For instance by developing new recursive formula the WeilPetersson volume of this space Mirzakhani was able to asymptotically count number simple prime geodesics length up some threshold hyperbolic surface more precisely she obtained asymptotics such given orbit mapping class group answer turns out polynomial contrast much larger nonsimple whose are exponential theorem developed classic series works Delsart Huber Selberg and Margulis also used establish proof conjecture Witten intersection numbers that first proven Kontsevich. Ratner s theorems are incredibly useful for all sorts of problems connected to homogeneous dynamics and analogous established by Mirzakhani Eskin Mohammadi have similarly broad range applications instance counting periodic billiard trajectories rational polygons. . The problem involves perhaps simplest example of Schr dinger operator with nontrivial spectral properties namely almost Mathieu defined for parameters and by discrete onedimensional cosine potential This bounded selfadjoint thus has spectrum that compact subset real line arises number physical contexts notably theory integer quantum Hall effect though will discuss these applications here

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Read More →H. It is expected that for any reasonable choice of initial data the solution to this equation should asymptotically be distributed according Kolmogorov power law discussed previous post. The public was invited to participate for first time. On the other hand professionals working applied form formulate mathematical models problem solving realworld situations. HO math

Read More →Prior to the work of Avila and Jitormiskaya there were number partial results on this problem notably Puig establishing Cantor spectrum for full measure set parameters well requiring perturbative hypothesis such being very small large. HO math . RT math

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Read More →Manjul Bhargava found methods to compose forms of higher order with exponents larger than in the polynomials similar way as Gauss did quadratic and even described these by means general theory. Besides the Fields Medal Bhargava received many other prizes and awards for instance Blumenthal Advancement of Research Pure Mathematics Packard Foundation Fellowship Science Engineering Merten . Unfortunately the strong Feller property is unlikely to hold in phase space NavierStokes but Hairer and Mattingly develop clean abstract substitute this which they call asymptotic again regularity transition operator turn then demonstrated by careful application of Malliavin calculus

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Ratner s theorems are already notoriously difficult prove and rely very much on polynomial stability properties of unipotent flows in this even more complicated setting longer tractable Mirzakhani instead uses recent exponential drift method Benoist Quint as substitute. a Liouville number or very far from rational Diophantine although the analyses for these two cases failed to meet middle leaving some untreated. The ten martini problem asserts that for every irrational and choice of coupling constant spectrum is homeomorphic to Cantor set. people attended the Decatur offsite