Mathematics, Computer Science and Statistics

Nicholas Scoville

My research interest is in topology. Topology studies the shape of spaces like in geometry, but in a much more general way. Basically topology is concerned with the number and type of holes in an object. Topology allows one to smoothly deform one object into another or to show why such a smooth deformation is not possible. The world of topology is vast and has many applications in physics, robotics, sensor networks, and big data analysis.

When I am not counting holes, I love to cook Indian food and BBQ (low and slow) while listening to old school country music (think Hank Williams and Marty Robbins), sew and tailor clothes while listening to Italian opera (but we’ll make an exception for Bizet), play ragtime piano, and read classic literature. I am the father of 6 daughters so far, and consider myself an amateur scholastic.

Some people read the bible every day and some read Dante. I listen to this every day: fairly unrelated.

My new book on discrete Morse theory is now available from amazon or the AMS.

Department

Mathematics and Computer Science

Degrees

  • B.S, Western Michigan University
  • M.S, Western Michigan University
  • M.A, Dartmouth College
  • Ph.D., Dartmouth College

Teaching

CIE 100 (CIE 100)

Problem Solving [Putnam prep in fall, GRE prep in spring] (Math 010)

Calculus II (Math 112)

Linear Algebra (Math 235)

Discrete Math (Math 236W)

Modern Geometry (Math 322)

Probability (Math 341)

Graph Theory (Math 361)

Topology (Math 421)

Discrete Morse theory (Math 451)

Mathematics for Human Flourishing (Math 400)

Website

https://nscoville.github.io/website/

Research Interests

Algebraic topology

(simplicial) Lusternik-Schnirelmann category

Discrete Morse theory

Simplicial complexes

Digital homotopy theory

complex of discrete Morse functions

Cech closure spaces

Homology

Recent Work

“Discrete Morse theory for open complexes,” (with Kevin P. Knudson), submitted

“A Second Homotopy Group for Digital Images,” (with Gregory Lupton, Oleg Musin, P. Christopher Staecker, Jonathan Trevino-Marroquin,), submitted

“On cycles and merge trees,” (with Julian Bruggemann), submitted

“The DOPE Distance is SIC: A Stable, Informative, and Computable Metric on Time Series And Ordered Merge Trees,” (with Christopher J. Tralie, Zachary Schlamowitz, Jose Arbelo, Antonio I. Delgado, and Charley Kirk), submitted

“Star clusters in the Matching, Morse, and Generalized Morse complex,” (with Connor Donovan), New York J. Math., 29 (2023) 1393–1412.

“The digital Hopf construction,” (with Greg Lupton and John Oprea) Topology and its Applications, 2023, 108405, ISSN 0166-8641, https://doi.org/10.1016/j.topol.2022.108405.

“Merge trees in discrete Morse theory,” (with Benjamin Johnson) Research in the Mathematical Sciences 9, 49 (2022). https://doi.org/10.1007/s40687-022-00347-x

“Digital Fundamental Groups and Edge Groups of Clique Complexes,” (with Greg Lupton) Journal of Applied and Computational Topology, (2022). https://doi.org/10.1007/s41468-022-00095-5

“Fundamental Theorems of Morse theory on posets,” (with D. Fernandez-Ternero, E. Macias-Virgos, D. Mosquera-Lois, and J. A. Vilches), AIMS Mathematics, 2022, Volume 7, Issue 8: 14922-14945. doi: 10.3934/math.2022818

“Subdivision of Maps of Digital Images,” (with Greg Lupton and John Oprea), Discrete and computational geometry,  67, 698–742 (2022). https://doi.org/10.1007/s00454-021-00350-z

“Higher connectivity of the Morse complex,” (with Matthew C.B. Zaremsky) Proceedings of the AMS Series B, 9 (2022), 135–149

“On the homotopy and strong homotopy type of complexes of discrete Morse functions,” (with Connor Donovan Max Lin) Canadian Mathematical Bulletin, 1-19, 2022, doi:10.4153/S0008439522000121

“Homotopy Theory in Digital Topology,” (with Greg Lupton and John Oprea), Discrete and computational geometry, 67 (2022), no. 1, 112–165, doi.org/10.1007/s00454-021-00336-x

“A Fundamental Group for Digital Images,” (with Greg Lupton and John Oprea) Journal of Applied and Computational Topology, 5 (2021), no. 2, 249–311.

“On the automorphism group of the Morse complex,” (with M. Lin) Advances in Applied Mathematics, Volume 131, October 2021, 102250

“Strong discrete Morse theory and simplicial Lusternik–Schnirelmann category: A discrete version of the Lusternik-Schnirelmann Theorem,” (with D. Fernandez-Ternero, E. Macias-Virgos, and J. A. Vilches), Discrete and Computational Geometry, 62, (2020), no. 3, 607-623

“Discrete Morse functions, vector fields, and homological sequences on trees,” (with Ian Rand), Involve, A Journal of Mathematics, 13-2 (2020), 219–229. DOI 10.2140/involve.2020.13.219

“Persistence equivalence of discrete Morse functions on trees,” (with  Yuqing Liu), Algebra Colloquium27 : 3 (2020) 455-468 DOI: 10.1142/S1005386720000371

“Homological Sequences in discrete Morse theory” (with Michael Agiorgousis, Brian Green, Alex Onderdonk, and Kim Rich), Topology Proceedings, 54, (2019), 283–294

“Knots related by Knotoids” (with Colin Adams, Kate Kearney, and Allison Henrich), American Mathematical Monthly, Volume 126, 2019 - Issue 6, 483–490

“A Persistent Homological Analysis of Network Data Flow Malfunctions” (with Karthik Yegnesh) Journal of Complex Networks, Issue 6, 1 December 2017, Pages 884-892, https://doi.org/10.1093/comnet/cnx038 

“On the Lusternik-Schnirelmann category of a simplicial map” (with Willie Swei), Topology and its applications, 216 (2017), 116-–128

“Estimating the discrete Lusternik-Schnirelmann category” (with Mimi Tsuruga and Brian Green) Topological Methods in Nonlinear Analysis, 45, No. 1 (2015), 103–116

 “Graph Isomorphisms in Discrete Morse Theory” (with Seth Aaronson, Marie Meyer, Mitchell T. Smith, and Laura M. Stibich) AKCE International Journal of Graphs and Combinatorics, 11, No. 2 (2014), 163-176

 “Metric Structures for CW Complexes” Topology Proceedings, Volume 44 (2014) 117-131

“A Distributed Greedy Algorithm for Constructing Connected Dominating Sets in Wireless Sensor Networks” (with Akshaye Dhawan and Michelle Tanco) 3rd International Conference on Sensor Networks (SENSORNETS), Lisbon, Portugal, January, 2014

“Lusternik-Schnirelmann category for cell complexes,” Illinois J. of Mathematics, 57, No. 3 (2013), 743-753

“Georg Cantor at the Dawn of Point-Set Topology” Loci (March 2012), DOI: 10.4169/loci003861

 “Lusternik–Schnirelmann Category and the Connectivity of X” Algebraic & Geometric Topology 12 (2012) 435-448 

Years of Service to Ursinus

2010-