As part of a more general conjecture by Katok and Spatzier, it was asked if all smooth Anosov Z^r-actions on tori, nilmanifolds and infranilmanifolds without rank-1 factor actions are, up to smooth conjugacy, actions by automorphisms. In this talk, we will discuss a recent joint work with Federico Rodriguez Hertz that affirmatively answers this question.
I'll report on joint work with Nick Sheridan (Princeton/IAS) about mirror symmetry for Calabi-Yau (CY) manifolds. Kontsevich's homological mirror symmetry (HMS) conjecture proposes that the Fukaya category of a CY manifold (viewed as a symplectic manifold) is equivalent to the derived category of coherent sheaves on its mirror. We show that if one can prove an apparently weaker fragment of this conjecture, for some mirror pair, then one can deduce HMS for that pair. We expect this fragment to be amenable to proof for the mirror pairs constructed in the Gross-Siebert program, for example. We also show that the "closed-open string map" is an isomorphism, thereby opening a channel for proving the "closed string" predictions of mirror symmetry.
We construct the Eichler-Shimura morphisms for families of overconvergent modular forms via Scholze's theory of pro-etale site, as well as the Hodge-Tate period maps on modular curves of infinite level. We follow some of the main ideas in the work of Andreatta-Iovita-Stevens. In particular, we reprove the main result in their paper. Since we work entirely on the generic fiber of the modular curve, log structures will not be needed if we only consider the Eichler-Shimura morphism for cusp forms. Moreover, the well-established theory of the Hodge-Tate period map for Shimura varieties of Hodge type may allow us to generalize the construction to more general Shimura varieties. This is a joint work with Hansheng Diao.
We consider direct product of finitely many Young towers with the tails decaying at certain rate and show that the product map admits a Young tower whose tail can be estimated in terms of the rates of component towers. It has been shown that many systems admit such a towers and our results therefore imply statistical properties such as decay of correlations, central limit theorem, large deviations, local limit theorem for large class of product systems.
A (complex) projective structure is a geometric structure
on a real surface, and it is a refinement of a complex structure.
In addition each projective structure enjoys a homomorphism of the
fundamental group of the surface into PSL(2,C), which is called
holonomy representation.
We discuss about some well-known results and basic examples of
complex projective structures. In addition, we talk about different
projective structures sharing such a homomorphism.
A number is called badly approximable if there is a positive constant c such that |x-p/q| > c/q^2 holds for all rationals p/q, so that close approximation by rationals requires relatively large denominators. The set of such numbers is Lebesgue-null but has full Hausdorff dimension. This set can be viewed as the union over c of the set BA(c) of numbers which satisfy the above inequality for the fixed constant c. J. Kurzweil obtained dimension bounds on BA(c), which were later improved by D. Hensley. We will discuss recent work, joint with D. Kleinbock, in which we use homogeneous dynamics to produce dimension bounds for a higher-dimensional analog.