About

Edward Swartz is a professor in the Department of Mathematics at Cornell University, affiliated with the College of Arts and Sciences. He holds a B.A. in Mathematics from the University of Rochester, an M.S. in Mathematics from Yale University, and a Ph.D. in Mathematics from the University of Maryland. His research centers on the interplay between combinatorics, geometry, topology, and algebra, with a special emphasis on matroids and the combinatorial properties of simplicial complexes. His work includes significant contributions to the understanding of projection volumes of hyperplane arrangements, Buchsbaum modules, face enumeration of spheres and manifolds, topological representations of matroids, and the relationships between matroids and quotients of spheres.

Research topics

• Mathematics
• Mathematical analysis
• Combinatorics
• Computer Science
• Pure mathematics
• Discrete mathematics

Selected publications

• Polymatroids are to finite groups as matroids are to finite fields

  arXiv (Cornell University) · 2024

  1st authorCorresponding
  • Computer Science
  • Mathematics
  • Computer Science

  Given a subgroup $\mathcal{H}$ of a product of finite groups $\mathcal{G} = \displaystyle\prod^n_{i=1} Γ_i$ and $b>1,$ we define a polymatroid $P(\mathcal{H},b).$ If all of the $Γ_i$ are isomorphic to $\mathbb{Z}/p\mathbb{Z},$ $p$ a prime, and $b=p,$ then $P(\mathcal{H},b)$ is the usual matroid associated to any $\mathbb{Z}/p\mathbb{Z}$-matrix whose row space equals $\mathcal{H}.$ In general, there are many ways in which the relationship between $P(\mathcal{H},b)$ and $\mathcal{H}$ mirrors that of the relationship between a matroid and a subspace of a finite vector space. These include representability by excluded minors, the Crapo-Rota critical theorem, the existence of a concrete algebraic object representing the polymatroid dual of $P(\mathcal{H},b),$ analogs of Greene's theorem and the MacWilliams identities when $\mathcal{H}$ is a group code over a nonabelian group, and a connection to the combinatorial Laplacian of a quotient space determined by $\mathcal{G}$ and $\mathcal{H}.$ We use the group Crapo-Rota critical theorem to demonstrate an extension to hypergraphs of the classical duality between proper colorings and nowhere-zero flows on graphs.

  DOI

• Three-dimensional normal pseudomanifolds with relatively few edges

  Advances in Mathematics · 2020 · 9 citations

  Senior authorCorresponding
  • Mathematics
  • Combinatorics
  • Mathematical analysis
  DOI

• <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> -vectors of manifolds with boundary

  Algebraic Combinatorics · 2020 · 3 citations

  Senior authorCorresponding
  • Mathematics
  • Combinatorics
  • Pure mathematics

  We extend several <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>g</mml:mi> </mml:math> -type theorems for connected, orientable homology manifolds without boundary to manifolds with boundary. As applications of these results we obtain Kühnel-type bounds on the Betti numbers as well as on certain weighted sums of Betti numbers of manifolds with boundary. Our main tool is the completion <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>Δ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> of a manifold with boundary <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> ; it is obtained from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> by coning off the boundary of <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>Δ</mml:mi> </mml:math> with a single new vertex. We show that despite the fact that <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover accent="true"> <mml:mi>Δ</mml:mi> <mml:mo>^</mml:mo> </mml:mover> </mml:math> has a singular vertex, its Stanley–Reisner ring shares a few properties with the Stanley–Reisner rings of homology spheres. We close with a discussion of a connection between three lower bound theorems for manifolds, PL-handle decompositions, and surgery.

  DOI

Recent grants

• From Topology to Combinatorics and Back

  NSF · $150k · 2009–2012

• From Topology to Combinatorics and Back

  NSF · $118k · 2006–2009

Frequent coauthors

• Isabella Novik

  University of Washington

  20 shared

• T. Sulanke

  Indiana University Bloomington

  11 shared

• Frank H. Lutz

  Technische Universität Berlin

  11 shared

• R. O. Pohl

  8 shared

• Caroline J. Klivans

  3 shared

• Ezra Miller

  3 shared

• Henry E. Fischer

  Institut Laue-Langevin

  3 shared

• P. Türkes

  Infineon Technologies (Germany)

  2 shared

Similar researchers at Cornell University

• Edward Bucklerh-182
• Robert Sternbergh-168
• John P. Mooreh-153
• Olivier Elementoh-150
• Andy Clarkh-142