Recent progress on the viana conjecture

Marcelo Miranda Viana da Silva born 4 March is a Brazilian mathematician working in dynamical systems theory. He was a Guggenheim Fellow in Viana was vice-president of the International Mathematical Union in —, and president of the Brazilian Mathematical Society — Inhe was a plenary speaker at the International Congress of Mathematiciansin Berlin. Viana is director elected of the IMPA for the period — Viana is a columnist for Folha de S.

He is the chair of the executive committee for the International Congress of MathematiciansRio de Janeiro. Viana was born in Rio de JaneiroBrazil, his parents being Portuguese. He grew up in Portugal, and his undergraduate studies were at the University of Porto. He received his Ph. He is now director at IMPA.

Viana's work concerns chaotic dynamical systems and strange attractors. From Wikipedia, the free encyclopedia. Marcelo Viana. Rio de JaneiroBrazil [1]. Archived from the original on The World Academy of Sciences. Bielefeld Extra Vol. ICM Berlin,vol. Paulo — Colunistas".

recent progress on the viana conjecture

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Mathematician Proves Huge Result on ‘Dangerous’ Problem

IMPA Ph. Jacob Palis [1]. This article about a mathematician is a stub. You can help Wikipedia by expanding it.Conjecture 1: Let be a prime and be the set of squares. Then either or. As the squares are a multiplicative subgroup, it is natural to guess they cannot be written additively as a sumet.

Conjecture 1 is in a similar spirit of a long list of conjectures concerning the independence of multiplication and addition, such as the twin prime conjecturethe abc-conjectureand the sum-product conjecture.

Progress towards Conjecture 1, as of last week, was summarized in this mathoverflow post and this other mathoverflow post and the papers referenced within. In fact, Shparlinski in Theorem 7 improved 1 and then later Shkredov showed in Corollary 2. Brandon Hanson and Giorgis Petridisutilizing the polynomial methodrecently made significant progress towards Conjecture 1. Theorem 1 Hanson-Petridis : Suppose. In particular every element of has a unique representation of the form.

Their techniques handle the case where is replaced by any nontrivial multiplicative subgroup, but we focus on the squares for simplicity. In particular, if is a prime, then Conjecture 1 is established. This implies Conjecture 1 is true for infinitely many primes, provided there are infinitely many Sophie Germain primes yet another conjecture based on the independence of multiplication and addition.

Making use of 1 we are able to prove this unconditionally. Here is the prime counting function. Corollary 1: For all but primes less thanConjecture 1 holds. Proof: Let be a prime. Suppose with. Then by Theorem 1 and 1has a divisor between and. Corollary 1 then follows from the prime number theorem. Ifwe always have trivial bound. Theorem 2: Let be the quadratic character modulo. In particular ifthen. The proof of Theorem 2, which we give, is standard Fourier manipulations see chapter 4 of Tao and Vu for more details.

As we will see below, Hanson and Petridis make no use of this perspective.Skip to search form Skip to main content You are currently offline. Some features of the site may not work correctly. Tao Published Mathematics.

The purpose of these notes is describe the state of progress on the restriction problem in harmonic analysis, with an emphasis on the developments of the past decade or so on the Euclidean space version of these problems for spheres and other hypersurfaces. As the field is quite large and has so many applications, it will be impossible to completely survey the field, but we will try to at least give the main ideas and developments in this area.

View PDF on arXiv. Save to Library. Create Alert. Launch Research Feed. Share This Paper. Top 3 of 38 Citations View All Extension and averaging operators in vector spaces over finite fields. Koh, Chun-Yen Shen From harmonic analysis to arithmetic combinatorics: a brief survey. Izabella Laba Bochner-riesz Multipliers. Kinnear Figures from this paper. Paper Mentions. What's new.

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Citation Type. Has PDF. Publication Type. More Filters. Extension and averaging operators in vector spaces over finite fields. View 1 excerpt, cites background. Research Feed. View 1 excerpt, cites methods. A note on the Stein restriction conjecture and the restriction problem on the torus. Highly Influenced. The restriction and Kakeya conjectures. Optimal constants and maximising functions for Strichartz inequalities.We report on recent progress on both the local and global Gross—Prasad conjectures for unitary groups.

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With an appendix by the authors and Eitan Sayag. Arthur, J. Beuzart-Plessis, R. Preprint Deligne, P. Lecture Notes in Mathematics, vol.

Springer, Berlin Google Scholar. Gan, W. Parts 1, 2.

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Ginzburg, D. Ohio State Univ. Automorphic forms and L -functions I. Global aspects.

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recent progress on the viana conjecture

Annals of Mathematics Studies, vol. Princeton University Press, Princeton Henniart, G. De paires. Ichino, A. Jacquet, H.Take a number, any number. Do all starting numbers lead to 1? Experienced mathematicians warn up-and-comers to stay away from the Collatz conjecture. The Collatz conjecture is quite possibly the simplest unsolved problem in mathematics — which is exactly what makes it so treacherously alluring. Earlier this year one of the top mathematicians in the world dared to confront the problem — and came away with one of the most significant results on the Collatz conjecture in decades.

Lothar Collatz likely posed the eponymous conjecture in the s. The problem sounds like a party trick. Pick a number, any number. Now you have a new number. Apply the same rules to the new number. The conjecture is about what happens as you keep repeating the process. Intuition might suggest that the number you start with affects the number you end up with.

Maybe some numbers eventually spiral all the way down to 1. Maybe others go marching off to infinity. He conjectured that if you start with a positive whole number and run this process long enough, all starting values will lead to 1.

And once you hit 1, the rules of the Collatz conjecture confine you to a loop: 1, 4, 2, 1, 4, 2, 1, on and on forever. The internet is awash in unfounded amateur proofs that claim to have resolved the problem one way or the other.

Other results have similarly picked at the problem without coming close to addressing the core concern.

The Poincaré conjecture - Relativity 21

Lagarias first became intrigued by the conjecture as a student at least 40 years ago. For decades he has served as the unofficial curator of all things Collatz.

And what he realized was that the Collatz conjecture was similar, in a way, to the types of equations — called partial differential equations — that have featured in some of the most significant results of his career. Partial differential equations, or PDEs, can be used to model many of the most fundamental physical processes in the universe, like the evolution of a fluid or the ripple of gravity through space-time. But Tao realized there was something similar about them.

With a PDE, you plug in some values, get other values out, and repeat the process — all to understand that future state of the system. For any given PDE, mathematicians want to know if some starting values eventually lead to infinite values as an output or whether an equation always yields finite values, regardless of the values you start with.This year saw a steady stream of answers or at least partial answers to tough questions that had puzzled mathematicians for decades, as well as new techniques that captured our attention in a big way.

Here are the numbers—and the minds behind them—that mattered most this year. The Riemann Hypothesis is generally seen as the biggest open problem in current mathematics.

Standing sinceit relates to how prime numbers work, and connects to many other branches of math. Researchers this year proved something directly related to the Riemann hypothesis.

Their proof is both insightful toward solving the big question, and fascinating in its own right.


This one is some seriously ancient math. Diophantine equations are named after Diophantus of Alexandria, a 3rd century mathematician. Two particular Diophantine equations, including the one seen in this photo, evaded mathematicians until The breakthrough was enabled by the latest tech in shared computer power.

The improved results posted by prolific mathematician Terrence Tao rocked the math community. Even after Dr. Posed inthe Sensitivity Conjecture became a major unresolved question in mathematical computer science.

In a frenzied few weeks following the initial announcement, scientists digested Dr. Mathematicians are always looking for ways to help in the fight against cancer.

The year started with this joint work by mathematicians and biologists. Innovative math modeling helped guide their experiments on cell growth. Then came this researchwhich used math models to gain new insight on how breast cancer metastasizes. Harvard researchers mastered the math of kirigami this year, illuminating new frontiers in manufacturing and materials sciences.

In Ramsey Theory, mathematicians look for predictable patterns amidst large amounts of chaos. Professor Po-Shen Loh of Carnegie Mellon University made waves this year, popularizing an alternative way of approaching quadratic equations.

Uhlenbeck invented enough math to literally fill books. Her name is foremost in some super advanced math subjects, like geometric analysis and gauge theory.The scope of the conference will be fairly wide, but it will emphasize specific problems in dynamical systems, smooth ergodic theory, and related topics, including differential geometry, group theory and number theory, to name a few, that have seen much progress, but where significant problems vital to the field remain open.

Specific examples are the Katok entropy rigidity conjecture for geodesic flows of negatively curved manifolds, the Boltzmann ergodic hypothesis, Liouvillean phenomena, the construction of metrics with ergodic geodesic flow, smooth rigidity of actions of abelian groups of higher rank. This workshop will include a session to discuss open problems and to produce a list of these, with the intent of producing a concrete agenda for further research on several fronts.

The following are some of the subjects related to dynamical systems that have seen outstanding advances recently, and that will be significant components of this workshop. The workshop will be held on the occasion of the 60th birthday of Anatole Katok. We will take this occasion to honor Katok during the conference for his significant influence on almost all areas in dynamical systems through his research, collaboration, teaching, organization and exposition.

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Additional lodging options may be found on our short term housing page. Billiards, and the question of their hyperbolicity and ergodicity. The theory of partially hyperbolic systems and nonuniformly hyperbolic systems. The close relations between dynamical systems and differential especially Riemannian geometry.

recent progress on the viana conjecture

The significant interactions that have developed between dynamical systems and number theory. The extension of methods from the study of invariant measures for homogeneous and affine actions of higher rank abelian groups, initiated by Furstenberg's x2,x3-question, to other areas. The study of actions of lattices in semisimple Lie groups. The impact on statistical physics of Ruelle's work showing that a generalized Sinai--Ruelle--Bowen-state depends differentiably on the hyperbolic diffeomorphism for which it is defined.

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Visa Information. Hyperbolic Dynamics and Riemannian Geometry. Gerhard Knieper. Symbolic Extension and Entropy Structure. Mike Boyle. Long time Asymptotics in Averaging. Dmitry Dolgopyat. Five Most Resistant Problems in Dynamics. Anatole Katok Pennsylvania State University.

RigidityKAM and other Methods.


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