DIMACS | Center for Discrete Mathematics and Theoretical Computer Science
Let X and Y be two finite-area hyperbolic Riemann surfaces. Since X and Y are both hyperbolic, we know that they must both be uniformized by ℍ ; however, the maps ℍ → X and ℍ → Y are not in general of finite degree. In general, guaranteeing that X and Y share conformally equivalent finite covers is too much to hope. The Ehrenpreis conjecture states that we can achieve something close: in particular, for any K>1, we can find finite covers X' → X and Y' → Y such that X' and Y' are K-quasiconformally equivalent; in other words, we can always find finite covers which have little distortion of angles relative to each other. The conjecture was recently settled in the case of closed surfaces in 2015 by Kahn and Markovic, but there has been no such resolution for punctured Riemann surfaces, and in fact the truth of the conjecture remains contested. The modular surface is a model example of a punctured Riemann surface of finite area. In this project, we numerically find and study finite covers of the modular surface constructed by gluing together immersed hyperbolic pants and "degenerate pants", in which we allow one cuff to degenerate into a cusp, subject to similar technical constraints. We hope that the work will help in generating insight towards an angle of attack in a proof of the cusped Ehrenpreis conjecture inspired by the methods used in the proof of the case of closed surfaces.
We consider a variant of a problem posed by Lubetzky: given an arbitrary vertex coloring of a graph, if we randomly update the color of each successive vertex in a given sequence, so that at each step, each color has probability proportional to λx with x its count among the neighbors, then what initial coloring maximizes the probability of all vertices being the same color, say blue, at the end? If there are only two colors, there is a trivial proof by coupling that the all blue initial coloring is best. For more colors, we show that this is still true under certain assumptions on the graph, but we construct a counterexample in general. We build our counterexample by first appealing to a generalized setting, where the graph and the update function λx change at each step, and then repeatedly modifying it until it works in the original setting. The techniques we will use include identifying and exploiting ergodic Markov chains within our process and implementing a digraph within a standard graph by ensuring that the neighbors that we want to be the inneighbors vastly outnumber the neighbors that we want to be the outneighbors for every vertex.
Graphs are everywhere and are growing increasingly compact with data. Understanding the data found within requires a summarization process to dwindle the information down to a human digestible amount. When summarizing text, translating how humans build connections and realize the semantic meaning of phrases to a computer is far from trivial. Some methods involve computing scores for each word based on how often they occur in a given text and computing word vectors to determine similarity between words. However, none of these processes by themselves work well to accurately summarize text. The result of my process shows that combining techniques can produce promising summarized representations of information. These results demonstrate the potential that finding the perfect combination of techniques has. I anticipate my process to be a starting point for more sophisticated combinations of methods to develop.
Our work focuses on building preparedness and resilience within vulnerable communities, specifically religious communities and houses of worship, against violent extremism. In today's society, the threat of mass casualty attacks and targeted violence is a global issue, with a notable increase in anti-Islamic and anti-Semitic attacks, as well as an increase in attacks by white nationalists. Many houses of worship have been targets of these attacks with numerous lives lost and many others left physically and emotionally impaired. With anti-Semitic and anti-Islamic incidents occurring globally, it is important for houses of worship to have the tools and guidance to prepare for and prevent further damage; however, there is no widely available, comprehensive resource of best practices or methods for religious communities. The R.E.S.I.L.I.E.N.C.E. model was created as a basis for our study. We researched case studies that highlighted best practices that can educate vulnerable communities about resources and methods to better protect themselves. We have found many strategies to do so, such as building connections between the public and enlisted guardians, sharing information through constant communications, implementing the best practices and plans for a situation, providing sufficient resources for said plans, and neutralizing negative mindsets through encouragement and empowerment. Previously, these strategies were not compiled into a single resource that would be available to the public and vulnerable communities, but with this research, that can change. There are many roles and responsibilities community members can assume to build readiness. While analyzing these roles and responsibilities, we also found many exemplary ways to utilize the aforementioned strategies and build resiliency against any form of violence within any vulnerable community.
We studied a new method for simulating T(n)-time-bounded Turing machines by unbounded circuits with low depth and size polynomial in T. Our construction offers lower depth circuits than the ones by Ryan Williams in the case that the total number of reversals of every tape head is less than O(T=logT). This continues a research direction of Borodin to study the connections between simultaneous Turing machine time and space complexity and simultaneous circuit size and depth complexity, which is heavily connected to the research of Hopcroft, Paul, and Valiant on time and space. Finally, reducing the depth of circuits for Turing machines has applications to parallelizing sequential computations, as discussed in Ryan Williams' aforementioned paper.
The Apollonian circle packing, generated from three mutually-tangent circles in the plane, has inspired over the past half-century the study of other classes of space-filling packings, both in two and in higher dimensions. Recently, Kontorovich and Nakamura introduced the notion of crystallographic sphere packings, n-dimensional packings of spheres with symmetry groups that are isometries of Hn+1, which can be related to configurations of planes in Hn+1 for various quadratic forms in n+2 variables. When applied in conjunction with the Koebe-Andreev-Thurston Theorem, Kontorovich and Nakamura's Structure Theorem guarantees crystallographic packings to be generated from polyhedra in n=2. The Structure Theorem similarly allows us to generate packings from the reflective extended Bianchi groups in n=2 by applying Vinberg's algorithm to obtain the appropriate Coxeter diagrams. In n>2, the Structure Theorem when used with Vinberg's algorithm allows us to explore whether certain Coxeter diagrams in Hn+1 for a given quadratic form admit a packing at all. Kontorovich and Nakamura's Finiteness Theorem shows that there exist only finitely many classes of superintegral such packings, all of which exist in dimensions n<21. In this work, we systematically determine and enumerate crystallographic sphere packings arising from polyhedra on up to seven vertices, all crystallographic packings arising from Bianchi groups, and all known examples of crystallographic packings arising from higher dimensional quadratic forms.
