The Topology of Tile Invariants

In this note we use techniques in the topology of 2-complexes to recast some tools that have arisen in the study of planar tiling questions. With spherical pictures we show that the tile counting group associated to a set $T$ of tiles and a set of regions tileable by $T$ is isomorphic to a quotient of the second homology group of a 2-complex built from $T$. In this topological setting we derive some well-known tile invariants, one of which we apply to the solution of a tiling question involving modified rectangles.Comment: 25 pages, 24 figure

Tilings of Annular Region

We present our summer research on mathematical tiling. We classified which rectangular annular regions are tileable by the set of T and skew tretrominoes. We present a partial proof of this result, and discuss some of the context for this problem

Competitive Tiling

Competitive tiling consists of two players, a tile set, a region, and a non-negative integer d. Alice and Bob, our two players, alternate placing tiles on the untiled squares of the region. They play until no more tiles can be placed. Alice wins if at most d squares are untiled at the end of the game, and Bob wins if more than d squares are untiled. For given regions and tile sets we are interested in the smallest value of d such that Alice has a winning strategy. We call this the game tiling number. In this project, we focus on finding the game tiling number for the game played with dominoes on 2 x n rectangles, modified 2 x n rectangles, and rectangular annular regions

Mortality and pulmonary complications in patients undergoing surgery with perioperative SARS-CoV-2 infection: an international cohort study

Background: The impact of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) on postoperative recovery needs to be understood to inform clinical decision making during and after the COVID-19 pandemic. This study reports 30-day mortality and pulmonary complication rates in patients with perioperative SARS-CoV-2 infection. Methods: This international, multicentre, cohort study at 235 hospitals in 24 countries included all patients undergoing surgery who had SARS-CoV-2 infection confirmed within 7 days before or 30 days after surgery. The primary outcome measure was 30-day postoperative mortality and was assessed in all enrolled patients. The main secondary outcome measure was pulmonary complications, defined as pneumonia, acute respiratory distress syndrome, or unexpected postoperative ventilation. Findings: This analysis includes 1128 patients who had surgery between Jan 1 and March 31, 2020, of whom 835 (74·0%) had emergency surgery and 280 (24·8%) had elective surgery. SARS-CoV-2 infection was confirmed preoperatively in 294 (26·1%) patients. 30-day mortality was 23·8% (268 of 1128). Pulmonary complications occurred in 577 (51·2%) of 1128 patients; 30-day mortality in these patients was 38·0% (219 of 577), accounting for 81·7% (219 of 268) of all deaths. In adjusted analyses, 30-day mortality was associated with male sex (odds ratio 1·75 [95% CI 1·28–2·40], p\textless0·0001), age 70 years or older versus younger than 70 years (2·30 [1·65–3·22], p\textless0·0001), American Society of Anesthesiologists grades 3–5 versus grades 1–2 (2·35 [1·57–3·53], p\textless0·0001), malignant versus benign or obstetric diagnosis (1·55 [1·01–2·39], p=0·046), emergency versus elective surgery (1·67 [1·06–2·63], p=0·026), and major versus minor surgery (1·52 [1·01–2·31], p=0·047). Interpretation: Postoperative pulmonary complications occur in half of patients with perioperative SARS-CoV-2 infection and are associated with high mortality. Thresholds for surgery during the COVID-19 pandemic should be higher than during normal practice, particularly in men aged 70 years and older. Consideration should be given for postponing non-urgent procedures and promoting non-operative treatment to delay or avoid the need for surgery. Funding: National Institute for Health Research (NIHR), Association of Coloproctology of Great Britain and Ireland, Bowel and Cancer Research, Bowel Disease Research Foundation, Association of Upper Gastrointestinal Surgeons, British Association of Surgical Oncology, British Gynaecological Cancer Society, European Society of Coloproctology, NIHR Academy, Sarcoma UK, Vascular Society for Great Britain and Ireland, and Yorkshire Cancer Research

Sabbatical Leave Report

This sabbatical provided Dr. Hitchman with the opportunity to work on two projects: research in the mathematics of tiling, and the creation of an open-content edition of his textbook Geometry with an Introduction to Cosmic Topology. Work on both projects began in the fall of 2016, when Dr. Hitchman took a courtesy appointment in the Department of Mathematics at the National University of Ireland, Galway. Dr. Hitchman’s efforts led to the contribution of a chapter in the text A Primer for Undergraduate Research: From Groups and Tiles to Frames and Vaccines, to be published by Birkhaüser in February 2018, and the completion of his open-content geometry text. In September 2017 the American Institute of Mathematics added his text to its list of approved textbooks as part of their Open Textbook Initiative

Faculty Development Grant Report

This grant provided Dr. Hitchman with funding to help facilitate a sabbatical courtesy appointment at The National University of Ireland, Galway (NUIG). At NUIG, Dr. Hitchman focused on generating questions for undergraduate research in the mathematics of tiling. Dr. Hitchman also engaged colleagues on a question in low-dimensional topology that arose from one of Dr. Hitchman’s previous papers. The primary product of this grant to date is the publication of a chapter in the text A Primer for Undergraduate Research: From Groups and Tiles to Frames and Vaccines, to be published by Birkhaüser in February 2018

Geometry with an Introduction to Cosmic Topology

Geometry with an Introduction to Cosmic Topology is motivated by questions that have ignited the imagination of stargazers since antiquity. What is the shape of the universe? Does the universe have an edge? Is it infinitely big? Dr. Hitchman aims to clarify this fascinating area of mathematics and focuses on the mathematical tools used to investigate the shape of the universe. The text follows the Erlangen Program, which develops geometry in terms of a space and a group of transformations of that space. This approach to non-Euclidean geometry provides excellent material by which students can learn the more sophisticated modes of thinking necessary in upper-division mathematics courses.https://digitalcommons.linfield.edu/linfauth/1012/thumbnail.jp

Geometry, Topology, and the Shape of Space

What is the shape of the universe? Michael Hitchman investigates this question, considers its ties to geometry and topology, and discusses some strategies in cosmic topology for (possibly) answering it. This presentation is part of a five-lecture series focusing on different aspects of astronomy and cosmology

The Willamette Valley REU in Mathematics

During summer 2013, Linfield hosted a team of undergraduate research students as part of the NSF-sponsored Willamette Valley Mathematics Consortium REU-RET (Research Experience for Undergraduates—Research Experience for Teachers) program. In addition, two Linfield students participated through funding from the Linfield Student-Faculty Collaborative Research Grant Program. The team consisted of two faculty members, Chuck Dunn and Mike Hitchman, and six undergraduate students: Levi Altringer (Linfield College), Amanda Bright (Westminster College–Missouri), Greg Clark (Westminster College–Pennsylvania), Kyle Evitts (Linfield College), Brian Keating (University of California, San Diego), and Brian Whetter (University of Puget Sound). The team worked on new questions in the area of mathematical tiling. As part of the Consortium, the team participated in five mini-conferences held at each of the participating institutions: Willamette University (two conferences), Lewis and Clark College, the University of Portland, and Linfield College

Extended X-ray absorption fine structure, crystal structures at 295 and 173 K, and electron paramagnetic resonance and electronic spectra of bis[tris(2-pyridyl)-methane]copper(II) dinitrate

The crystal structure of bis[tris(2-pyridyl)methane]copper(II) dinitrate, [Cu{(C₅H₄N)₃CH}₂][NO₃]₂ has been determined. At 295 K the Cu atom lies on a special position so that all six Cu–N bonds are crystallographically equivalent [Cu–N 2.103(4)Å]. The structure at 173 K is very similar [Cu–N 2.095(3)Å]. However, the electronic spectrum suggests that the Cu²⁺ ion experiences a ligand field of tetragonal symmetry. This has been confirmed by the EXAFS of the compound, which shows four nitrogen atoms at 2.04 Å and two at 2.25 Å from the copper. The apparent trigonal symmetry revealed by the X-Ray analysis is thus due to disorder of the long and short Cu–N bonds about the three-fold axis. The EPR spectrum shows an isotropic signal at 295 K, but a signal characteristic of a tetragonally elongated octahedral complex at 150 K. This suggests that the directions of the long and short bonds interchange rapidly on the EPR time-scale at room temperature, but that the complexes become frozen into particular orientations on cooling