About Me

Orbits of a one-parameter group of Möbius transformations z ↦ (az + b)/(cz + d) with two fixed points (filled: attracting, ring: repelling). Particles follow the Steiner circles as the flow moves between elliptic, hyperbolic, and loxodromic. Click or drag to move a fixed point; double-click to let them drift.

I’m Patrick Dugan, a student at the University of Pennsylvania in the submatriculation program: an M.S.E. and B.S.E. in Computer Science and a B.A. in Mathematics, expected December 2027. My work sits where math and CS meet: proving things about combinatorial objects, and building systems that use machine learning and compilers.

Research

Real roots of chain polynomials. My preprint Root Bounds for Chain Polynomials Beyond the Cohen–Macaulay World proves that every finite graded poset has all the real roots of its chain polynomial in [−4, 0], and that the constant 4 is sharp. The usual tools for bounds like this are topological (shellability, Cohen–Macaulayness, h-vectors), and graded posets need not satisfy them. The paper uses a sign condition on the inverse of I + aA instead, which can be checked one pair of vertices at a time. The same idea gives shorter intervals for subspace lattices and for Eulerian and k-Eulerian posets, and it even covers relations that aren’t partial orders, such as chains of normal subgroups. Submitted to the Electronic Journal of Combinatorics. (PDF · watch the theorem run live)

Current interests

Algebraic and enumerative combinatorics: real-rootedness, Möbius functions, and the poset side of the paper above. The open case I keep returning to is the coprime graph on the integers.
Learning theory: the PAC model, VC dimension, and sample complexity. I’m taking Theory of Machine Learning with Michael Kearns this fall.
Compilers for ML: MLIR/LLVM tooling and LLM agents for hardware design (Concurrent EDA, summer 2026), and C2NN, a compiler that turns C programs into neural networks.
Machine perception: projective geometry, camera models, and multi-view geometry.

Elsewhere

I TA CIS 1600 (discrete math) at Penn. Before that I was a machine learning engineer at Palm Cosmetics (computer vision) and a data science intern in CMU’s REUSE program. The blog has older writeups, mostly mathematical oddities from high school onward.

Full details are on my CV.