UW Bothell and Seattle University join forces to host a 2017 research symposium that will showcase a plenary talk and the results from various groups from the National Science Foundation Research Experiences for Undergraduates (REU) sites at that two campuses. The full program for August 10 is below.

**All events will be in Discovery Hall 162. For directions to campus click here.
**

9:00-9:20 **Continental breakfast**

9:20-9:30 **Welcome**

Jennifer McLoud-Mann, Professor, University of Washington Bothell

9:30-9:50

**Vn Property Tiles and 3D Heesch Numbers
**

Jadie Adams, Gabriel Lopez, Nhi Tran

Abstract: This presentation focuses on two properties of tilings: the Vn property and Heesch number.

H. Voderberg constructed a tile with the property that two copies of the tile can enclose one or two other copies. This tile’s design can be extended to a general form which has the property that any number of copies can be enclosed within just two tiles. This property is known as the Vn property, where n is the number of tiles enclosed. Grunbaum and Shephard pose the question: Does there exists a tile with property Vn for each n ≥ 3 that admits a monohedral tiling? We settle this question by describing how any general form of the Voderberg tile can be used to construct a periodic as well as a nonperiodic monohedral tiling. Furthermore, we use one such tiling to disprove another conjecture posed by Grunbaum and Shephard: the neighborhood of any given tile in a monohedral tiling will be equivalent to the patch generated by that tile. We demonstrate that double spiral tilings generated by Voderberg tiles with the Vm property (where m ≥ 3) provide counterexamples. Such tilings contain tiles with cavity neighborhoods (neighborhoods that are not simply connected) so the patches these tiles generate are not equivalent to their neighborhoods.

The Heesch number of a given tile T is the number of coronas that can be formed around a centrally placed copy of T without gaps or overlaps. Previously, there have been very few examples of 3D solids with positive ﬁnite Heesch number. We found the Heesch number of cubes and rhombic dodecahedra with all possible combinations of single protrusions and indentations on their faces through exhaustive computer search. This search revealed that several of these modiﬁed solids have Heesch number 1. Thus we supply examples of 3D tiles with nontrivial Heesch number, and provide a feasible, eﬀective method of discovering more later on.

10:00-10:20

**Enumerating Minimal Length Lattice Walks
**

Jack Evoniuk and Van Magnan

Abstract: If we are given a set of vectors, we can consider lattice paths, which are composed as sequences of vector “steps” from that set. For instance, if the allowable steps are <1,0> and <0,1>, then the number of ways to reach any point (a,b) is equal to the binomial coefficient ‘a+b choose a’.

Depending on the vectors in the set, there may be different numbers of steps in two paths terminating at the same point. We will focus on paths using a minimal number of steps and enumerate such paths for several sets of allowable vectors.

10:30-10:50

**Knotty Outcomes in Billiard Boards of Hexagonal Tiles
**

John Bush, Patty Commins, Tamara Gomez

Abstract: Square mosaic knots have many applications in algebra, such as modeling quantum states. We continue the work of the previous REU cohort in extending mosaic knot theory to a theory of hexagonal mosaic knots, which are knots and links embedded in a plane tiling of regular hexagons. We define a new knot invariant, the *corona number*, by restricting the placement of tiles. We establish the corona number for knots of nine or fewer crossings, excluding $9_{16}$. We also examine tile patches with a high number of link crossings, which we describe as *saturated link diagrams*. Considering patches of varying size and shape, we identify the number of components and what knots are produced in these saturated link diagrams, with a particular focus on patches circumscribed by regular and irregular hexagons. We also examine how the class of $(2, q)$ torus knots can be constructed on a hexagonal mosaic board, and produce an upper limit on the corona number of these knots. Finally, we discuss open questions relating to the saturated link diagrams and bounds on the corona number for torus knots.

11:00-12:00 **Plenary Talk **(Introduction by Casey Mann, Professor, UW Bothell)

**Partial symmetries, inverse semigroups, and inverse semigroup algebras**

David Milan, Associate Professor, University of Texas at Tyler

Abstract: Groups are well known to be the algebraic objects of interest in the study of symmetry. Inverse semigroups are less well known, but can be used to model “partial symmetry” and they also appear in the theory of operators on infinite dimensional spaces.

This talk will introduce congruences on inverse semigroups and how they play a role similar to that of normal subgroups in group theory. We will describe work done by REU students at the University of Texas at Tyler to determine some congruences on the inverse semigroups of groups acting on directed graphs. We will briefly discuss how inverse semigroups play a part in the larger theory of rings and operator algebras.

12:00-1:00 **Lunch
**

Directions provided by Eric Bahuaud, Assistant Professor, Seattle University

1:00-1:20

**A Closer look at Taxicab Geometry: The Parallelogram Law, Conic ****Sections, and Apollonian Sets**

Anna Miller, Freddy Nungaray, Suki Shergill

Abstract: What happens when we change how we think about distance? We explore this question by taking Euclidean geometric ideas and analyzing them in terms of the taxicab metric. We begin by introducing taxicab geometry, and we explain the failure of the parallelogram law. Next, we discuss all of the familiar conic sections in terms of distance formulas and compare them to the ones found by slicing cones. Finally, we delve into the exploration of Apollonian sets.

1:30-1:50

**Downtown Distances and Taxi Hubs: Triangle Centers in Taxicab Geometry**

Shana Crawford, AJ Fish, Julian Tiffay, Niscasio Velez

Abstract: The Euclidean metric is not the only way to measure distance. Using the Taxicab metric, which measures distance as the sum of the horizontal and vertical displacements, we can develop an entirely different geometry. Here we explore a synthetic approach to some triangle centers common in Euclidean geometry. We look for centers analogous to the circumcenter, incenter, Miquel points, and Fermat point, and develop tools for a synthetic analysis of Taxicab geometry.

**Nonsmooth Spectral Gradient Methods on Convex Sets**

2:30-3:00 **Tea Time & Snacks
**

Directions provided by Pietro Paparella, Assistant Professor, UW Bothell

3:00-3:20

**On the realizability of the critical points of a realizable multiset**

Sarah Hoover, Daniel McCormick, Amber Thrall

Abstract: The longstanding nonnegative inverse eignevalue problem (NIEP) is to characterize the spectra of nonnegative matrices. A multiset of complex numbers is called realizable if it is the spectrum of a nonnegative matrix. Monov conjectured that the k-th moments of a multiset of critical points of a realizable multiset are nonnegative. Johnson further conjectured that the multiset of critical points must be realizable. In this work, Johnson’s conjecture, and consequently Monov’s conjecture, is established for a variety of important cases including Ciarlet spectra, Suleimanova spectra, and spectra realizable via Hadamard or DFT similarities. It is noted that the validity of Johnson’s conjecture yields a novel necessary condition for the NIEP (following a majorization result due to Schmeisser.

3:30-3:50

**The Combinatorics of Factorial Base Representation
**

Joanne Beckford, Paul Dalenberg, and Tina Rajabi

Abstract: Every nonnegative integer can be written using what is known as the factorial base representation. We define this notion and explore certain combinatorial structures arising from the arithmetic of these representations. In particular, we will investigate the sum-of-digits function, carry sequences, and a partial order referred to as digital dominance. Finally, we describe an analog of a classical theorem due to Kummer that relates the combinatorial objects of interest by constructing a variety of new integer sequences.

4:00-4:20

**The Combinatorics of Zeckendorf Representations
**

Rachel Chaiser, Dean Dustin, Paul Lagarde

Abstract: In this talk we acknowledge the idea that natural numbers exist regardless of their representation. We explore the properties of a representation that utilizes the Fibonacci sequence as the base, which is called the Zeckendorf representation. We examine the combinatorics arising from the arithmetic of these representations, with a particular emphasis on understanding the “Zeckendorf Tree” that encodes them. We introduce several new results related to the tree, allowing us to develop a partial analog to Kummer’s classical theorem.

4:25 **Closing remarks
**

Milagros Loreto, Assistant Professor, University of Washington Bothell