We study Euler solutions via novel function spaces constructed using sparseness and local approximations. In particular, we incorporate Tadmor's scale of regularity spaces (2001) to our framework and applying interpolation/extrapolation methods we give a new approach to convergence of approximate Euler solutions. This is joint work with Mario Milman.
The first-order theories of local fields of positive characteristic, i.e. fields of Laurent series over finite fields, are far less well understood than their characteristic zero analogues: the fields of real, complex and p-adic numbers. On the other hand, the existential theory of an equicharacteristic henselian valued field in the language of valued fields is controlled by the existential theory of its residue field. One is decidable if and only if the other is decidable. When we add a parameter to the language, things get more complicated. Denef and Schoutens […]
Olga Paris-Romaskevich Un piano parfait ou une introduction aux mots sturmiens Prenez le clavier d’un piano et écrivez (mentalement !) sur chacune de ses touches les mois de l’année, en commençant par le mois janvier sur la note fa. Fa dièse serait annotée comme février, puis sol comme mars, etc. Vous bouclerez sans surprise, comme il y a 12 notes dans une octave. Mais vous vous apercevrez que les cinq mois courts de l’année se retrouveront tous sur les notes noires. Dans cet exposé, nous allons voir que cela ne […]
Gerard Ben Arous (New York University) Title: Effective dynamics and critical scaling for Stochastic Gradient Descent in high dimensions Abstract: SGD in high dimension is a workhorse for high dimensional statistics and machine learning, but understanding its behavior in high dimensions is not yet a simple task. We study here the limiting 'effective' dynamics of some summary statistics for SGD in high dimensions, and find interesting and new regimes, i.e. not the expected one given by the population gradient flow. We find that a new corrector term is needed and that the phase […]
To a complex of l-adic sheaves on a quasi-projective variety one associate an integer, its complexity. The main result on the complexity is that it is continuous with tensor product, pullback and pushforward, providing effective version of the constructibility theorems in l-adic cohomology. Another key feature is that the complexity bounds the dimensions of the cohomology groups of the complex. This can be used to prove equidistribution results for exponential sums over finite fields. This is due to Will Sawin, written up in collaboration with Javier Fresán and Emmanuel Kowalski.
A σ-variety over a difference field (K,σ) is a pair (X,φ) consisting of an algebraic variety X over K and φ:X → X^σ is a regular map from X to its transform Xσ under σ. A subvariety Y ⊆ X is skew-invariant if φ(Y) ⊆ Y^σ. In earlier work with Alice Medvedev we gave a procedure to describe skew-invariant varieties of σ-varieties of the form (𝔸^n,φ) where φ(x_1,...,x_n) = (P_1(x_1),...,P_n(x_n)). The most important case, from which the others may be deduced, is that of n = 2. In the present […]
Over the last 15 years a remarkable link between o-minimality and algebraic/arithmetic geometry has been unfolding following the discovery of Pila-Wilkie's counting theorem and its applications around unlikely intersections, functional transcendence etc. While the counting theorem is nearly optimal in general, Wilkie has conjectured a much sharper form in the structure R_exp. There is a folklore expectation that such sharper bounds should hold in structures "coming from geometry", but for lack of a general formalism explicit conjectures have been made only for specific structures. I will describe a refinement of […]
Les solutions de petites amplitudes d'équations aux dérivées partielle nonlinéaires dispersives sur un compact sans bord (par exemple un tore ou une sphère) sont soumises à deux effets concurrents : la dispersion des ondes, conséquence du fait que les ondes planes solutions de la partie linéaire de l'équation voyagent avec des vitesses différentes (les ondes s'éloignent les unes des autres) la compacité du domaine qui incite à l'interaction via la non-linéarité (les ondes sont amenées à se revoir souvent !). Qui gagne ? La dynamique en temps long va-t-elle vers […]
Le théorème du corps est l'observation qu'un groupe de rang de Morley fini connexe, résoluble, et non nilpotent, interprète un corps infini. Par d'autres résultats classiques, le corps est commutatif et même algébriquement clos. Le théorème du corps est souvent vu comme corollaire du «théorème d'engendrement par des indécomposables» mais c'est une erreur car il en est indépendant. Il a quelques variantes, des théorèmes de linéarisation d'actions de groupes. Je donnerai un énoncé qui généralise naturellement tous les résultats «à la Zilber». C'est un résultat de linéarisation de bimodules, dans […]
