Each participant in our program produces a paper describing their results after just eight weeks of work. The research represented by these papers was made possible by NSF-REU grants DMS-9987955, DMS-0139426, DMS-0453605, DMS-0850959, DMS-1156608, DMS-1461286, DMS-1758020 as well as the generous support of California State University, San Bernardino.
Below are some of the papers from each of the topics we have studied: Geometry, and Knot Theory. Although combinatorics is not necessarily one of the topics we still offer as part of this REU, we list the work of our former participants in that field as well.
Summer 2023 Geometry Projects
The Category of Model Spaces and Applications to Structure Groups of Algebraic Curvature Tensors, by Logan Borghi
The Denseness of Canonical Algebraic Curvature Tensors and a Revision to the Signature Conjecture, by Kieran Favazza
On the Ricci Map of Algebraic Curvature Tensors, by Alice Postovskiy
Summer 2023 Knot Theory Projects
Symmetries of Augmented Links: On FALs, Chain Links, and their FAL-Equivalent Counterparts, by Annette Belleman
Classifying Cuboctahedral FALs, by Annie Givens and Regan Richardson
b-prime Fully Augmented Links and Complete Augmenation, by Ethan Sherman
Crushtaceans: A Graph Theoretic Representation of Fully Augmented Links, by Cole Twogood
Summer 2022 Geometry Projects
Kernels of Algebraic Curvature Tensors of Mixed Symmetric and Skew-Symmetric Build, by Basia Klos
Structure Groups of Algebraic Curvature Tensors Containing Rotation Actions, by Jeffrey Nubla
Decomposability and Linear Independence of Canonical Algebraic Curvature Tensors Expressed Through Chain Complexes Over Pseudo-Euclidean Vector Spaces, by Tommy Zamroz
On the Construction of Canonical Algebraic Curvature Tensors Using a Chosen Rank, by Callie Reimann
Summer 2022 Knot Theory Projects
All Links are Sublinks of Cuboctahedral Arithmetic Links, by Luisa Boateng
Supercoiled DNA Tangles and Stick Number of Montesinos Links, by Armando Felix
Categorizing Embedded Totally Geodesic Surfaces Within Chain Links, by Amy Earley
On Characterizing Cuboctahedral Fully Augmented Links, by Emerson Worrell
Summer 2021 Geometry Projects
G-Modeling and Warped Product Curvature Models, by Rasiel Chishti
Linear Dependence of Canonical Algebraic Curvature Tensors as Described by Weighted Directed Graphs, by Shira Julie
Constructing Spanning Sets of Affine Algebraic Curvature Tensors, by Stephen Kelly
Summer 2021 Knot Theory Projects
Using the Crushtaceans of Fully Augmented Links to Investigate Cheeger Constants, by Audrey Baumheckel
On the Classification of Belted Sum Decomposition of Nested Links, by Brandon Castillo
Method to Generate b-Prime Fully Augmented Links, by Leah Mork
On the Arithmeticity of FALs, by Sadie Mills
Summer 2020 Geometry Projects
An Examination of Linear Combinations of Skew-Adjoint Build Algebraic Curvature Tensors, by Joey Brundan
k-Plane Scalar Curvature and Structure Groups of Model Spaces, by Preston Kelley
On the geometric realization of irreducible manifolds by decomposable model spaces, by Charlie McMenomy
Assessing the Geometric Realizations of Hermitian Manifolds, by Frank Pryor
Algebraic Requirements for k-Curvature Homogeneous Spaces, by Zoë Smith
Summer 2020 Knot Theory Projects
Nested Links, Linking Matrices, and Crushtaceans, by Madeline Brown
Topological Realizations of Fully Augmented Links and Polyhedral Partners, by Lisa Cenek
Crushtaceans and Complements of Fully Augmented and Nested Links, by Matty Zevenbergen
Summer 2019 Geometry Projects
General Curvature Homogeneity Theories, by Alex Luna
Canonical Expressions of Algebraic Curvature Tensors, by Katie Ragosta
Linear Independence of Algebraic Curvature Tensors in the Higher-Signature Setting, by Chris Tripp
Decomposable Model Spaces and a Topological Approach to Curvature, by Kevin Tully
A Method to Finding Invariants Using Sectional Curvature, by Montana Williams
Summer 2019 Knot Theory Projects
Belted Sum Decompositions in Nested Links, by Maddy Howard
Volume Bounds for Trivalent Planar Graphs, by Anita Koh
Embedded Totally Geodesic Surfaces in Fully Augmented Link Complements, by Sierra Knavel
Summer 2018 Geometry Projects
Covariant Derivative Lengths in Curvature Homogeneous Manifolds, by Gabby Konyndyk
An upper bound on \eta(n), by Gabe Lopez
Putting the "k" in Curvature: k-Plane Constant Curvature Conditions, by Maxine Calle
Summer 2018 Knot Theory Projects
Systole Length and Preservation Under Belt-Sums of the Borromean Rings, by Amanda Cowell
Factorization of Fully Augmented Links via Belted Sum Decomposition, by Brian Ransom
Volume Bounds of T-links, by Maya Klaib
Geometric Exploration of T-Links using Lorenz Link Correspondence, by Ryan Rhoades
Generalized Cell Decompositions of Nested Lorenz Links, by Wil Coenen
Summer 2017 Geometry Projects
Constant Vector Curvature for Skew-Adjoint Linear Transformations , by Mack Beveridge
A complete description of constant vector curvature in the 3-dimensional setting , by Andrew Lavengood-Ryan
Algebraic curvature tensors of Einstein and weakly Einstein model spaces, by Roberta Shapiro
Summer 2017 Knot Theory Projects
Belted sum decomposition of fully augmented links, by Porter Morgan, Dean Spyropoulos, Cameron Ziegler (the "Belt Summers")
Geometrically realizing nested torus links, by Amie Bray
Analysis of length spectra of geodesics in the knot complement of the Borromean Rings, by Kayla Neal
Summer 2016 Geometry Projects
Constant Vector Curvature in 3-Dimensional Lorentzian Space with Diagonalized Ricci Tensor , by Ana Doktorova
1-Homothety Curvature Homogeneity and 1-Curvature Homogeneity, by Rory Martin-Hagemeyer
A Generalization of Various Theories of Curvature Homogeneity, by Sammy Sbiti
Summer 2016 Knot Theory Projects
Cusp Density in Nested Octahedral Links, by Gabrielle Angeloro
Nested and Fully Augmented Links, by Hayley Olson
Iterative Extension on Volume Altering Hyperbolic Tangle Surgeries, by Lucas Everham
Volume Preserving Surgeries on Hyperbolic 3-Manifolds, by Peter Rudzis
Volume Comparison of 3-Braided Links with Dehn Fillings, by Vanessa Vega
Summer 2015 Geometry Projects
Curvature Homogeneous Types in Generlized Dunn Manifolds , by Daniel Winney
The Study of the Constant Vector Curvature Curvature Condition for Model Spaces of Three Dimensions in the Lorentzian Setting , by Angela Peng
Structure Groups and Linear Transformations , by Darien Elderfield
Linear Dependence of Canonical Algebraic Curvature Tensors with Associated Chain Complexes, by David Williams
Linear Dependence in Sets of Three Canonical Algebraic Curvature Tensors, by Susan Ye
Summer 2015 Knot Theory Projects
Relating Hyperbolic Braids and PSL(2,Z), by Cat Weiss
Fully Augmented Links, by John Harnios
The Bracket Polynomial and Genus of Alternating Links, by Karina Novoa
Summer 2014 Geometry Projects
Linearly Dependent Sets of Algebraic Curvature Tensors , by Alberto Acevedo
Linear Dependence of Canonical Algebraic Curvature Tensors of Symmetric and Anti-Symmetric Builds , by Elise McMahon
A Look at Constant Vector Curvature on Three-Dimensional Model Spaces according to the Curvature Tensor , by Albany Thompson
Kernels of Canonical Algebraic Curvature Tensors , by Adam Williams
Summer 2014 Knot Theory Projects
On Relating Linking Number and Stick Number , by Jackson Hsu
Three-Variable Bracket Polynomial for Three Braid Knots and Links , by Madeleine Johnson
Recovering Link Data From the B-Graph of an Alternating Diagram , by Maggie Miller
Three-Variable Bracket Polynomial for Algebraic Links , by Tyler Underwood
Summer 2013 Geometry Projects
Curvature homogeneity of type (1,3) in pseudo-Riemannian manifolds , by Cullen McDonald
The study of the constant vector curvature condition for model spaces in dimension three with positive definite inner product , by Kelci Mumford
Structure groups of algebraic curvature tensors in dimension three , by Lea Beneish
Summer 2013 Knot Theory Projects
Torified Rational Links , by Karly Brint
The three-variable bracket polynomial for reduced, alternating links , by Kelsey Lafferty
Three-variable bracket polynomial for two-bridge knots , by Matthew Overduin
Summer 2012 Geometry Projects
Linear Dependence and Hermitian Geometric Realizations of Canonical Algebraic Curvature Tensors , by Dan Diroff
Jordan Normal Forms for a Skew-Adjoint Operator in the Lorentzian Setting , by Lily Saunders
Is Every Invertible Linear Map in the Structure Group of some Algebraic Curvature Tensor? , by Lisa Kaylor
On the Computation and Dimension of Structure Groups of Algebraic Curvature Tensors , by Malik Obeidin
Signature Change and Linear Dependence of Curvature Tensors , by Sarah Schmidt
Summer 2012 Knot Theory Projects
Ramsey Numbers and Other Knot Invariants , by Maribeth Johnson
Structure of K Braid Satellites Within a Type K Torus , by Michael Lessley
Proving the absence of certain totally geodesic surfaces , by Warren Shull
Summer 2011 Geometry Projects
Linear Independence of Sets of Three Algebraic Curvature Tensors , by Britany Lovell
On the Structure Groups of Decomposable Algebraic Curvature Tensors , by Cole Franks
Linearly Dependent Sets of Algebraic Curvature Tensors with a Nondegenerate Inner Product , by Evan Smothers
An Upper Bound on antisymmetric Algebraic Curvature Tensors , by Joey Thomas
Summer 2011 Knot Theory Projects
Volume of Hyperbolic Closed 3-Braids , by Aly Bittner
Linking Number of a Linear Embedding with Two Components , by Julie Wasiuk
Classifying Closed Three- and Four-Braids as Hyperbolic , by Marjorie Kasten
Summer 2010 Geometry Projects
Structure Groups of Pseudo-Riemannian Algebraic Curvature Tensors, by Joseph Palmer
Skew-Tsankov Algebraic Curvature Tensors in the Lorentzian setting, by Kaetlin Taylor
Algebraic Curvature Tensors and Antisymmetric Forms, by Blake Treadway
Summer 2010 Knot Theory Projects
Second Twist Number of 2-bridge knots, by Jessica Ceniceros
Composing two non-tricolorable knots, by Kelly Harlan
Ropelength of Twists and Alternating Knots, by William Kanegis
Lower Bounds for the Ropelength of Reduced Conformations, by Robert McGuigan
Linear Ropelength upper bounds of twist knots, by Michael Swearingin
Summer 2009 Geometry Projects
The linear independence of sets of two and three canonical algebraic curvature tensors, by Alexander Diaz
A Spanning Set for Algebraic Covariant Derivative Curvature Tensors, by Bronson Lim
Constructing Algebraic Curvature Tensors Using Symmetric Bilinear Forms, by John Reynolds
The decomposability of full and weak model spaces, by Danielle Strieby
Summer 2009 Knot Theory Projects
Ropelength and Lissajous Diagrams, by Katie Agle
Ropelength, Connect-Sum, and Cabling of Paired Knots, by Jessica Alley
Lower bound for ropelength of m-almost alternating knots, by Charley Mathes
Dowker Notation and Arc Presentation, by Jennifer Winters
Summer 2007 Knot Theory Projects
Convexity and Minimum Distance Energy, by Rosanna Speller
Ropelength of Braids and Tangle Decomposition of Knots, by Sarah Ruether
On Ropelength of Alternating Links, by Shahla Sadjadi
Summer 2007 Combinatorics Projects
Tree Congestion for Complete n-Partite Graphs, by Allen Cox
Spanning Tree Congestion Critical Graphs, by Daniel Tanner
On Spanning Tree Congestion of Product Graphs, by Rachel Hunter
On 3-cyclic Cutwidth Critical Graphs, by Matthew Straughn
On 3-cyclic Bandwidth Critical Graphs, by Eric Bucher
Summer 2006 Knot Theory Projects
The Crossing Probability Knot Energy, by Blake Mitchell
Upper Bound for Ropelength of Pretzel Knots, by Safiya Moran
The Minimum Distance Energy and Midpoint Insertion, by Stephanie Proksch
The Minimum Distance Energy for Polygonal Unknots, by Johanna Tam
Summer 2006 Combinatorics Projects
Two-Dimensional Bandwidth, by Chris Duran
Edge Bandwidth, by Taijiro Bernstein
3 and 4-Bandwidth Critical Graphs, by Ann Kilzer
Investigation of 4-Cutwidth Critical Tree Graphs and Complete Cyclic Cutwidth Critical Graphs, by Ruffin Swain
Summer 2005 Knot Theory Projects
The Twist Numbers of Graphs and the Tutte Polynomial, by Thomas Albertson
An Upper Bound on Stick Numbers for Links, by Erik Insko
On the Second Twist Number, by Gabriel Murillo
Twist Numbers of Links from the Jones Polynomial, by Matthew Williamson
Summer 2005 Combinatorics Projects
The Tree Congestion of Graphs, by Diana Carr
On the Cyclic Cutwidth of Complete Tripartite and n-Partite Graphs, by Heather Allmond
On 3-bandwidth Critical Graphs, by Holly Westerfield
The Linear Cutwidth of Complete n-Partite Graphs, by Chelsea Weitzel
Summer 2004 Knot Theory Projects
The Twisted Torus and Knots, by Jenny Buontempo
The Supercrossing Index of Torus Knots, by Ryan Petitfils
Knotted Ribbons and Ribbon Length, by Maria Salcedo
The Stick Number of Tr,4r, by Ivan Ventura
Summer 2004 Combinatorics Projects
The Linear Cutwidth of Complete Bipartite and Tripartite Graphs, by Stephanie Bowles
The Linear and Cyclic Wirelength of Complete Bipartite Graphs, by Liz Hartung
On Tree Congestion of Graphs, by Stephen Hruska
Minimum Grid Cutwidth of Complete Bipartite Graphs, by Karl Quick
Summer 2003 Knot Theory Projects
Star Move and Construction of a 10-Stick 11-Crossing Knot, by Naomi Davis
The n-iterated Clasp Move and Torus Links, by Stacy Nicole Mills
Iterated Clasp Move and Upper Bounds for 2-bridge Links, by Tim Nawojski
Self-Linking Number as a Geometric Invariant through Geo(8), by Joseph Rhoads
Summer 2003 Combinatorics Projects
The Cyclic Cutwidth of Qn, by Jason Erbele
A Proof for the Cyclic Cutwidth of Q6, by Candi Castillo
The Embedding of Complete Bipartite Graphs onto 2´n Grids with Minimum Grid Cutwidth, by Carlos Ibarra
The Cyclic Cutwidth of Complete Bipartite Graphs, by Megan Holben
Summer 2002 Knot Theory Projects
On the Lower Bounds of Ramsey Numbers of Knots, by Joe Clark
Knots in a Cubic Lattice, by Marta Kobiela
Stick Numbers of Links with Two Trivial Components, by Alexa Mater
Summer 2002 Combinatorics Projects
On Which Graphs Have Linear Cutwidth Equal to Cyclic Cutwidth, by Melissa Banister
The Linear and Cyclic Cutwidth of the Complete Bipartite Graph, by Matt Johnson
Cutwidth of a Complete Graph embedded on an m × n Grid, by Brad Marchand
Proof of the Grid Cutwidth for a Complete Graph, by Alvin Sacdalan
The Cyclic Cutwidth of a P2 × P2 × P n Mesh, by Victor Sciortino
Summer 2001 Knot Theory Projects
Clasping Within Polygonal Links, by Adam O'Connor
Minimum Stick Number for Knots and Links, by Bart Podlesny
Upper Bounds for the Stick Number of Three-Braids, by Nerissa Soriano
Upper Bounds for the Stick Numbers of Three Braids, by Diana Wall
Summer 2001 Combinatorics Projects
A Proof for the Cyclic Cutwidth of Q5, by Ryan Aschenbrenner
Bandwidths, Cutwidths, and Wirelengths of Complete Graphs of triangular and related numbers embedded on triangular lattice grid host graphs, by LeeAnn Feathers
The cutwidth of the complete graph on 2n vertices when embedded into a 2×n grid, by Annie Wang