Dynamical Properties of Weierstrass Elliptic Functions on Square Lattices

In this dissertation we prove that the Julia set of a Weierstrass elliptic function on a square lattice is connected. We further show that the parameter space contains an infinite number of Mandelbrot sets. As a consequence, this proves the existence of Siegel disks and gives a description of the bifurcation locus about super-attracting parameters corresponding to super-attracting fixed points. We conclude with a description of a family of rational maps that approximate the Weierstrass elliptic function on a square lattice

Dynamics of SL2(ℝ) Over Moduli Space in Genus Two

This paper classifies orbit closures and invariant measures for the natural action of SL2(ℝ) on Ωℳ2, the bundle of holomorphic 1-forms over the moduli space of Riemann surfaces of genus two.Mathematic

Teichmüller Curves in Genus Two: Discriminant and Spin

Mathematic

Braid Groups and Hodge theory

This paper gives an account of the unitary representations of the braid group that arise via the Hodge theory of cyclic branched coverings of $\mathbb{P}^1$ , highlighting their connections with ergodic theory, complex reflection groups, moduli spaces of 1-forms and open problems in surface topology.Mathematic

Mahler Measures of Hypergeometric Families of Calabi-Yau Varieties

The logarithmic Mahler measure of a nonzero n-variable Laurent polynomial P ∈ C [X_1^(±1),…,X_n^(±1) ], denoted by m(P), is defined to be the arithmetic mean of log⁡|P| over the n-dimensional torus. It has been proved or conjectured that the logarithmic Mahler measures of some classes of polynomials have connections with special values of L-functions. However, the precise interpretation of m(P) in terms of L-values is not clearly known, so it has become a new trend of research in arithmetic geometry and number theory to understand this phenomenon. In this dissertation, we study Mahler measures of certain families of Laurent polynomials of two, three, and four variables, whose zero loci define elliptic curves, K3 surfaces, and Calabi-Yau threefolds, respectively. On the one hand, it is known that these Mahler measures can be expressed in terms of hypergeometric series and logarithms. On the other hand, we derive explicitly that some of them can be written as linear combinations of special values of Dirichlet and modular L-functions, which potentially carry some arithmetic information of the corresponding algebraic varieties. Our results extend those of Boyd, Bertin, Lalín, Rodriguez Villegas, Rogers, and many others. We also prove that Mahler measures of those associated to families of K3 surfaces are related to the elliptic trilogarithm defined by Zagier. This can be seen as a higher dimensional analogue of relationship between Mahler measures of bivariate polynomials and the elliptic dilogarithm known previously by work of Guillera, Lalín, and Rogers

A walk in the noncommutative garden

This text is written for the volume of the school/conference "Noncommutative Geometry 2005" held at IPM Tehran. It gives a survey of methods and results in noncommutative geometry, based on a discussion of significant examples of noncommutative spaces in geometry, number theory, and physics. The paper also contains an outline (the ``Tehran program'') of ongoing joint work with Consani on the noncommutative geometry of the adeles class space and its relation to number theoretic questions.Comment: 106 pages, LaTeX, 23 figure