Vic Reiner- Math Latin Honors theses

• Hans Christianson conjectured and proved the structure of the critical group of a threshold graph in the "generic" case, building on REU work of P. Bendich and T. Bogart. This later led to the paper "The critical group of a threshold graph", (Linear Algebra Appl. 349 (2002), 233–244).

• Brian Jacobson solved completely the problem of computing the critical group of all threshold graphs. The other part of his thesis proves a conjecture of Kuperberg about the critical group of a planar graph and its isomorphism with a certain Kasteleyn-Percus matrix for the graph.
  His thesis in PDF.

• David Treumann explored functoriality of the critical group, that is, in what situations to expect a homomorphism between the critical group of two graphs. His work turned out to be crucial for a later paper of Berget, Manion, Maxell, Potechin, and Reiner on critical groups of line graphs.
  His thesis in PDF.

• Andy Berget explored the relationship between the critical group of a regular graph and the critical group of its line graph. He computed the critical groups for the line graphs of complete graphs, and began making some conjectures, involving functorial maps of the type explored in David Treumann's thesis, which were later resolved in the paper he wrote jointly with Manion, Maxwell, Potechin and Reiner.
  His thesis in PDF.

• Alex Miller continued his REU work on the Smith normal forms of the up-down and down-up maps in differential posets, eventually leading to a co-authored paper that appeared in the journal ORDER.
  His thesis in PDF.

• John Machacek followed up on the work of Berget, Manion, Maxell, Potechin, and Reiner. The latter had exhibited an exact sequence relating the critical groups of a regular graph and its line graph, and had refined this to give the exact relation between the two groups in the nonbipartite case. Machacek explored several infinite families of bipartite examples, in order to gain data on how a conjecture should look in this case. Although he conjectured and proved what the critical group structures in many cases, and there are tantalizing patterns, nothing conclusive has emerged yet from the data.
  His thesis in PDF.
  His oral presentation in PDF.

• Patrick Floryance gave an exposition of the well-known connection between random walks on graphs, theory of electrical networks, and the problem of "perfectly squared" rectangles and squares.
  His thesis in PDF.

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