Thu, 04 Mar 2021
11:30
Virtual

Non-archimedean analogue of Wilkie's conjecture, and, point counting from Pfaffian over subanalytic to Hensel minimal

Raf Cluckers
(University of Lille)
Abstract

Point counting on definable sets in non-archimedean settings has many faces. For sets living in Q_p^n, one can count actual rational points of bounded height, but for sets in C((t))^n, one rather "counts" the polynomials in t of bounded degree. What if the latter is of infinite cardinality? We treat three settings, each with completely different behaviour for point counting : 1) the setting of subanalytic sets, where we show finiteness of point counting but growth can be aribitrarily fast with the degree in t ; 2) the setting of Pfaffian sets, which is new in the non-archimedean world and for which we show an analogue of Wilkie's conjecture in all dimensions; 3) the Hensel minimal setting, which is most general and where finiteness starts to fail, even for definable transcendental curves! In this infinite case, one bounds the dimension rather than the (infinite) cardinality. This represents joint work with Binyamini, Novikov, with Halupczok, Rideau, Vermeulen, and separate work by Cantoral-Farfan, Nguyen, Vermeulen.

Mon, 15 Feb 2010
14:15
Eagle House

Fractional Stockastic Fields and Wavelet Methods

Antoine Ayache
(University of Lille)
Abstract

Abstract: The goal of this talk is to discuss threeproblems on fractional and related stochastic fields, in which wavelet methodshave turned out to be quite useful.

  The first problemconsists in constructing optimal random series representations of Lévyfractional Brownian field; by optimal we mean that the tails of the seriesconverge to zero as fast as possible i.e. at the same rate as the l-numbers.Note in passing that there are close connections between the l-numbers of aGaussian field and its small balls probabilities behavior.

  The secondproblem concerns a uniform result on the local Hölder regularity (the pointwiseHölder exponent) of multifractional Brownian motion; by uniform we mean thatthe result is satisfied on an event with probability 1 which does not depend onthe location.

  The third problemconsists in showing that multivariate multifractional Brownian motion satisfiesthe local nondeterminism property. Roughly speaking, this property, which wasintroduced by Berman, means that the increments are asymtotically independentand it allows to extend to general Gaussian fields many results on the localtimes of Brownian motion.

 

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