Existence and minimizing properties of retrograde orbits to the three-body...
Poincaré made the first attempt in 1896 on applying variational calculus to the three-body problem and observed that collision orbits do not necessarily have higher values of action than classical...
View ArticleThe primes contain arbitrarily long arithmetic progressions
We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemerédi’s theorem, which asserts that any subset of the integers of...
View ArticleCyclic homology, cdh-cohomology and negative $K$-theory
We prove a blow-up formula for cyclic homology which we use to show that infinitesimal $K$-theory satisfies ${\rm cdh}$-descent. Combining that result with some computations of the ${\rm...
View ArticleThe Poincaré inequality is an open ended condition
Let $p >1$ and let $(X,d,\mu)$ be a complete metric measure space with $\mu$ Borel and doubling that admits a $(1,p)$-Poincaré inequality. Then there exists $\varepsilon >0$ such that $(X,d,\mu)$...
View ArticleGrowth and generation in $\mathrm{SL}_2(\mathbb{Z}/p \mathbb{Z})$
We show that every subset of $\mathrm{SL}_2(\mathbb{Z}/p\mathbb{Z})$ grows rapidly when it acts on itself by the group operation. It follows readily that, for every set of generators $A$ of...
View ArticleUniform expansion bounds for Cayley graphs of $\mathrm{SL}_2(F_p)$
We prove that Cayley graphs of $\mathrm{SL}_2(\mathbb{F}_p)$ are expanders with respect to the projection of any fixed elements in $\mathrm{SL}(2, \mathbb{Z})$ generating a non-elementary subgroup, and...
View ArticleAlmost all cocycles over any hyperbolic system have nonvanishing Lyapunov...
We prove that for any $s>0$ the majority of $C^s$ linear cocycles over any hyperbolic (uniformly or not) ergodic transformation exhibit some nonzero Lyapunov exponent: this is true for an open dense...
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