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, Maxwell, 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.
- Christian Gaetz studied critical groups of
faithful finite group representations, as introduced by
Benkart, Klivans, and Reiner.
He proved a formula for the cardinality of the critical
group in terms of the character values
for the representation. He also proved some results on their
structure, including determining the exact structure in the case of
the irreducible reflection representation of S_{n}.
His thesis in PDF.
His oral presentation in PDF.
The much more abbreviated
arXiv version, to appear in Lin. Alg. Appl..
- Yao-Rui Yeo introduced a graphical invariant of bicolored trees embedded in the plane, considered as dessins d'enfants
of genus one with one face. This invariant is the rank r as a free abelian
group of the associated superpotential algebra for the quiver which is
planar dual to the tree. He shows that the superpotential algebra in
this case is always a commutative Z-algebra of the form
Z[x], Z[x,x^{-1}] or Z[x]/[x^r-1] for some r > 1. He also
conjectures that the rank r is a Galois invariant of the dessin, and
verifies this for bicolored trees having up to 10 edges, using known
classifications.
His thesis in PDF.
His oral presentation in PDF.
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