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A list of all the posts and pages found on the site. For you robots out there is an XML version available for digesting as well.
Pages
Posts
Complex Image Mapping
Published:
In this blog post I will be talking about a piece of software I made for my final project in my Senior year highschool art class.
Dirichlet Convolutions
Published:
An introduction to Dirichlet convolution, written the summer after my junior year of high school. I knew the Möbius function existed but deliberately didn’t look up its definition, so I could try to reconstruct it from what it had to do: invert the constant-one function under convolution. The write-up builds up from that.
Fields
Published:
This is part 3 of a three-part series, written in my junior year of high school, that works up to proving that (Z/pZ)× is cyclic: the multiplicative group of integers mod a prime has a generator. I was teaching myself group theory at the time and worked out the route on my own, so the proofs are mine rather than the textbook’s.
Factoring Polynomials in Arbitrary Fields
Published:
This is part 2 of a three-part series, written in my junior year of high school, that works up to proving that (Z/pZ)× is cyclic: the multiplicative group of integers mod a prime has a generator. I was teaching myself group theory at the time and worked out the route on my own, so the proofs are mine rather than the textbook’s.
Some Results with Kernels
Published:
This is part 1 of a three-part series, written in my junior year of high school, that works up to proving that (Z/pZ)× is cyclic: the multiplicative group of integers mod a prime has a generator. I was teaching myself group theory at the time and worked out the route on my own, so the proofs are mine rather than the textbook’s.
portfolio
Complex image mapping
Published:
Warping images through complex-valued functions: each pixel at x + yi moves to f(x + yi).
Social platform with an LLM assistant
Published:
Full-stack social platform with face matching, retrieval-backed chat, and Spark-based feed ranking.
C2NN: a C-to-neural-network compiler
Published:
An LLVM-based compiler that lowers C programs into equivalent neural networks.
LLM agent for FPGA design (Concurrent EDA)
Published:
An agent that runs the FPGA loop end to end: HDL generation, synthesis, optimization.
publications
Application of Machine Learning for Assessing Movement Disorders
Published in New Jersey Governor's School in the Sciences (student report), 2022
Student research report (NJ Governor's School in the Sciences) on using ML to detect walking irregularities.
Root Bounds for Chain Polynomials Beyond the Cohen–Macaulay World
Published in Preprint, submitted to the Electronic Journal of Combinatorics, 2026
All real roots of the chain polynomial of every finite graded poset lie in [−4, 0]. The proof replaces shellability and h-vectors with a sign condition on a matrix inverse.
Recommended citation: P. Dugan. "Root bounds for chain polynomials beyond the Cohen–Macaulay world." Preprint, 2026.
Download Paper
talks
teaching
Teaching Assistant, CIS 1600: Mathematical Foundations of Computer Science
Undergraduate course, University of Pennsylvania, Department of Computer and Information Science, 2024
TA for CIS 1600 (discrete mathematics) on Professor Rajiv Gandhi’s staff, Fall 2024 to present.