Physical Sciences and Mathematics Commons^™

Bicategorical Traces And Cotraces, Justin Barhite

Theses and Dissertations--Mathematics

Familiar constructions like the trace of a matrix and the Euler characteristic of a closed smooth manifold are generalized by a notion of trace of an endomorphism of a dualizable object in a bicategory equipped with a piece of additional structure called a shadow functor. Another example of this bicategorical trace, in the form of maps between Hochschild homology of bimodules, appears in a 1987 paper by Joseph Lipman, alongside a more mysterious ”cotrace” map involving Hochschild cohomology. Putting this cotrace on the same category-theoretic footing as the trace has led us to propose a ”bicategorical cotrace” in a closed …

Aspects Of Topology In Moiré Graphene, Ahmed Khalifa

Theses and Dissertations--Physics and Astronomy

Moiré materials, such as twisted bilayer graphene, have provided a rich platform for fundamental physics and potential technological applications. Superconductivity,
correlated insulators, and Chern insulators are examples of phenomena that have been found experimentally in moiré systems. The interplay of strong electron-electron interactions and topology lies at the heart of the mechanism driving these phenomena. In this work, we study the topological aspects of moiré graphene materials, such as the valley Chern and Chern insulating phases. To study the topological response of these phases, we construct models to describe the edge states which are the telltale signs of nontrivial topology. …

Periodic Points On Tori: Vanishing And Realizability, Shane Clark

Theses and Dissertations--Mathematics

Let $X$ be a finite simplicial complex and $f\colon X \to X$ be a continuous map. A point $x\in X$ is a fixed point if $f(x)=x$. Classically fixed point theory develops invariants and obstructions to the removal of fixed points through continuous deformation. The Lefschetz Fixed number is an algebraic invariant that obstructs the removal of fixed points through continuous deformation. \[L(f)=\sum_{i=0}^\infty (-1)^i \tr\left(f_i:H_i(X;\bQ)\to H_i(X;\bQ)\right). \] The Lefschetz Fixed Point theorem states if $L(f)\neq 0$, then $f$ (and therefore all $g\simeq f$) has a fixed point. In general, the converse is not true. However, Lefschetz Number is a complete invariant …

Three Dimensional Modeling And Animation Of Facial Expressions, Alice J. Lin

University of Kentucky Doctoral Dissertations

Facial expression and animation are important aspects of the 3D environment featuring human characters. These animations are frequently used in many kinds of applications and there have been many efforts to increase the realism. Three aspects are still stimulating active research: the detailed subtle facial expressions, the process of rigging a face, and the transfer of an expression from one person to another. This dissertation focuses on the above three aspects.

A system for freely designing and creating detailed, dynamic, and animated facial expressions is developed. The presented pattern functions produce detailed and animated facial expressions. The system produces realistic …

Topological And Combinatorial Properties Of Neighborhood And Chessboard Complexes, Matthew Zeckner

University of Kentucky Doctoral Dissertations

This dissertation examines the topological properties of simplicial complexes that arise from two distinct combinatorial objects. In 2003, A. Björner and M. de Longueville proved that the neighborhood complex of the stable Kneser graph SG_n,k is homotopy equivalent to a k-sphere. Further, for n = 2 they showed that the neighborhood complex deformation retracts to a subcomplex isomorphic to the associahedron. They went on to ask whether or not, for all n and k, the neighborhood complex of SG_n,k contains as a deformation retract the boundary complex of a simplicial polytope. Part one of this dissertation …