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John Peca-Medlin
- Postdoctoral Research Associate I
- (520) 621-6892
- Mathematics, Rm. 115
- Tucson, AZ 85721
- johnpeca@arizona.edu
Biography
I (he/him) am the Richard Pierce Postdoctoral Fellow in the Department of Mathematics at the University of Arizona, mentored by Nick Ercolani. My primary research is in random matrix theory, with an emphasis on applications to problems in numerical analysis, machine learning, data science, statistical modeling, random graphs and number theory. I also am involved in educational research and outreach relating to equal access issues in STEM. I completed my PhD in mathematics from the University of California, Irvine in 2021, under the advisement of Mike Cranston and Tom Trogdon (University of Washington). My dissertation, entitled "Numerical, spectral, and group properties of random butterfly matrices", can be found here.
I completed my AB in mathematics from the University of Chicago in 2006. I then worked for 9 years in the "real world", first as a research analyst with the Social and Economic Sciences Research Center with Washington State University and then as a data analyst/BI engineer with the Office of Research Information Services at the University of Washington. I started my PhD program one month after the birth of my first son (not pictured).
Outside of math, I enjoy running, watching movies, doing trivia, and spending as much time as possible with my family (me+wife+two sons+two dogs). My most recent hobby involves relocating scorpions found in my backyard using a blacklight, although I much prefer not finding scorpions in my backyard using a blacklight.
Degrees
- Ph.D. Mathematics
- University of California, Irvine, Irvine, California, United States
- Numerical, spectral, and group properties of random butterfly matrices
- M.S. Mathematics
- University of California, Irvine, Irvine, California, United States
- B.A. Mathematics
- University of Chicago, Chicago, Illinois, United States
Work Experience
- University of California, Irvine, Irvine, California (2020 - 2021)
- Georgia Institute of Technology (2018)
- University of California, Irvine, Irvine, California (2015 - 2020)
- University of Washington, Seattle, Washington (2011 - 2015)
- Social and Economic Science Research Center - Washington State University (2007 - 2010)
- University of Chicago, Chicago, Illinois (2005 - 2006)
Licensure & Certification
- Mentoring Excellence Certificate, UCI Graduate Division (2021)
- Inclusive Excellence Certificate, UCI Office of Inclusive Excellence (2021)
- Data Science Certification Program, UCI Graduate Division (2021)
- Course Design Certificate, UC Irvine Division of Teaching Excellence and Innovation (2021)
Interests
Teaching
Topics in probability, statistics, and data science.Topics in algebra, including linear algebra, group theory, ring theory, and representation theory.Active learning techniques in the classroom (online)Virtual instructionCourse design best practices
Research
Random matrix theoryStatistical modelingNumerical analysisRandom graphsCohen-Lenstra heuristicsEquity in STEM education
Courses
2023-24 Courses
-
Intro to Linear Algebra
MATH 313 (Summer I 2024) -
Honors Thesis
DATA 498H (Spring 2024) -
Honors Thesis
DATA 498H (Fall 2023) -
Intro to Linear Algebra
MATH 313 (Fall 2023) -
Major Colloquium
DATA 195M (Fall 2023) -
Major Colloquium
MATH 195M (Fall 2023)
2022-23 Courses
-
Intro to Linear Algebra
MATH 313 (Summer I 2023) -
Directed Research
MATH 492 (Spring 2023) -
Preceptorship
MATH 491 (Spring 2023) -
Directed Research
MATH 492 (Fall 2022) -
Intro to Linear Algebra
MATH 313 (Fall 2022) -
Preceptorship
MATH 491 (Fall 2022)
2021-22 Courses
-
Intro Statistical Method
DATA 363 (Summer I 2022) -
Intro Statistical Method
MATH 363 (Summer I 2022) -
Theory of Statistics
MATH 466 (Spring 2022) -
Calculus I
MATH 125 (Fall 2021) -
Preceptorship
MATH 491 (Fall 2021)
Scholarly Contributions
Journals/Publications
- Peca-Medlin, J. (2024). Distribution of the number of pivots needed using Gaussian elimination with partial pivoting on random matrices. The Annals of Applied Probability (to appear).More infoGaussian elimination with partial pivoting (GEPP) is a widely used method to solve dense linear systems. Each GEPP step uses a row transposition pivot movement if needed to ensure the leading pivot entry is maximal in magnitude for the leading column of the remaining untriangularized subsystem. We will use theoretical and numerical approaches to study how often this pivot movement is needed. We provide full distributional descriptions for the number of pivot movements needed using GEPP using particular Haar random ensembles, as well as compare these models to other common transformations from randomized numerical linear algebra. Additionally, we introduce new random ensembles with fixed pivot movement counts and fixed sparsity, $\alpha$. Experiments estimating the empirical spectral density (ESD) of these random ensembles leads to a new conjecture on a universality class of random matrices with fixed sparsity whose scaled ESD converges to a measure on the complex unit disk that depends on $\alpha$ and is an interpolation of the uniform measure on the unit disk and the Dirac measure at the origin. [Journal_ref: ]
Presentations
- Peca-medlin, J. (2022, August). Using self-assessments to measure program success in Math Circles. MathFest 2022. Philadelphia: MAA.
- Peca-medlin, J. (2022, January). Random butterfly matrices and growth factors. Matrix Theory SeminarUniversity of Nevada, Reno.
- Peca-medlin, J. (2022, January). Random butterfly matrices and growth factors. Matrix analysis seminar. University of Nevada, Reno (virtual).
- Peca-medlin, J., & Pantano, A. (2022, October). Developing a culturally responsive math afterschool program for underserved students. Mathematics Instruction ColloquiumUA.
- Peca-medlin, J. (2021). Random butterfly matrices and growth factors. Analysis, dynamics, and applications seminar. University of Arizona.
- Peca-medlin, J. (2021, December). Random butterfly matrices and growth factors. Frontier Probability Days. UNLV.
Poster Presentations
- Peca-medlin, J., Isham, K., & Kreger, J. (2021, August). COMP: Graduate students building and promoting community and inclusivity during a pandemic. MathFest 2021. Virtual: MAA.More infoPoster presentation regarding student run organization started at UC Irvine to better support graduate students within the math department through mentoring, building community, and actively seeking to broaden participation among underrepresented groups with targeted recruitment events.