Professor, Mathematics
- PhD, University of California, Santa Barbara; Mathematics
- MA, University of California, Santa Barbara; Mathematics
- BS, Virginia Tech, Mathematics
Dr. Diane Hoffoss has been a member of the University of San Diego faculty since 2001, following postdoctoral positions at Rice University and Colorado College. Her mathematical research focuses on 3-manifold topology, hyperbolic geometry, and flows and foliations. Her work and insights have been featured in Forbes, local San Diego news programs, an award winning documentary about the idea of beauty, and on the cover of Notices of the American Mathematical Society. Dr. Hoffoss has taught nearly all of the mathematics courses offered at USD, with particular enthusiasm for Calculus III, Abstract Algebra, Geometry, Topology as Knot Theory, Real Analysis, and the Problem Solving Seminar.
Scholarly Work
Dr. Hoffoss's mathematical research is driven by her fascination with shape visualization and its inherent challenges. Her field of topology is the study of shapes without regard to distortions such as stretching or shrinking.
Her early research explored 3-manifolds constructed as mapping tori of hyperbolic surfaces with pseudo-Anosov monodromy, demonstarting that the flow lines within these manifolds are quasigeodesic, or approximately straight, in the hyperbolic metric. More recently, her work has compared two definitions of "width" for three-dimensional shapes: one topological and the other geometric, as defined by Gromov.
Dr. Hoffoss has mentored students on a wide array of research projects, including:
- Classifying 4- and 5-component links of convex planar knot components
- Investigating whether Dürer’s unfolding polyhedra conjecture holds for hyperbolic polyhedra
- Defining and analyzing a new category of knots, called semi-rigid knots
In addition to her mathematical pursuits, Dr. Hoffoss is a dedicated artist. As a founding member of the artist collective ArtBuilds, an interdisciplinary team of USD faculty, students, and San Diego community members, she creates large-scale interactive sculptures. She has crafted innovative installations that merge art, mathematics, and interactivity for Burning Man annually since 2017. In addition, several of her sculptures have been commissioned by the City of San Diego and installed at various locations across the city.
On USD’s campus, her artwork Reflexion (originally installed at La Jolla Cove as part of the City of San Diego’s Park Social Initiative) was installed near Plaza de San Diego during Homecoming 2023. Another local example of her work can be found in the USD Mathematics Department: a glowing pentagon that was a prototype for part of a larger sculpture called Unfolding Humanity.
Areas of Interest
Dr. Hoffoss is deeply committed to encouraging and enabling students to actively contribute to and take ownership of their own learning. Her teaching philosophy emphasizes:
- Encouraging meaningful classroom participation
- Leading students to participate in the development of new ideas
- Allocating time for in-class problem solving under supervision to better prepare students for independent work
Beyond acquiring valuable mathematical knowledge and skills, her classes are designed to cultivate these essential human learning outcomes.
- Bravery, via facing difficulties and risking making mistakes;
- Resilience and comfort with temporary failure, taking mistakes in stride, and springing back ready to try again to solve a problem or gain understanding of a mathematical concept;
- Persistence, Determination, and Endurance, via facing many challenging problems which may not yield on your first attempt, and which may require incremental progression;
- Public Speaking Skills facilitated by asking questions in class;
- Effective ways to communicate mathematics in the form of logically complete, well organized and clearly worded solutions to problems;
- Higher order logical reasoning skills, including understanding definitions and statements of theorems, and knowing when various theorems apply;
- Understanding the connection and transition between previously studied mathematics and more advanced mathematics.
Dr. Hoffoss is a relentless cheerleader for the beauty and excitement of mathematics, and has endless energy for supporting and encouraging her students as they go through their mathematical studies.