# Paul Horn

## Associate Professor

303-871-2911 (Office)

http://www.cs.du.edu/~paulhorn/

Clarence M. Knudson Hall, 2390 S. York St. Denver, CO 80210

### What I do

I am an Associate Professor of Mathematics at the University of Denver; my role here involves research, teaching classes and advising graduate and undergraduate students. I am also currently the graduate coordinator in the mathematics department.### Specialization(s)

combinatorics, graph theory, probability

### Professional Biography

My research interests are in combinatorics. Specifically, I am interesting in using ideas from probability, linear algebra, and geometry methods to understand graphs, which are a mathematical abstraction of networks.

I received my PhD in 2009 from the University of California at San Diego, advised by Fan Chung, and joined the faculty of DU in the fall of 2013 after postdoctoral positions at Emory University and Harvard University.

I am also closely involved with several research workshops. In particular I co-organize the Rocky Mountains-Great Plains Graduate Research Workshop in Combinatorics (GRWC), which is an annual research workshop for graduate students in combinatorics. I also help organize the graph theory section Masamu Advanced Studies Institute, an annual research workshop in southern Africa.

I received my PhD in 2009 from the University of California at San Diego, advised by Fan Chung, and joined the faculty of DU in the fall of 2013 after postdoctoral positions at Emory University and Harvard University.

I am also closely involved with several research workshops. In particular I co-organize the Rocky Mountains-Great Plains Graduate Research Workshop in Combinatorics (GRWC), which is an annual research workshop for graduate students in combinatorics. I also help organize the graph theory section Masamu Advanced Studies Institute, an annual research workshop in southern Africa.

### Degree(s)

- Ph.D., Mathematics, University of California, San Diego, 2009

### Professional Affiliations

- American Mathematical Society
- Society for Industrial and Applied Mathematics

### Research

My research interests are in combinatorics. Specifically, I focus on the use of probabilistic, spectral, and geometric methods to understand graphs, which are a mathematical abstraction of networks. My particular work includes using ideas from 'continuous' mathematics, for instance the 'diffusion of heat' on a network, to understand structural properties of networks, for instance whether there are 'bottlenecks' in the network. I also use structural information about networks to understand random processes on graphs.

One recent project I've been involved in is the development of notions of 'curvature' for graphs. Curvature, in the study of manifolds (which are objects that 'locally' look like Euclidean space, in the way the surface of the earth 'locally looks like a plane), is a measure of how a space expands 'locally.' The notion of curvature we introduced has allowed me and my collaborators to prove graph theoretical analogues of a number of results from Riemannian geometry; specifically a graph theoretical version of the 'Li-Yau inequality' along with many consequences.

One recent project I've been involved in is the development of notions of 'curvature' for graphs. Curvature, in the study of manifolds (which are objects that 'locally' look like Euclidean space, in the way the surface of the earth 'locally looks like a plane), is a measure of how a space expands 'locally.' The notion of curvature we introduced has allowed me and my collaborators to prove graph theoretical analogues of a number of results from Riemannian geometry; specifically a graph theoretical version of the 'Li-Yau inequality' along with many consequences.

#### Areas of Research

probabilistic methods

spectral graph theory

theoretical computer science

analysis on graphs.

#### Key Projects

- Collaboration on problems in graph theory, and geometric analysis on graphs
- Collaborative Research: Rocky Mountains-Great Plains Graduate Research Workshops in Combinatorics
- Curvature and Geometric Analysis of graphs

### Featured Publications

(2020). Volume doubling, Poincare inequality and Gaussian heat kernel estimate for non-negatively curved graphs. Journal fur die reine und andgenwandte mathematik (Crelle's Journal), 757, 89-130.

. (2018). Rainbow spanning trees in complete graphs colored by one-factorizations. Journal of Graph Theory, 87(3), 333-346.

. (2016). Isomorphic edge disjoint subgraphs of hypergraphs. Random Structures and Algorithms, 48(4), 767-793.

. (2016). Graphs with many strong orientations. SIAM Journal of Discrete Mathematics, 30(2), 1269-1282.

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