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.
- Maria Gilbert verified several instances of the following conjecture by Joel Lewis and myself: in a well-generated complex reflection, two factorizations of a Coxeter element into reflections will lie in the same Hurwitz orbit if and only they use the same multiset of conjugacy classes of reflections. She and her fellow student Xuan Liu used SAGE code that computes the Hurwitz orbit of a factorization, and compared orbits sizes to a generating function of Chapuy and Stump that conveniently computes the total number of reflection factorizations of a given length. They checked the conjecture holds up through reflection length at least four (and in some cases longer) for the complex reflection groups G_{4}, G_{5}, G_{8}, G_{20}.
Her thesis in PDF.
- Owen Levin was mainly supervised by Volkan Isler in our Computer Science department, and I mostly co-signed the thesis for the School of Mathematics. He worked on a problem that arises in robotics and drones: given a collection of n points in Euclidean space that can form an edge for communication when they are within distance 1, find an algorithm that lets them move the smallest amount until this graph of edges connects them all. "Smallest" here means in such a way that the maximum movement of all of the nodes is minimized. It is known that there is no algorithm to find the optimal solution which runs in polynomial time in n, but he worked on several approximation algorithms, and compared them to a previously known algorithm.
His thesis in PDF.
- Nhung Pham studied a famously difficult discrete dynamical system called Bulgarian solitaire. It can be viewed as a self-map on the set of integer partitions of n. When one iterates this map, the long term recurrent behavior and the elements of the recurrent set within each orbit are well-understood, parametrized by necklaces of black and white beads. However, the size of the orbit parametrized by a necklace is almost completely unknown. Pham gave the first results analyzing these orbit sizes for necklaces that are powers P^{k} of a fixed primitive necklace P, including a full analysis when P has length at most 3 (so P = W or BW or BWW or BBW). Furthermore, she conjectured a statement about exact geometric growth in k for P^{k} when P has length at least 3. She also considered for P^{k} the generating function counting elements in its orbit by their distance to the recurrent set, showing that as k grows, this generating function approaches a rational function f(x)/g(x), with bounds on the degree of f(x) and g(x).
Her thesis in PDF.
Her oral presentation in PDF.
Her paper that appeared in INTEGERS Vol 23 (2023); see also the arXiv version.
- AJ Harris studied some variations on the Bulgarian solitaire discrete dynamical system that Nhung Pham had worked on. There is a known variant called Carolina Solitaire, where the piles are ordered, having the same structure for its recurrent cycles and orbits parametrized by necklaces, but where the transient structure and orbit sizes seem to behave very differently. He introduced some new variations, where even the orbits and recurrent cycles are harder to characterize, and some non-deterministic versions, where instead of the recurrent cycles, one examines the digraph "sinks" in their poset of strongly connected components.
His thesis, his oral presentation, and poster.
- Son Nguyen studied face numbers for the relatively recently defined poset associahedra of Pavel Galashin. Together with collaborator Andrew Sack, he showed that the face numbers depend only upon the comparability graph of the poset. The two also studied a family of poset associahedra, indexed by what they call broom posets, that interpolate between permutohedra and associahedra. They computed their f-vectors and h-vectors in terms of descents of permutations defined via the stack-sorting map, and in some cases proved the strong properties of gamma-nonnegativity and real-rootedness.
His thesis,
his oral presentation, and poster.
.
Other MN undergrad supervised papers
- Kaustubh Verma
worked on a URS project related to double-coverings of graphs. It is known that when one has such a double cover, the number of spanning trees for the base graph divides the number for the covering graph. Verma investigated whether one can have a finer divisibility of Tutte polynomials T_{G}(x,y), and
showed that at least one gets divisibility of the specializations
T_{G}(x,1) for a certain special family of graphs
called flowers, but not for general graphs.
Here is his resulting paper that appeared
in the Minn. J. Undergrad Math.
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