Mathematics
http://hdl.handle.net/10211.3/141103
Thu, 06 Aug 2020 16:20:31 GMT2020-08-06T16:20:31ZAlgebraic and combinatorial aspects of polytopes and domino tilings
http://hdl.handle.net/10211.3/214080
Algebraic and combinatorial aspects of polytopes and domino tilings
Yamzon, Nicole
Two classical objects of study in combinatorics are polytopes and domino tilings. In the 1990s William Thurston proved that the set of domino tilings of a simply- connected region is connected by flips. The first part of this thesis will introduce definitions associated to the underlying grid graph of a collection of domino tilings. We will then provide an analogous proof of Thurston’s result by using the language of toric ideals.
The second half of this thesis, a joint project with Anastasia Chavez, focuses on d-dimensional simplicial polytopes. In particular, we introduce the associated face vector i.e. the f-vector of a d-dimensional polytope. In conclusion, we prove a correspondence between the maximal linearly independent subsets of the f-vectors of simplicial polytopes and the Catalan numbers.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10211.3/2140802019-01-01T00:00:00ZSpectra of tropical Laplacians of classical root polytopes
http://hdl.handle.net/10211.3/213976
Spectra of tropical Laplacians of classical root polytopes
Perez, Maria Isabel
Farhad Babaee and June Huh introduced the tropical Laplacian of a tropical surface
and used it to disprove a generalized Hodge conjecture. We study four families
of tropical surfaces arising from the root polytopes of types A ,B ,C and D. We
compute the spectra of their tropical Laplacians for type A, and describe them conjecturally
for types B, C and D. Our results confirm that these tropical Laplacians
have exactly one negative eigenvalue, as anticipated by Babaee and Huh. Moreover,
we give an explicit eigenbasis for the tropical Laplacian of An- i , proving its
diagonalizability.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10211.3/2139762019-01-01T00:00:00ZEhrhart quasipolynomials of coxeter permutahedra
http://hdl.handle.net/10211.3/213961
Ehrhart quasipolynomials of coxeter permutahedra
McWhirter, Jodi
The Ehrhart polynomial counts lattice points in a dilated lattice polytope. The
Ehrhart polynomials of permutahedra of types A, B, C, and D have been calculated
by Federico Ardila, Federico Castillo, and Michael Henley (2015). However, when a
type B permutahedron is shifted so that its center is the origin, it Decomes a halfintegral
polytope, and its Ehrhart quasipolynomial was previously unknown. The
same is true of odd-dimension type A permutahedra. We use signed graphs that
arise from the generating vectors of each permutahedron to determine which sets of
vectors are linearly independent and thus which form parallelepipeds that are a part
of a zonotopal decomposition, as well as wmch of these parallelepipeds stays on the
lattice when the permutahedron is shifted. This yields new approaches/formulas for
Ehrhaxt quasipolynomials for these rational permutahedra.
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10211.3/2139612019-01-01T00:00:00ZUnderstanding the complexity of the domain of approximation of a rational tuple
http://hdl.handle.net/10211.3/213935
Understanding the complexity of the domain of approximation of a rational tuple
Knitter, Oliver Y.
In continued fraction theory, a convergent is an irreducible rational number that best
approximates a particular real number when compared to all rationals of smaller denominator.
Generalizing this definition to include rational tuples of arbitrary length
d shows that every point in Qd determines a domain of approximation: the set of
all points in Rd that axe best approximated by this particular rational. It is known
that the domain of approximation is rectilinear, with its sides parallel to the coordinate
axes. We shall examine the geometric properties of a carefully constructed
set of sheared lattices in order to characterize the “corner points” of the domain of
approximation, with a particular emphasis placed on identifying “primary” corners.
We then show that for any 2-tuple ( ^ , ^ ) € Q2, the number of primary corners in
its corresponding domain of approximation is O(logg).
Tue, 01 Jan 2019 00:00:00 GMThttp://hdl.handle.net/10211.3/2139352019-01-01T00:00:00Z