This dissertation addresses the structure of the group of interval exchange transformations. The two primary topics considered are:
a) the classification of interval exchange actions for certain groups; and b) properties of the interval exchange group which are reflected in the dynamics of interval exchange maps.
In Chapter 3 a...
In this thesis, we study arithmetic phenomena exhibited by polynomial dynamical systems on the projective line. Specifically, given a number field $K$, we are interested in the arithmetic of orbits of points $\alpha\in K$ under polynomials $\phi\in K[z]$. Given such a polynomial $\phi$ of degree $d\ge2$, we prove a lower...
The Witten Laplacian corresponding to a Morse function on the circle is studied using methods of complex WKB and resurgent analysis. It is shown that under certain assumptions the low-lying eigenvalues of the Witten Laplacian are resurgent.
Let X be a quasi-projective complex variety. It follows from the work of Voevodsky that the motivic cohomology of X, denoted as $H^{p,q}(X)$ where q and p are integers with q nonnegative, can be represented in the triangulated category of motives over the field of complex numbers, denoted as $DM^{eff,-}_{Nis}$....
This dissertation contains three results related to modular forms and Galois representations of low weight. In chapter 1, we prove that the Galois pseudo-representation valued in a Hecke algebra which acts faithfully on a space of weight one Katz modular forms of level prime to p is unramified at p....
Algebras and their bimodules form a 2-category in which 2-morphisms are certain zero-th Hochschild cohomology groups. When we derive this structure (i.e., use Hochschild cochains instead of HH^0 for 2-morphisms), we find that algebras form a category in dg cocategories. The Hochschild-Kostant-Rosenberg theorem and non-commutative calculus give a rich algebraic...