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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.

Series: Dissertation Defense

The talk consists of two parts.The first part is devoted to results in Discrepancy Theory. We consider geometric discrepancy in higher dimensions (d > 2) and obtain estimates in Exponential Orlicz Spaces. We establish a series of dichotomy-type results for the discrepancy function which state that if the $L^1$ norm of the discrepancy function is too small (smaller than the conjectural bound), then the discrepancy function has to be very large in some other function space.The second part of the thesis is devoted to results in Additive Combinatorics. For a set with small doubling an order-preserving Freiman 2-isomorphism is constructed which maps the set to a dense subset of an interval. We also present several applications.

Series: Dissertation Defense

We say that a cover of surfaces S-> X has the Birman--Hilden property if the subgroup of the mapping class group of X consisting of mapping classes that have representatives that lift to S embeds in the mapping class group of S modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a cover has this property. We give new explicit examples of irregular branched covers that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.

Series: Dissertation Defense

Series: Dissertation Defense

Advisor: Dr. Matthew Baker

We study various binomial and monomial ideals related to the theory of
divisors, orientations, and matroids on graphs. We use ideas from potential
theory on graphs and from the theory of Delaunay decompositions for lattices
to describe minimal polyhedral cellular free resolutions for these ideals.
We will show that the resolutions of all these ideals are closely related
and that their Betti tables coincide. As corollaries we give conceptual
proofs of conjectures and questions posed by Postnikov and Shapiro, by
Manjunath and Sturmfels, and by Perkinson, Perlman, and Wilmes. Various
other results related in the theory of chip-firing games on graphs --
including Merino's proof of Biggs' conjecture and Baker-Shokrieh's
characterization of reduced divisors in terms of potential theory -- also
follow immediately from our general techniques and results.

Series: Dissertation Defense