties and/or performance of advanced material systems. This presentation will describe the foundational AI/ML (artificial intelligence/machine learning) concepts and frameworks needed to formulate and continuously update the DT of a selected material system. The central challenge comes from the need to establish reliable models for predicting the effective (macroscale) functional response of the heterogeneous material system, which is expected to exhibit highly complex, stochastic, nonlinear behavior. This task demands a rigorous statistical treatment (i.e., uncertainty reduction, quantification and propagation through a network of human-interpretable models) and fusion of insights extracted from inherently incomplete (i.e., limited available information), uncertain, and disparate (due to diverse sources of data gathered at different times and fidelities, such as physical experiments, numerical simulations, and domain expertise) data used in calibrating the multiscale material model. This presentation will illustrate with examples how a suitably designed Bayesian framework combined with emergent AI/ML toolsets can uniquely address this challenge.

Randomized Kaczmarz methods — popular special case of the sketch-and-project optimization framework — solve linear systems through iterative projections onto randomly selected equations, resulting in exponential expected convergence via cheap, local updates. While known to be effective in highly overdetermined problems or under the restricted data access, identifying generic scenarios where these methods are advantageous compared to classical Krylov subspace solvers (e.g., Conjugate Gradient, LSQR, GMRES) remained open. In this talk, I will present our recent results demonstrating that properly designed randomized Kaczmarz (sketch-and-project) methods can outperform Krylov methods for both square and rectangular systems complexity-wise. In addition, they are particularly advantageous for approximately low-rank systems common in machine learning (e.g., kernel matrices, signal-plus-noise models) as they quickly capture the large outlying singular values of the linear system. Our approach combines novel spectral analysis of randomly sketched projection matrices with classical numerical analysis techniques, such as including momentum, adaptive regularization, and memoization.