Alexander Diaz-Lopez
Villanova University, PA·Mathematics & Statistics
Permutations, the different ways to order a set of objects, describe symmetry and arrangement. They show up in sorting algorithms in computer science, quantum particle states in physics and RNA sequences in biology.
- Fields
Mathematics & Statistics
- Topics
combinatorics
permutations
peak sets
- Grants
- 1 known grant, through Aug 2027
What they work on
- Describe the structure coefficients of the peak algebra, the numbers governing how its elements multiply, to better understand permutations grouped by their peak sets.
- Develop a combinatorial reciprocity theory for peak and descent polynomials, which count permutations with a given set of peaks or descents.
- Describe the polytopes, geometric shapes in higher dimensions, formed by permutations that share the same peak set or the same descent set.
- Count the arithmetical structures, special number labelings, on the family of Y-graphs and describe how a smoothing operation on the graph changes them.
More about the research
Graphs, networks of points joined by lines, describe relationships between objects. This project studies both from the angles of algebra and geometry. It asks how permutations that share certain properties are built and how they are arranged in space, and how many special number labelings certain graphs allow.
Why it matters
This is basic mathematics. Clearer theory for counting permutations by their peaks and descents, and for counting arithmetical structures on graphs, adds to the toolkit of combinatorics. Because permutations underlie sorting algorithms, quantum particle states and RNA sequence descriptions, the results could eventually inform those areas. The summer research retreat gives students at Villanova and the University of Puerto Rico hands-on research experience.
Known grants
From NIH and NSF research grants. Labs often have other funding we can't see.
Closest labs
Jacob Fox
Stanford University, CA·Mathematics & Statistics
Determine how the number of colors and the structure of hypergraphs, networks whose edges can join more than two points, govern the off-diagonal growth of hypergraph Ramsey numbers.
Jesus De Loera
University of California-Davis, CA·Mathematics & Statistics
Study slices and shadows of polytopes, meaning their cross-sections and projections, using enumerative and extremal combinatorics with computer-aided experimentation.
Bhargav Narayanan
Rutgers University-New Brunswick, NJ·Mathematics & Statistics·Early-career
Study up-sets and down-sets in the Boolean hypercube from a probabilistic viewpoint, where they encode events of interest in probabilistic combinatorics and statistical physics.
The school
Villanova University: 4 labs on PI Pages, $33M a year on research, 32 research doctorates a year.
Reach out
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