Mathematics
Mathematics Bio
Diane Hoffoss has been a member of University of San Diego’s Mathematics Department faculty since 2001, after obtaining her PhD at University of California Santa Barbara then holding postdocs at Rice University and Colorado College.
Her mathematical research focuses on the areas of 3-manifold topology, hyperbolic geometry and flows and foliations. Some works and thoughts have appeared in the Journal of Differential Geometry, the cover of the Notices of the American Mathematical Society, Forbes, and local San Diego news programs.
She has taught nearly every mathematics course at USD, with a particular fondness for Calculus III, Abstract Algebra, Geometry, Topology as Knot Theory, Real Analysis, and our Problem Solving seminar.
Scholarly Work
The excitement and challenges in shape visualization motivates much of Dr. Hoffoss’s mathematical work. Her research area Topology is the study of the basic underlying shapes of objects, without taking into account differences created by stretching, shrinking, and distortion. In our internal departmental classifications of mathematical subjects, Topology is in the Shape area along with geometry.
Her earliest research studied particular class of 3-manifolds (ones constructed as mapping tori of hyperbolic surfaces with pseudo-Anosov monodromy), showing that the flow lines arising from this construction are quasigeodesic, or approximately straight, in the hyperbolic metric on that manifold. Her most recent paper compared two definitions of width of three-dimensional shapes, one of which comes from a topological description of the space whereas the other is a geometric definition of width due to Gromov.
Papers and Publications
Morse Functions to Graphs and Topological Complexity for Hyperbolic 3-Manifolds, with J. Maher, Communications in Analysis and Geometry, Vol. 30, No. 4 (2022), pp. 843-868
At the Intersection of Mathematics, Art and Religious Studies: Unfolding Humanity,with S. Babka and S. Devadoss, preprint (2020).
Interviewed for cerebral beauty sections of multiple award winning documentary It’s Beautiful by Melibe Productions, (2020).
Makerspaces on the Continuum: Examining Undergraduate Student Learning in Formal and Informal Settings, with Gordon Hoople, Alex Mejia, and Sat36, No. 4 (2020) 1184–1195.
About Time: Visualizing Time at Burning Man, with G. Hoople, A. Choi-Fitzpatrick, N. Parde, M. Mellette, R. Nishimura, V. Gutman, Spotlight Article in The STEAM Journal, Volume 4, Issue 1 (2019).
Unfolding Humanity: Mathematics at Burning Man, with S. Devadoss, Notices of the American Mathematical Society, 66 (2019) 572–575
Unfolding Humanity episode for Burner Podcast, with M. Elliott, Episode 97, April 3, 2019.
Morse Area and Scharlemann-Thompson Width for Hyperbolic 3-Manifolds, with J. Maher, Pacific Journal of Mathematics 281-1 (2016), 83–102 .
Problems in Groups, Geometry, and Three-Manifolds, with Kelly Delp and Jason Manning, (2015) arXiv.org:1512.04620
Suspension Flows are Quasigeodesic, Journal of Differential Geometry, Vol. 76, No. 2 (2007) 215–248.
Resources from my Keynote Address to Rice University School Mathematics Program’s Spring Networking Conference were incorporated into a Geometry Module for comprehensive teacher training, with funding from the Texas Education Agency and the Texas Higher Education Coordinating Board (2004).
Boundary Slopes – This computer program calculates boundary slopes of surfaces in cusped manifolds using degenerations of an ideal triangulation of a manifold (1992).
LAURE Integrity Tester – A computer program written in the language LAURE which checks whether the language is still valid after changes in implementation; Published as part of the LAURE computer language by Bellcore (1990).
LAURE Tutorial – A tutorial introducing users to the computer language LAURE; Published as part of documentation for LAURE by Bellcore (1990)
Length Spectra – A computer program which calculates complex length spectra of a manifold given matrix generators for its fundamental group. Included in ( SnapPy ), a program for creating and studying hyperbolic 3-manifolds (1989).
Student Research and Project Involvement
Unfolding Humanity Renaissance Fabrication
Re:Emergence Fabrication
SURE scholars Navin Rai and Ysabel Yu joined a group of 7 USD engineering students who assisted ArtBuilds with the engineering, fabrication, and test install of our sculpture Re:Emergence
Re:Emergence Fabrication II
Co-mentored high school students, Jose Barcelo and Charlie Chavez,as they assisted in the fabrication of this art project, under the Bridging the Gap mentorship program.
Configuration Space of a Winged Cube
SURE scholar Giacomo Radaelli investigated flexible polyhedra and determined the configuration space of a winged cube. USD Alum Jordan Matuszewski joined this research without pay, and Jenny Lee from Oberlin and Emily Zhang from Wellesley participated part time.
Virtual Reality Configuration Space
Fletcher Jones scholar Phillip Miller created a virtual realityrepresentation of the configuration space of the winged cube, in which physically moving in theconfiguration space in VR caused the configuration of the cube change in real time based on your location.
Emergence Art Installation
I co-supervised engineering students Nicolas De La Fuente, Jane Kim, and Rosie Pham on their design of the sculpture Emergence, from giving art critique and suggestions throughout the summer, to building a narrative and descriptive video of the piece, to fabrication and installation at the art festival Everywhen in October.
Hyperbolic Durer’s Conjecture Presentation
Fletcher Jones 2020 students Sean Kim and Jordan Matuszewski presented their research with my support on Hyperbolic Durer’s Conjecture at Northern California Undergraduate Mathematics Conference
Hyperbolic Durer’s Conjecture
Fletcher Jones Scholars Jordan Matuszewski and Sean Kim translated Durer’s Unfolding Conjecture into hyperbolic space, and investigated special classes of polyhedra.
COMAP’s Mathematical Contest in Modeling
Recruited and supported 81 teams of students to participate in COMAP’s annual International Mathematical Modeling Contest. Three of our teams have earned a “Meritorious” score and five more have earned “Honorable Mention.
Dodecahedron Mirror Lamps
Kate Rumann and Kiana Guastaferro designed and constructed five dodecahedron model lamps with infinite internal mirror reflections as artistic gifts for our highest contributing donors to Unfolding Humanity, exhibited at North County Maker Faire
Engineering Problem Solving for Unfolding Humanity
SURE scholar Sydney Platt spent the summer solving fabrication and engineering problems for Unfolding Humanity.
Fabrication of Unfolding Humanity
At least 12 additional students and alums (Quinn Pratt, Michael Sween, Kate Rumann, Kiana Guastaferro, Melissa Carin, Glenn Moss, Sarina Haghiat, Nat Yee, Ava Bellizzi, Gabby Goerke, James Enders, D.D. Latimore, Kelli Kufta, Sean Hough) helped with the fabrication and later refurbishing of Unfolding Humanity.
Lighting Animation for The Journey Project
Computer Science majors Erick Perez, Alexander Alvarez, and Seth Nakanishi contributed lighting animation routines for The Journey Project.
Building Topology exhibit Taping Shape
Five of my topology students helped build Taping Shape, a multi-room, large scale topological surface, large enough to walk through, out of packing tape, at the Reuben H Fleet Science Center
COMAP’s Mathematical Modeling Competition
Diane has served as faculty advisor to over 100 students in COMAP’s 4 day international Mathematical Modeling Competition.
Regular Stick Number
Cryptography Honors Thesis
Advisor for Kelly Fromm’s honors thesis entitled “The Applications of Algorithms and Mathematics in the Military: Cryptography of Past, Present, and Future Warfare
Knot Theory and DNA Project
Brent Allman presented his class project on Knot Theory and DNA at the Pacific Coast Undergraduate Math Conference
Spread of Yellow Fever in Senegal
Sami Armstrong presented the research project she did for our Biomathematics class at USD’s Undergraduate Research Conference, and also at the local Mathematical Association of America meeting (where she earned the Best Poster award).
Putnam Competition
Increased Putnam Competition participation from 3-5 per year to 11 students my first year and 14 the next.
COMAP MCM Solution Presentation
Veronica Rindge presented her team’s solution to COMAP’s International Mathematical Modeling Contest at Pacific Coast Undergraduate Math Conference
Brunnian Links Presentation
Carolyn Yarnall presented her proving the non-existence of convex planar Brunnian Links of 5 components at Pacific Coast Undergraduate Math
Conference
COMAP MCM Solution Presentation
Sabrina Pierard presented her team’s solution to COMAP’s International Mathematical Modeling Contest at Pacific Coast Undergraduate Math Conference.
Convex Planar Brunnian Links
Carolyn Yarnall proved that there do not exist any convex planar Brunnian links of 5 components.
Eigenvectors and Google PageRank
Carolyn Yarnall presented at Pacific Coast Undergraduate Math Conference about how Google uses Eigenvectors to help rank web pages in search.
Shape of Space
Justin Webster and Carolyn Yarnall worked through Jeff Weeks’ book Shape of Space.
Modeling Clam Evolution
Senior Thesis Committee Member for Kevin Brink’s Applied Math Thesis entitled Clam Evolution Using a Continuous Age and Time Structured Model.
Convex Planar Brunnian Links
Michelle Wilkerson filled in Hugh Howards’ proof that there exists no 4 component, planar convex Brunnian links, and began generalizing to the 5 component case.
Error Correcting Codes and Sphere Packings
Michelle Wilkerson studied the relationship between error correcting codes and sphere packings.
Mathematical Association of America Section Meeting Posters
Advised 6 groups of students presenting posters at the MAA Section Meeting at USD. One student’s poster received the Best Poster award, and three groups were from my 100 level Investigations in Mathematics course.
Teaching
Dr. Hoffoss is strongly committed to encouraging and enabling students to actively participate in their own learning by encouraging meaningful classroom participation, leading students to join in the development of new ideas being presented, and reserving class time for students to work through problems under the professor’s supervision before facing their homework alone.
Calculus Course Content Videos
Calculus II Lectures
The Definite Integral & Fundamental Theorem of Calculus (Section 5.2-5.3)
U-Substitution (Section 5.5)
Areas Between Curves (Section 6.1)
Volumes Lecture 1 (Section 6.2)
Volumes Lecture 2 & Cylindrical Shells Lecture 1 (Section 6.2 – 6.3)
Applications to Physics & Engineering Lecture 1 (Section 6.4)
Applications to Physics and Engineering Lecture 2 (Section 6.4)
Techniques of Integration (Section 7.1)
Integration by Parts (Section 7.1)
Trigonometric Integrals (Section 7.2)
Trigonometric Substitution Lecture 1 (Section 7.2)
Trigonometric Substitution Lecture 2 (Section 7.2)
Integration of Rational Functions by Partial Fractions (Section 7.3)
Integration Using Tables and Technology (Section 7.6)
Approximate Integration Lecture 1 (Section 7.7)
Approximate Integration Lecture 2 (Section 7.7)
Approximate Integration Lecture 3 (Section 7.7)
Improper Integrals Lecture 1 (Section 7.8)
Improper Integrals Lecture 2 (Section 7.8)
Improper Integrals Lecture 3 (Section 7.8)
Sequences Lecture 1 (Section 11.1)
Sequences Lecture 2 & Series Lecture 1 (Section 11.1-11.2)
Series Lecture 2 (Section 11.2)
Series Lecture 3 & The Integration Test and Estimates of Sums Lecture 1 (Section 11.2-11.3)
The Integral Test and Estimates of Sums Lecture 2 (Section 11.3)
The Comparison Test Lecture 1 (Section 11.4)
The Comparison Test Lecture 2 (Section 11.4)
Alternating Series and Absolute Convergence Lecture 1 (Section 11.5)
Alternating Series/Absolute Convergence Lecture 2 & Ratio/Root Tests Lecture 1 (Section 11.5-11.6)
The Ratio and Root Tests Lecture 2 (Section 11.6)
Power Series & Representing Functions as Power Series (Section 11.8-11.9)
Representation of Functions as Power Series Lecture 2 & Taylor Series Lecture 1 (Section 11.9-11.10)
Taylor and Maclaurin Series Lecture 2 & Applications (Section 11.10-11.11)
Differential Equations Lecture 1 (Section 6.5)
Differential Equations Lecture 2 (Section 6.5)
Calculus III Lectures
16.3 Fundamental Theorem for Line Integrals – Lecture 1
16.4 Green’s Theorem – without full meaning
16.6 Parametric Surfaces and Surface Area – Part 1
16.6 Parametric Surfaces & Surface Area Part 2; and 16.7 Surface Integrals Part 1
16.7 Surface Integrals Part 2; and 16.8 Stokes Theorem, How to Use a Theorem
16.9 Divergence Theorem
Diane Hoffoss – All rights reserved