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Series: Dissertation Defense

The work in this dissertation is mainly focused on three subjects which are
essentially related to linear systems on metric graphs and its application: (1)
rank-determining sets of metric graphs, which can be employed to actually compute
the rank function of arbitrary divisors on an arbitrary metric graph, (2) a
tropical convexity theory for linear systems on metric graphs, and (3) smoothing of
limit linear series of rank one on refined metrized complex (an intermediate object
between metric graphs and algebraic curves),

Series: Dissertation Defense

In this work, we numerically studied the effect of the vorticity on the
enhancement of heat transfer in a channel flow. Based on the model we
proposed, we find that the flow exhibits different properties depending on
the value of four dimensionless parameters. In particularly, we can
classify the flows into two types, active and passive vibration, based on
the sign of the incoming vortices. The temperature profiles according to
the flow just described also show different characteristics corresponding
to the active and passive vibration cases. In active vibration cases, we
find that the heat transfer performance is directly related to the strength
of the incoming vortices and the speed of the background flow. In passive
vibration cases, the corresponding heat transfer process is complicated and
varies dramatically as the flow changes its properties. Compared to the
fluid parameters, we also find that the thermal parameters have much less
effect on the heat transfer enhancement. Finally, we propose a more
realistic optimization problem which is to minimize the maximum temperature
of the solids with a given input energy. We find that the best heat
transfer performance is obtained in the active vibration case with zero
background flow.

Series: Dissertation Defense

Robertson and Seymour proved that graphs are well-quasi-ordered by the
minor relation and the weak immersion relation. In other words, given
infinitely many graphs, one graph contains another as a minor (or a
weak immersion, respectively). An application of these theorems is
that every property that is closed under deleting vertices, edges, and
contracting (or "splitting off", respectively) edges can be
characterized by finitely many graphs, and hence can be decided in
polynomial time.
In this thesis we are concerned with the topological minor relation.
We say that a graph G contains another graph H as a topological
minor if H can be obtained from a subgraph of G by repeatedly
deleting a vertex of degree two and adding an edge incident with the
neighbors of the deleted vertex. Unlike the relation of minor and weak immersion,
the topological minor relation does not well-quasi-order graphs in general.
However, Robertson conjectured in the late 1980's that for every positive integer
k, the topological minor relation well-quasi-orders graphs that do not contain a
topological minor isomorphic to the path of length k with each edge duplicated.
This thesis consists of two main results. The first one is a structure theorem for
excluding a fixed graph as a topological minor, which is analogous to a cornerstone
result of Robertson and Seymour, who gave such structure for graphs that exclude a
fixed minor. Results for topological minors were previously obtained by Grohe and
Marx and by Dvorak, but we push one of the bounds in their theorems to the
optimal value. This improvement is needed for the next theorem.
The second main result is a proof of Robertson's conjecture. As a
corollary, properties on certain graphs closed under deleting
vertices, edges, and "suppressing" vertices of degree two can be
characterized by finitely many graphs, and hence can be decided in
polynomial time.

Series: Dissertation Defense

We consider a class of dynamical systems with random switching with the following specifics: Given a finite collection of smooth vector fields on a finite-dimensional smooth manifold, we fix an initial vector field and a starting point on the manifold. We follow the solution trajectory to the corresponding initial-value problem for a random, exponentially distributed time until we switch to a new vector field chosen at random from the given collection. Again, we follow the trajectory induced by the new vector field for an exponential time until we make
another switch. This procedure is iterated. The resulting two-component process whose first component records the position on the manifold, and whose second component records the driving vector field at any given time, is a Markov process.
We identify sufficient conditions for its invariant measure to be unique and absolutely continuous. In the one-dimensional case, we show that the invariant densities are smooth away from critical points of the vector fields and derive asymptotics for the invariant densities at critical points.

Series: Dissertation Defense

We study the structure of the stable coefficients of the Jones polynomial of an alternating link. We start by identifying the first four stable coefficients with polynomial invariants of a (reduced) Tait graph of the link projection. This leads us to introduce a free polynomial algebra of invariants of graphs whose elements give invariants of alternating links which strictly refine the first four stable coefficients. We conjecture that all stable coefficients are elements of this algebra, and give experimental evidence for the fifth and sixth stable coefficient. We illustrate our results in tables of all alternating links with at most 10 crossings and all irreducible planar graphs with at most 6 vertices.

Series: Dissertation Defense

In this thesis we study topology of symplectic fillings of contact manifolds supported by planar open books. We obtain results regarding geography of the symplectic fillings of these contact manifolds. Specifically, we prove that if a contact manifold $(M,\xi)$ is supported by a planar open book, then Euler characteristic and signature of any Stein filling of $(M,\xi)$ is bounded. We also prove a similar finiteness result for contact manifolds supported by spinal open books with planar pages. Moving beyond the geography of Stein fillings, we classify fillings of some lens spaces.In addition, we classify Stein fillings of an infinite family of contact 3-manifolds up to diffeomorphism. Some contact 3-manifolds in this family can be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian 2-bridge knots. We also classify Stein fillings, up to symplectic deformation, of an infinite family of contact 3-manifolds which can be obtained by Legendrian surgeries on $(S^3,\xi_{std})$ along certain Legendrian twist knots. As a corollary, we obtain a classification of Stein fillings of an infinite family of contact hyperbolic 3-manifolds up to symplectic deformation.

Series: Dissertation Defense

The first result of this thesis is a partial result in the direction of Steinberg's Conjecture. Steinberg's Conjecture states that any planar graph without cycles of length four or five is three colorable. Borodin, Glebov, Montassier, and Raspaud showed that planar graphs without cycles of length four, five, or seven are three colorable and Borodin and Glebov showed that planar graphs without five cycles or triangles at distance at most two apart are three colorable. We prove a statement that implies the first of these theorems and is incomparable with the second: that any planar graph with no cycles of length four through six or cycles of length seven with incident triangles distance exactly two apart are three colorable. We are next concerned with the study of Pfaffian orientations. A theorem proved by William McCuaig and, independently, Neil Robertson, Paul Seymour, and Robin Thomas provides a good characterization for whether or not a bipartite graph has a Pfaffian orientation as well as a polynomial time algorithm for that problem. We reprove this characterization and provide a new algorithm for this problem. First, we generalize a preliminary result needed to reprove this theorem. Specifically, we show that any internally 4-connected, non-planar bipartite graph contains a subdivision of K3,3 in which each path has odd length. We then make use of this result to provide a much shorter proof of this characterization using elementary methods. In the final piece of the thesis we investigate flat embeddings. A piecewise-linear embedding of a graph in 3-space is flat if every cycle of the graph bounds a disk disjoint from the rest of the graph. We first provide a structural theorem for flat embeddings that indicates how to build them from small pieces. We then present a class of flat graphs that are highly non-planar in the sense that, for any fixed k, there are an infinite number of members of the class such that deleting k vertices leaves the graph non-planar.

Series: Dissertation Defense

This thesis proposes a novel and efficient method (Method of Evolving
Junctions)
for solving optimal control problems with path constraints, and whose
optimal
paths are separable. A path is separable if it is the concatenation of
finite
number of subarcs that are optimal and either entirely constraint active or
entirely constraint inactive. In the case when the subarcs can be computed
efficiently, the search for the optimal path boils down to determining the
junctions that connect those subarcs. In this way, the original infinite
dimensional problem of finding the entire path is converted into a finite
dimensional problem of determining the optimal junctions. The finite
dimensional
optimization problem is then solved by a recently developed global
optimization
strategy, intermittent diffusion. The idea is to add perturbations (noise)
to
the gradient flow intermittently, which essentially converts the ODE's
(gradient
descent) into a SDE's problem. It can be shown that the probability of
finding
the globally optimal path can be arbitrarily close to one. Comparing to
existing
methods, the method of evolving junctions is fundamentally faster and able
to
find the globally optimal path as well as a series of locally optimal
paths.
The efficiency of the algorithm will be demonstrated by solving path
planning
problems, more specifically, finding the optimal path in cluttered
environments
with static or dynamic obstacles.

Series: Dissertation Defense

Given a closed surface S_g of genus g, a mapping class f is said to be pseudo-Anosov if it preserves a pair of transverse measured foliations such that one is expanding and the other one is contracting by a number $\lambda$. The number $\lambda$ is called a stretch factor (or dilatation) of f. Thurston showed that a stretch factor is an algebraic integer with degree bounded above by 6g-6. However, little is known about which degrees occur. Using train tracks on surfaces, we explicitly construct pseudo-Anosov maps on S_g with orientable foliations whose stretch factor $\lambda$ has algebraic degree 2g. Moreover, the stretch factor $\lambda$ is a special algebraic number, called Salem number. Using this result, we show that there is a pseudo-Anosov map whose stretch factor has algebraic degree d, for each positive even integer d such that d is less than or equal to g. Our examples also give a new approach to a conjecture of Penner.

Series: Dissertation Defense

Chip-firing is a deceptively simple game played on the vertices of a graph, which was independently discovered in probability theory, poset theory, graph theory, and statistical physics. In recent years, chip-firing has been employed in the development of a theory of divisors on graphs analogous to the classical theory for Riemann surfaces. In particular, Baker and Norin were able to use this setup to prove a combinatorial Riemann-Roch formula, whose classical counterpart is one of the cornerstones of modern algebraic geometry. It is now understood that the relationship between divisor theory for graphs and algebraic curves goes beyond pure analogy, and the primary operation for making this connection precise is tropicalization, a certain type of degeneration which allows us to treat graphs as "combinatorial shadows" of curves. This tropical relationship between graphs and algebraic curves has led to beautiful applications of chip-firing to both algebraic geometry and number theory.
In this thesis we continue the combinatorial development of divisor theory for graphs.