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We formalize a complete proof of the regular case of Fermat's Last Theorem in the Lean4 theorem prover. Our formalization includes a proof of Kummer's lemma, that is the main obstruction to Fermat's Last Theorem for regular primes. Rather than following the modern proof of Kummer's lemma via class field theory, we prove it by using Hilbert's Theorems 90-94 in a way that is more amenable to formalization.

Randomized matrix algorithms have become workhorse tools in scientific computing and machine learning. To use these algorithms safely in applications, they should be coupled with posterior error estimates to assess the quality of the output. To meet this need, this paper proposes two diagnostics: a leave-one-out error estimator for randomized low-rank approximations and a jackknife resampling method to estimate the variance of the output of a randomized matrix computation. Both of these diagnostics are rapid to compute for randomized low-rank approximation algorithms such as the randomized SVD and randomized Nystr\"om approximation, and they provide useful information that can be used to assess the quality of the computed output and guide algorithmic parameter choices.

We study high-dimensional, ridge-regularized logistic regression in a setting in which the covariates may be missing or corrupted by additive noise. When both the covariates and the additive corruptions are independent and normally distributed, we provide exact characterizations of both the prediction error as well as the estimation error. Moreover, we show that these characterizations are universal: as long as the entries of the data matrix satisfy a set of independence and moment conditions, our guarantees continue to hold. Universality, in turn, enables the detailed study of several imputation-based strategies when the covariates are missing completely at random. We ground our study by comparing the performance of these strategies with the conjectured performance -- stemming from replica theory in statistical physics -- of the Bayes optimal procedure. Our analysis yields several insights including: (i) a distinction between single imputation and a simple variant of multiple imputation and (ii) that adding a simple ridge regularization term to single-imputed logistic regression can yield an estimator whose prediction error is nearly indistinguishable from the Bayes optimal prediction error. We supplement our findings with extensive numerical experiments.

The Gibbs sampler (a.k.a. Glauber dynamics and heat-bath algorithm) is a popular Markov Chain Monte Carlo algorithm which iteratively samples from the conditional distributions of a probability measure $\pi$ of interest. Under the assumption that $\pi$ is strongly log-concave, we show that the random scan Gibbs sampler contracts in relative entropy and provide a sharp characterization of the associated contraction rate. Assuming that evaluating conditionals is cheap compared to evaluating the joint density, our results imply that the number of full evaluations of $\pi$ needed for the Gibbs sampler to mix grows linearly with the condition number and is independent of the dimension. If $\pi$ is non-strongly log-concave, the convergence rate in entropy degrades from exponential to polynomial. Our techniques are versatile and extend to Metropolis-within-Gibbs schemes and the Hit-and-Run algorithm. A comparison with gradient-based schemes and the connection with the optimization literature are also discussed.

We study combinatorial properties of plateaued functions. All quadratic functions, bent functions and most known APN functions are plateaued, so many cryptographic primitives rely on plateaued functions as building blocks. The main focus of our study is the interplay of the Walsh transform and linearity of a plateaued function, its differential properties, and their value distributions, i.e., the sizes of image and preimage sets. In particular, we study the special case of ``almost balanced'' plateaued functions, which only have two nonzero preimage set sizes, generalizing for instance all monomial functions. We achieve several direct connections and (non)existence conditions for these functions, showing for instance that plateaued $d$-to-$1$ functions (and thus plateaued monomials) only exist for a very select choice of $d$, and we derive for all these functions their linearity as well as bounds on their differential uniformity. We also specifically study the Walsh transform of plateaued APN functions and their relation to their value distribution.

We propose a numerical algorithm for the computation of multi-marginal optimal transport (MMOT) problems involving general probability measures that are not necessarily discrete. By developing a relaxation scheme in which marginal constraints are replaced by finitely many linear constraints and by proving a specifically tailored duality result for this setting, we approximate the MMOT problem by a linear semi-infinite optimization problem. Moreover, we are able to recover a feasible and approximately optimal solution of the MMOT problem, and its sub-optimality can be controlled to be arbitrarily close to 0 under mild conditions. The developed relaxation scheme leads to a numerical algorithm which can compute a feasible approximate optimizer of the MMOT problem whose theoretical sub-optimality can be chosen to be arbitrarily small. Besides the approximate optimizer, the algorithm is also able to compute both an upper bound and a lower bound for the optimal value of the MMOT problem. The difference between the computed bounds provides an explicit sub-optimality bound for the computed approximate optimizer. We demonstrate the proposed algorithm in three numerical experiments involving an MMOT problem that stems from fluid dynamics, the Wasserstein barycenter problem, and a large-scale MMOT problem with 100 marginals. We observe that our algorithm is capable of computing high-quality solutions of these MMOT problems and the computed sub-optimality bounds are much less conservative than their theoretical upper bounds in all the experiments.

Markov chain Monte Carlo (MCMC) algorithms are based on the construction of a Markov chain with transition probabilities leaving invariant a probability distribution of interest. In this work, we look at these transition probabilities as functions of their invariant distributions, and we develop a notion of derivative in the invariant distribution of a MCMC kernel. We build around this concept a set of tools that we refer to as Markov chain Monte Carlo Calculus. This allows us to compare Markov chains with different invariant distributions within a suitable class via what we refer to as mean value inequalities. We explain how MCMC Calculus provides a natural framework to study algorithms using an approximation of an invariant distribution, and we illustrate this by using the tools developed to prove convergence of interacting and sequential MCMC algorithms. Finally, we discuss how similar ideas can be used in other frameworks.

We propose a CPU-GPU heterogeneous computing method for solving time-evolution partial differential equation problems many times with guaranteed accuracy, in short time-to-solution and low energy-to-solution. On a single-GH200 node, the proposed method improved the computation speed by 86.4 and 8.67 times compared to the conventional method run only on CPU and only on GPU, respectively. Furthermore, the energy-to-solution was reduced by 32.2-fold (from 9944 J to 309 J) and 7.01-fold (from 2163 J to 309 J) when compared to using only the CPU and GPU, respectively. Using the proposed method on the Alps supercomputer, a 51.6-fold and 6.98-fold speedup was attained when compared to using only the CPU and GPU, respectively, and a high weak scaling efficiency of 94.3% was obtained up to 1,920 compute nodes. These implementations were realized using directive-based parallel programming models while enabling portability, indicating that directives are highly effective in analyses in heterogeneous computing environments.

We study the decidability and complexity of non-cooperative rational synthesis problem (abbreviated as NCRSP) for some classes of probabilistic strategies. We show that NCRSP for stationary strategies and Muller objectives is in 3-EXPTIME, and if we restrict the strategies of environment players to be positional, NCRSP becomes NEXPSPACE solvable. On the other hand, NCRSP_>, which is a variant of NCRSP, is shown to be undecidable even for pure finite-state strategies and terminal reachability objectives. Finally, we show that NCRSP becomes EXPTIME solvable if we restrict the memory of a strategy to be the most recently visited t vertices where t is linear in the size of the game.

In this work, we develop Crank-Nicolson-type iterative decoupled algorithms for a three-field formulation of Biot's consolidation model using total pressure. We begin by constructing an equivalent fully implicit coupled algorithm using the standard Crank-Nicolson method for the three-field formulation of Biot's model. Employing an iterative decoupled scheme to decompose the resulting coupled system, we derive two distinctive forms of Crank-Nicolson-type iterative decoupled algorithms based on the order of temporal computation and iteration: a time-stepping iterative decoupled algorithm and a global-in-time iterative decoupled algorithm. Notably, the proposed global-in-time algorithm supports a partially parallel-in-time feature. Capitalizing on the convergence properties of the iterative decoupled scheme, both algorithms exhibit second-order time accuracy and unconditional stability. Through numerical experiments, we validate theoretical predictions and demonstrate the effectiveness and efficiency of these novel approaches.