Resolution of singularities and geometric proofs of the Lojasiewicz inequalities

The Łojasiewicz inequalities for real analytic functions on Euclidean space were first proved by Stanisław Łojasiewicz in [83, 84, 87] using methods of semianalytic and subanalytic sets, arguments later simplified by Bierstone and Milman [7]. In this article, we first give an elementary geometric, coordinate-based proof of the Łojasiewicz inequalities in the special case where the function is C^1 with simple normal crossings. We then prove, partly following Bierstone and Milman [9, Section 2] and using resolution of singularities for real analytic varieties, that the gradient inequality for an arbitrary real analytic function follows from the special case where it has simple normal crossings. In addition, we give elementary proofs of the the Łojasiewicz inequalities when the function is C^2 and Morse–Bott or C^N and Morse–Bott of order N ≥ 2

On the Effective Putinar's Positivstellensatz and Moment Approximation

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar's Positivstellensatz over a compact basic semialgebraicset S, with a new polynomial bound on the degree of the positivity certificates. This bound involves a Lojasiewicz exponent associated to the description of S. We show that if the gradients of the active constraints are linearly independent on S (Constraint Qualification condition),this Lojasiewicz exponent is equal to 1. We deduce the first general polynomial bound on the convergence rate of the optima in Lasserre's Sum-of-Squares hierarchy to the global optimum of a polynomial function on S, and the first general bound on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated pseudo-moment sequences, which are positive on the quadratic module of S

Singularities, Supersymmetry and Combinatorial Reciprocity

This work illustrates a method to investigate certain smooth, codimension-two, real submanifolds of spheres of arbitrary odd dimension (with complements that fiber over the circle) using a novel supersymmetric quantum invariant. Algebraic (fibered) links, including Brieskorn-Pham homology spheres with exotic differentiable structure, are examples of said manifolds with a relative diffeomorphism-type that is determined by the corresponding (multivariate) Alexander polynomial.Engineering and Applied Science

Integration of Oscillatory and Subanalytic Functions

We prove the stability under integration and under Fourier transform of a concrete class of functions containing all globally subanalytic functions and their complex exponentials. This paper extends the investigation started in [J.-M. Lion, J.-P. Rolin: "Volumes, feuilles de Rolle de feuilletages analytiques et th\'eor\`eme de Wilkie" Ann. Fac. Sci. Toulouse Math. (6) 7 (1998), no. 1, 93-112] and [R. Cluckers, D. J. Miller: "Stability under integration of sums of products of real globally subanalytic functions and their logarithms" Duke Math. J. 156 (2011), no. 2, 311-348] to an enriched framework including oscillatory functions. It provides a new example of fruitful interaction between analysis and singularity theory.Comment: Final version. Accepted for publication in Duke Math. Journal. Changes in proofs: from Section 6 to the end, we now use the theory of continuously uniformly distributed modulo 1 functions that provides a uniform technical point of view in the proofs of limit statement

Acceleration Methods

This monograph covers some recent advances in a range of acceleration techniques frequently used in convex optimization. We first use quadratic optimization problems to introduce two key families of methods, namely momentum and nested optimization schemes. They coincide in the quadratic case to form the Chebyshev method. We discuss momentum methods in detail, starting with the seminal work of Nesterov and structure convergence proofs using a few master templates, such as that for optimized gradient methods, which provide the key benefit of showing how momentum methods optimize convergence guarantees. We further cover proximal acceleration, at the heart of the Catalyst and Accelerated Hybrid Proximal Extragradient frameworks, using similar algorithmic patterns. Common acceleration techniques rely directly on the knowledge of some of the regularity parameters in the problem at hand. We conclude by discussing restart schemes, a set of simple techniques for reaching nearly optimal convergence rates while adapting to unobserved regularity parameters.Comment: Published in Foundation and Trends in Optimization (see https://www.nowpublishers.com/article/Details/OPT-036

Artin Approximation

In 1968, M. Artin proved that any formal power series solution of a system of analytic equations may be approximated by convergent power series solutions. Motivated by this result and a similar result of P{\l}oski, he conjectured that this remains true when the ring of convergent power series is replaced by a more general kind of ring. This paper presents the state of the art on this problem, aimed at non-experts.Comment: Final versio