Although there has been profound evidence showing the positive correlation between spatial abilities and math performances, we still know very little about how and why spatial thinking facilitates the learning of mathematics. This dissertation unpacks several aspects of mathematics that are embedded in learning and playing an ancient and rich...
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...
Interfacial phenomena play a considerable role in many physical, chemical and biological systems. Some of these systems exhibit interaction of several interfaces. This thesis contains a study of two types of systems characterized by interacting interfaces. First, a theory of the formation of nanoscale porous structures in oxides of metals...
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.
This paper covers three main topics. The first is addressing the question of interpolating between disparate index theorems on noncommutative two-tori. The second is to compute Hochschild cohomology for quantum special linear and special unitary groups. The third is producing an orthonormal basis for the vector space of matrix corepresentations...
We consider two problems in low Reynolds-number, interfacial fluid mechanics: the rupture of thin liquid films on chemically patterned solid substrates, and the engulfment of foreign particles by a solidification front progressing through a binary alloy.
First we investigate the stability and rupture of thin liquid films on patterned sub-...
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...
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 $DM^{eff,-}_{Nis}$....
Given any polyhedron in R3, we can cut it open along its edges, flatten it out, and obtain a polygon in the plane R2. In this project, we explored the opposite process, an open question that was first posed about 70 years ago: given a polygon in R2, what is...
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....