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...
Mixing of granular materials is an important engineering process, and a very complex problem. In this thesis, I use granular mixing in a half-full biaxial spherical tumbler (BST), a spherical container that rotates sequentially about two orthogonal axes, to motivate the study of mixing with piecewise isometries (PWIs), a rich...
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...
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....
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...
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 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...
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 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...
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.