Enrique G Alvarado

Welcome to my mathematics home page! I am a postdoc in the mathematics department at Iowa State University. My mentor, Eric Weber and I are currently working on applying topological data analysis (TDA) techniques to spatio-temporal datasets (NSF grant #2219959). Before ISU, I was a postdoc at UC Davis for 3 years, where I worked with my mentor Qinglan Xia on Plateau-type problems and ways of generalizing the notion of iterated function systems. I got my Mathematics PhD at Washington State University, where I was co-advised by Bala Krishnamoorthy and Kevin Vixie. During my PhD, I was an RA for Aurora Clark's computational chemistry lab. There, I applied geometric measure theory and TDA techniques to various types of chemical simulation datasets.

Helping Lola smell the flowers

Recent News

6/30 - 7/18. @ SLMath in Berkeley, CA for their Summer Collaborators Program with Lisa Napels and Silvia Ghinassi
6/9 - 13. @ the Topological Visualization workshop in University of Iowa where I will be giving a "Wild Mappers" talk
10/30/24. Gave a talk @ ISU's Analysis and Probability seminar on my Partial Plateau's Problem work
10/25/24. Co-organizing the Topological Visualization session at the Mathematics of Data Science SIAM conference in Atlanta.
10/7/24. Poster presentation at the 2024 Joint ATD/AMPS PI workshop on "Parameter Selection in Mapper."
10/2/24. Paper, "Partial Plateau's Problem with H-Mass" is accepted for publication in Calculus of Variations and PDEs.
9/17/24. Kick-off meeting for Projects in TDA 2024
8/26/24. Started first semester at ISU, where I will be teaching my first graduate level Algebraic Topology class.
8/16/24. Started my postdoc at Iowa State University!
7/23/24. Talk at the CodEx seminar on "Parameter Selection in Mapper."

Employment

Postdoc at the Department of Mathematics, Iowa State University (August 2024 - present)

Funded by NSF grant #2219959 and mentored by Eric Weber

Krener Assistant Professor at the Department of Mathematics, UC Davis (July 2021 - June 2024)

Mentored by Qinglan Xia

Research Assistant in Aurora Clark's computational chemistry group at Washinton State University (January 2019 - June 2021)

Funded by DOE and NSF

Education

Ph.D. in Mathematics (applied track) from Washington State University (June, 2021)

Co-advised by Kevin R. Vixie and Bala Krishnamoorthy. Aurora Clark and Matthew Sottile were also on my committee.

B.S. in Mathematics with a minor in Computer Science from Gonzaga University (2014)

Research

I am a mathematician with a variety of pure and applied interests. I enjoy thinking about analysis, topology, and their applications to the physical and data sciences. Within mathematical analysis, I am interested in geometric minimization problems that you find within the area of geometric measure theory. For example, the Maximum Distance Problem [ACMN, AKV1] and a type of Partial Plateau's Problem [AX1]. I am interested in the various ways that one can generate fractals [AX2] and have recently been interested in fractional rectifiability. Within the area of applied topology, I am interested in parameter selection problems for Mapper [A_1, A_2], and questions which relate properties of sets to their random samples [AKV2].

I love interdisciplinary projects. I have collaborated with chemists, physicists, and data scientists -- writing code to apply the tools from above areas to help them solve problems they think are interesting. I combined tools from persistent homology and geometric measure theory to find interesting features on liquid-liquid interfaces [ALSKC] and was part of a large group that analyzed energy landscapes via sublevelset persistent homology [MZY_].

Recently, with my group from the 2024 MRC in "Climate Science at the Interface Between TDA and Dynamical Systems", I have been investigating the topological stability of trajectories within dynamical systems. With Erik Carlsson, we have been looking at molecular conformation spaces, and with Eric Weber, I have been working on combining tools from Harmonic Analysis and TDA to investigate time-varying data coming from human migration and traffic data.

Mentorship

Undergraduate research was pivotal in my mathematical upbringing and helping undergraduate students (especially underrepresented students) find their mathematical curiosity and drive is a very important me. During my time at UC Davis, I lead a group of 3 undergraduates to investigate ways that one could program a computer to randomly sample shapes, and at Iowa State, Eric Weber and I are running an ISMaRT project for undergraduates investigating similar questions. In the Fall of 2024, Robin Belton and I lead a group of 2 undergraduates and 2 graduates in "Projects in TDA" to investigate questions related to parameter selection in Mapper.

Students -- If you are interested in exporing research in Analysis or Topological Data Analysis with me, please contact me via email!

Publications

Geometric Measure Theory

Partial Plateau's Problem with H-mass, Calc. Var. 64, 13 (2025)

with Q. Xia

General fractals represented by F-limit sets of compression maps (submitted)

with T. Lazarus, Q. Xia

Asymptotics of Maximum Distance Minimizers (submitted)

with L. Catalano, T. Merchan, L. Naples

The Maximal Distance Problem and Minimal Spanning Trees, Int. J. Anal. Appl. 19, 5, 633-659 (2021)

with B. Krishnamoorthy and K. R. Vixie

A lower bound for the reach of flat norm minimizers

with K. R. Vixie

On the Measure of the Midpoints of the Cantor set in R

with Y. Hu

Topological Data Analysis

Any Graph is a Mapper Graph (submitted)

with R. Belton, E. Fischer, K.-J. Lee, S. Palande, S. Percival, E. Purvine, S. Tymochko

G-Mapper: Learning a cover in the mapper construction (submitted)

with R. Belton, E. Fischer, K.-J. Lee, S. Palande, S. Percival, E. Purvine

- Box Filtration (submitted)

with P. Gupta and B. Krishnamoorthy

Geometry of a Set and its Random Covers (submitted)

with B. Krishnamoorhty and K. R. Vixie

Applications

Electron Density Transport During Chemical Reactions (submitted)

J. Elowitt, N. May, Y. Wei, B. Sadhu, E. Alvarado, B. Krishnamoorthy, A. Clark

Representations of Energy Landscapes by their Subleveset Persistent Homology: An Example with n-Alkanes, J. Chem. Phys. 154, 114114 (2021)

J. Mirth, Y. Zhai, J. Bush, H. Jordan, M. Heim, B. Krishnamoorthy, M. Pflaum, A. Clark, Y. Zhang, H. Adams

- Shear stress dependence of force networks in 3D dense suspensions, Soft Matter 17, 7476-7486 (2021)

L. Edens, E. Alvarado, A. Singh, J. Morris, G. Schenter, J. Chun, and A. Clark

A Geometric Measure Theory Approach to Identify Complex Structural Features on Soft Matter Surfaces, J. Chem. Theory Comput. 16, 7, 4579–4587 (2020)

E. Alvarado, Z. Liu, M. Servis, B. Krishnamoorthy, A. Clark

Undergraduate Research

Klein links and related torus links, Involve, 9, 2, 347–359 (2016)

E. Alvarado, S. Beres, V. Coufal, K. Hlavacek, J. Pereira, B. Reeves.

Teaching

Iowa State University

For the Fall 2024 semester, I am teaching a graduate course in Algebraic Topology. We are using Spanier's classic text on the subject.

University of California, Davis

At Davis, I mainly taught undergraduate analysis courses (127 A, B, and C). In the first two courses, we went through all of Abbott's Understanding Analysis textbook, and in the last course, C, we went through about half of Munkres' Analysis on Manifolds text. I absolutely loved teaching out of both of these textbooks. Here are my lecture notes for 127A, and my lecture notes for 127C in case you would like to take a look :)
I also taught an "Introduction to Proofs" class, Calculus II (honors), Vector Calculus, and Calculus for the Biosciences.

Washington State University

Throughout my time at WSU, I was a TA for the calculus series, I taught a class on probability, and a couple of other lower-division mathematics courses.

Contact

enrique3 at iastate dot edu

My UCD email (ealvarado at math dot ucdavis dot edu) has expired. Please reach me at my ISU email above, or at my personal gmail (ealvarado9611 at gmail dot com).